Conference Proceedings of the Society for Experimental Mechanics Series
For other titles published in this series, go to www.springer.com/series/8922
BookID 214574_ChapID FM_Proof# 1 - 23/04/2011
Tom Proulx Editor
Structural Dynamics, Volume 3 Proceedings of the 28th IMAC, A Conference on Structural Dynamics, 2010
Editor Tom Proulx Society for Experimental Mechanics, Inc. 7 School Street Bethel, CT 06801-1405 USA
[email protected]
ISSN 2191-5644 e-ISSN 2191-5652 ISBN 978-1-4419-9833-0 e-ISBN 978-1-4419-9834-7 DOI 10.1007/978-1-4419-9834-7 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011929047 © The Society for Experimental Mechanics, Inc. 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
BookID 214574_ChapID FM_Proof# 1 - 23/04/2011
Preface
Structural Dynamics represents one of five clusters of technical papers presented at the 28th IMAC, A Conference and Exposition on Structural Dynamics, 2010 organized by the Society for Experimental Mechanics, and held at Jacksonville, Florida, February 1-4, 2010. The full proceedings also include volumes on Structural Dynamics and Renewable Energy, Nonlinear Modeling and Applications, Dynamics of Bridges, and Dynamics of Civil Structures. Each collection presents early findings from experimental and computational investigations on an important area within Structural Dynamics. The current volume on Structural Dynamics includes studies on Modal Analysis, Experimental Techniques, Damage Detection, Modal Techniques (Excitation), Biodynamics, Damping, Modal Parameter Identification, Rotating Machinery, Model Validation, Uncertainty Quantification, Model Calibration, Substructuring Structural Health Monitoring, Parameter Estimation, Experimental Modeling Methods Shock and Vibration, Ares IX Launch Vehicle Modal Testing, Ground Vibration Testing System Identification, and Aircraft Loading IMAC covers the wide variety of subjects that are related to the broad field of Structural Dynamics. It is the Society’s mission to disseminate information on a broad selection of subjects. To this end, research and application papers in this volume relate to the broad field of Structural Dynamics. Modal Analysis is a major enabling technology in this area and consequently is a significant component of this volume. The organizers would like to thank the authors, presenters, session organizers and session chairs for their participation in this track. Bethel, Connecticut
Dr. Thomas Proulx Society for Experimental Mechanics, Inc
BookID 214574_ChapID FM_Proof# 1 - 23/04/2011
Contents
1
Simulation and Validation of ODS Measurements made Using a Continuous SLDV Method on a Beam Excited by a Pseudo Random Signal D. Di Maio, G. Carloni, D.J. Ewins
1
2
Comparison of Image Based, Laser, and Accelerometer Measurements C. Warren, P. Pingle, C. Niezrecki, P. Avitabile
15
3
Modal Parameter Estimation Using Acoustic Modal Analysis W. Elwali, H. Satakopan, V. Shauche, R. Allemang, A. Phillips
23
4
Mitigation of Vortex-induced Vibrations in Long-span Bridges A. Patil, S. Jung, S.-D. Kwon
35
5
Vibration and Acoustic Analysis of Brake Pads for Quality Control P. Castellini, P. Chiariotti, M. Martarelli, E.P. Tomasini
43
6
Vibration Monitoring of a Small Span Composite Bridge D. Siegert, M. Döhler, O. Ben Mekki, L. Mevel, M. Goursat, F. Toutlemonde
53
7
Detection of Damage in Lightning Masts and Loosening of Bolted Connections in Structures Using Changes in Natural Frequencies K. He, W.D. Zhu
63
8
Order Tracking with Multi-sine Excitation on a Ground Vibration Test G. Hoople, K. Napolitano
79
9
Impact Excitation Processing for Improved Frequency Response Quality A. Brandt, R. Brincker
89
10
Review of Recent Developments in Multiple Reference Impact Testing (MRIT) D.L. Brown, M.C. Witter
97
11
Fiber Optic Strain Gage Verification and Polyethylene Hip Liner Testing L. Chavez, M. Martin, S.O. Neidigk, P. Cornwell, R.M. Meneghini, J. Racanelli
115
12
Suggestion for Evaluation Methods of Ride Comfort at Low Frequencies T. Koizumi, N. Tsujiuchi, S. Okumura, S. Yamada, J. Ninomiya, T. Tajima
135
viii
13
Miniature Model Examination as Human-body Dynamics for Seat Durability Evaluations T. Koizumi, N. Tsujiuchi, M. Yamamoto, H. Ando
143
14
Tutorial Guideline VDI 3830: Damping of Materials and Members L. Gaul
151
15
Blind Modal Identification Applied to Output-only Building Vibration S.I. McNeill, D.C. Zimmerman
157
16
Second Order Blind Source Separation Techniques (SO-BSS) and Their Relation to Stochastic Subspace Identification (SSI) Algorithm J. Antoni, S. Chauhan
177
17
Utilization of Blind Source Separation Techniques for Modal Analysis B. Swaminathan, B. Sharma, S. Chauhan
189
18
Minimally Intrusive Torsional Vibration Sensing on Rotating Shafts M.W. Trethewey, M.S. Lebold, M.W. Turner
207
19
Zebra Tape Butt Joint Algorithm for Torsional Vibrations K. Janssens, P. Van Vlierberghe, P. D’Hondt, T. Martens, B. Peeters, W. Claes
213
20
Acoustic and Mechanical Measurements of an Hydraulic Turbine’s Generator in Relation to Power Levels and Excitation Forces F. Lafleur, S. Bélanger, L. Marcouiller, A. Merkouf
225
21
Contact-less Wind Turbine Utilizing Piezoelectric Bimorphs with Magnetic Actuation S. Bressers, D. Avirovik, M. Lallart, D.J. Inman, S. Priya
233
22
Effect of Functionally Graded Materials on Resonances of Rotating Beams A.J. Mazzei, Jr., R.A. Scott
245
23
A Comparison of Gear Mesh Stiffness Modeling Strategies J. Meagher, X. Wu, D. Kong, C.H. Lee
255
24
Using Operating Deflection Shapes to Detect Faults in Rotating Equipment S.N. Ganeriwala, V. Kanakasabai, A. Muthukumarasamy, T. Wolff, M. Richardson
265
25
Damping Methodology for Condensed Solid Rocket Motor Structural Models S. Fransen, H. Fischer, S. Kiryenko, D. Levesque, T. Henriksen
273
26
Analysis and Optimization of the Current Flowing Technique for Semi-passive Multi-modal Vibration Reduction S. Manzoni, M. Redaelli, M. Vanali
283
27
Some Passive Damping Sources on Flooring-systems Besides the TMD L. Pedersen
295
28
Constrained Layer Damping Test Results for Aircraft Landing Gear T. Collins, K. Kochersberger
303
29
A Sloshing Absorber with a Flexible Container M. Gradinscak, S.E. Semercigil, Ö.F. Turan
315
ix
30
Effective Vibration Suppression of a Maneuvering Two-link Flexible Arm with an Event-based Stiffness Controller A. Özer, S.E. Semercigil
323
31
Vibration Testing of Bridge Stay Cables to Obtain Damping Values M.E. Turek, C.E. Ventura, K. Shawwaf
331
32
Identification of Stiffness and Damping in Nonlinear Systems M.L.D. Lumori, J. Schoukens, J. Lataire
341
33
Truncation Effects on the Dynamic Damage Locating Vector (DDLV) Approach M. Maddalo, D. Bernal
349
34
Residual Analysis for Damage Detection: Effects of Variation on the Noise Model C. Wright, D. Bernal
357
35
Minimizing Distortions from Time Domain Zooming D. Abramo, D. Bernal
367
36
Process and Measurement Noise Estimation for Kalman Filtering Y. Bulut, D. Vines-Cavanaugh, D. Bernal
375
37
Model Calibration for Fatigue Crack Growth Analysis Under Uncertainty S. Sankararaman, Y. Ling, C. Shantz, S. Mahadevan
387
38
Bayesian Finite Element Model Updating Using Static and Dynamic Data B.A. Zárate, J.M. Caicedo, G. Wieger, J. Marulanda
395
39
Frequency Domain Test-analysis Correlation in the Presence of Uncertainty S. Nimityongskul, D.C. Kammer, S. Lacy, V. Babuska
403
40
Error Quantification in Calibration of AFM Probes Due to Non-uniform Cantilevers H. Frentrup, M.S. Allen
419
41
Demystifying Wireless for Real-world Measurement Applications K. Veggeberg
433
42
Data Merging for Multi-setup Operational Modal Analysis with Data-driven SSI M. Döhler, P. Andersen, L. Mevel
443
43
On the Operational Modal Analysis of Solid Rocket Motors S. Fransen, D. Rixen, T. Henriksen, M. Bonnet
453
44
Frequency-domain Modal Analysis in the OMAX Framework T. De Troyer, M. Runacres, P. Guillaume
465
45
Application of Principal Component Analysis Methods to Experimental Structural Dynamics R.J. Allemang, A.W. Phillips, M.R. Allemang
46
Application of Modal Scaling to the Pole Selection Phase of Parameter Estimation A.W. Phillips, R.J. Allemang
477 499
x
47
Quantification of Aleatoric and Epistemic Uncertainty in Computational Models of Complex Systems A. Urbina, S. Mahadevan
519
48
The Dangers of Sparse Sampling for Uncertainty Propagation and Model Calibration F.M. Hemez, H.S. Atamturktur
537
49
Geological Stress State Calibration and Uncertainty Analysis J. McFarland, A. Morris, B. Bichon, D. Riha, D. Ferrill, R. McGinnis
557
50
Error Quantification in Mechanics Computational Models B. Liang, S. Mahadevan
571
51
Effectiveness of Modeling Thin Composite Structures Using Hex Shell Elements R.M. Garcia, Jr., D.G. Tipton
581
52
Quantifying the Effect of Dynamic Boundary Condition Differences Between the Vibration Laboratory and the Field Operating Environment R.L. Mayes, F. Bitsie, D. Bridgers
589
53
A Dual Approach to Substructure Decoupling Techniques S.N. Voormeeren, D.J. Rixen
601
54
Using Component Modes in a System Design Process G. Vermot des Roches, J.-P. Bianchi, E. Balmes, R. Lemaire, T. Pasquet
617
55
Computation of Distributed Forces in Modally Reduced Mechanical Systems W. Witteveen
627
56
Substructuring Using Impulse Response Functions for Impact Analysis D.J. Rixen
637
57
Static Correction in Model Order Reduction Techniques for Multiphysical Problems A.M. Steenhoek, D.J. Rixen, P. Nachtergaele
647
58
Truncation Error Propagation in Model Order Reduction Techniques Based on Substructuring A.M. Steenhoek, D.J. Rixen
663
59
Investigation on the Use of Various Decoupling Approaches D. Cloutier, P. Avitabile
679
60
Input Estimation from Measured Structural Response D. Harvey, E. Cross, R.A. Silva, C. Farrar, M. Bement
689
61
Damage Detection with Ambient Vibration Data Using Time Series Modeling M. Gul, F.N. Catbas
709
62
Wind Load Observer for a 5MW Wind Energy Plant M. Klinkov, C.-P. Fritzen
719
63
An Integrated SHM Approach for Offshore Wind Energy Plants C.-P. Fritzen, P. Kraemer, M. Klinkov
727
xi
64
Preliminary Validation of a Complex Aerospace Structure M. Arviso, D.G. Tipton, P.S. Hunter
741
65
Modeling the Elastic Support Properties of Bernoulli-Euler Beams T.A.N. Silva, N.M.M. Maia
753
66
Noise Reduction of Continuosly Variable Transmission (CVT) for Automobile N. Okubo, H. Aikawa, T. Toi, Y. Hirabayashi, A. Tsuruta
763
67
Smirnov Stationarity Criterion Applied to Rocket Engine Test Data Analysis S. Muller, Y. Mauriot
771
68
Mode Shape Identification Using Mobile Sensors J. Marulanda, J.M. Caicedo
779
69
Determination of Dynamically Equivalent FE Models of Structures from Experimental Data T. Karaağaçli, E.N. Yildiz, H.N. Özgüven
785
70
Optimization of Substructure Dynamic Interface Forces by an Energetic Approach T. Weisser, L.-O. Gonidou, E. Foltête, N. Bouhaddi
801
71
Converting CT Scans of a Stradivari Violin to a FEM M. Pyrkosz, C. Van Karsen, G. Bissinger
811
72
LAN-XI – The Next Generation of Data Acquisition Systems N.J. Jacobsen
821
73
Ultrasonic Guided Wave Modal Analysis Technique (UMAT) for Defect Detection J.L. Rose, F. Yan, Y. Zhu
831
74
Damage Diagnosis of Beam-like Structures Based on Sensitivities of Principal Component Analysis Results V.H. Nguyen, J.-C. Golinval
839
75
Coriolis Flowmeter Verification via Embedded Modal Analysis M. Rensing, T.J. Cunningham
851
76
Bayesian Finite Element Model Updating for Crack Growth J.M. Caicedo, B.A. Zárate, V. Giurgiutiu, L. Yu, P. Ziehl
861
77
A Non-destructive, Contactless Technique for the Health Monitoring of Ancient Frescoes L. Collini
867
78
Adaptive Filter Feature Identification for Structural Health Monitoring in Aeronautical Panel S. da Silva, C.G. Gonsalez, V. Lopes, Jr.
875
79
Prediction of Full Field Dynamic Stress/Strain from Limited Sets of Measured Data P. Pingle, P. Avitabile
883
80
Origins and History of Shock & Vibration (S&V) Requirements J.E. Howell, III
897
xii
81
Spinning Projectile Wing Development Structural Dynamic Modeling and Analysis J.E. Alexander
899
82
Underwater Explosion Phenomena and Shock Physics F.A. Costanzo
917
83
Some Aspects of Using Measured Data as the Basis of a Multi-exciter Vibration Test M. Underwood, R. Ayres, T. Keller
939
84
On the New American National Standard for Shock Testing Equipment B.W. Lang
955
85
Shock Data Filtering Consequences H.A. Gaberson
961
86
Evaluation of an Alternative Composite Material Failure Model Under Dynamic Loading C.T. Key, R.S. Fertig, III, W. Gregory, J. Gorfain
979
87
Free Vibration Analysis of Pyroshock-loaded Hardened Structures J.R. Foley, L.M. Watkins, B.W. Plunkett, J.C. Wolfson, P.C. Gillespie, J.C. Van Karsen, A.L. Beliveau
991
88
Ares I-X Launch Vehicle Modal Test Overview R.D. Buehrle, P.A. Bartolotta, J.D. Templeton, M.C. Reaves, L.G. Horta, J.L. Gaspar, R.A. Parks, D.R. Lazor
999
89
Identifying Goals for Ares 1-X Modal Testing R.E. Tuttle, J.S. Hwung, J.A. Lollock
1011
90
Ares I-X Launch Vehicle Modal Test Measurements and Data Quality Assessments J.D. Templeton, R.D. Buehrle, J.L. Gaspar, R.A. Parks, D.R. Lazor
1017
91
Modal Test Data Adjustment for Interface Compliance R.E. Tuttle, J.A. Lollock
1029
92
Finite Element Model Calibration Approach for Ares I-X L.G. Horta, M.C. Reaves, R.D. Buehrle, J.D. Templeton, D.R. Lazor, J.L. Gaspar, R.A. Parks, P.A. Bartolotta
1037
93
Model Reduction and Substructuring Techniques for the Vibro-acoustic Simulation of Automotive Piping and Exhaust Systems J. Herrmann, M. Junge, L. Gaul
1055
Examples of Hybrid Dynamic Models Combining Experimental and Finite Element Substructures R.L. Mayes, M. Ross, P.S. Hunter
1065
94
95
Propagation of Uncertainty in Test/Analysis Correlation for Substructured Spacecraft D.C. Kammer, S. Nimityongskul
96
Experimental Modal Substructuring to Extract Fixed-base Modes from a Substructure Attached to a Flexible Fixture M.S. Allen, H.M. Gindlin, R.L. Mayes
1083
1085
xiii
97
How Bias Errors Affect Experimental Dynamic Substructuring D. de Klerk
1101
98
A Benchmark Test Structure for Experimental Dynamic Substructuring P.L.C. van der Valk, J.B. van Wuijckhuijse, D. de Klerk
1113
99
Modal Test and Suspension Design for the Orion Launch Abort System D. Osterholt, D. Linehan
1123
100 Modal Survey Test and Model Correlation of the CASSIOPE Spacecraft V. Wickramasinghe, Y. Chen, D. Zimcik, P. Tremblay, H. Dahl, I. Walkty
1137
101 Assessment of Nonlinear Structural Response in A400M GVT J. Rodríguez Ahlquist, J. Martinez Carreño, H. Climent, R. de Diego, J. de Alba
1147
102 Application of PVDF Foils for the Measurements of Unsteady Pressures on Wind Tunnel Models for the Prediction of Aircraft Vibrations W. Luber, J. Becker
1157
103 Constrained Viscoelastic Damping, Test/Analysis Correlation on an Aircraft Engine E. Balmes, M. Corus, S. Baumhauer, P. Jean, J.-P. Lombard
1177
104 Identification of Material Damping in Vibrating Plates Using Full-field Measurements A. Giraudeau, F. Pierron
1187
105 A State Observer for Speed Regulation in Rolling Mill Drives F. Mapelli, E. Ruspini, E. Sabbioni, D. Tarsitano
1193
106 Vibrations Control in Cruise Ships Using Magnetostrictive Actuators F. Braghin, S. Cinquemani, F. Resta
1207
107 A Pedagogical Image Processing Tool to Understand Structural Dynamics J. Morlier
1215
108 Modal-based Camera Correction for Large Pitch Stereo Imaging P. Lanier, N. Short, K. Kochersberger, L. Abbott
1225
109 Structural Damage Identification Based on Multi-objective Optimization S. Jung, S.-Y. Ok, J. Song
1239
110 Effects of Shaker Test Set Up on Measured Natural Frequencies and Mode Shapes C. Warren, P. Avitabile
1245
111 Experimental Modal Analysis of Non-self Adjoint Systems: Inverse Problem Regularization M. Ouisse, E. Foltê te
1251
112 Frequency Domain Tracking of Time-varying Modes J. Lataire, R. Pintelon
1261
113 Variability in Natural Frequencies of Railroad Freight Car Components W.C. Shust, D. Iler
1273
xiv
114 Identification of Nonlinear Systems with a State-dependent ARX Model R.W. Clark, Jr., D.C. Zimmerman
1287
115 Model Identification for a Modal State Estimator from Output-only Data S. Engelke, C. Schaal, L. Gaul
1305
116 Data Acquisition for a Bridge Collapse Test K. Veggeberg
1313
117 Lake Mead Low Lift Pump Riser Failure Analysis D.H. Duffner, P.E. Duffner
1325
118 Beamforming for Quality Control in Industrial Environment P. Castellini, A. Sassaroli, N. Paone
1347
119 Noise Source Localization on Washing Machines by Conformal Array Technique and Near Field Acoustic Holography P. Chiariotti, M. Martarelli, E.P. Tomasini, R. Beniwal
1355
120 Design of an Active Seat Suspension for Agricultural Vehicles F. Braghin, F. Cheli, A. Facchinetti, E. Sabbioni
1365
121 Calibration and Processing of Geophone Signals for Structural Vibration Measurements R. Brincker, R. Bolton, A. Brandt
1375
122 Wake Penetration Effects on Dynamic Loads and Structural Design of Military and Civil Aircraft W. Luber
1381
123 Numerical and Experimental Modeling for Bird and Hail Impacts on Aircraft Structure M.-A. Lavoie, A. Gakwaya, M.J. Richard, D. Nandlall, M.N. Ensan, D.G. Zimcik
1403
124 A High Frequency Stabilization System for UAS Imaging Payloads K.J. Stuckel, W.H. Semke, N. Baer, R.R. Schultz
1411
125 Crystal Clear SSI for Operational Modal Analysis of Aerospace Vehicles M. Goursat, M. Dö hler, L. Mevel, P. Andersen
1421
126 Implementation of Multi-sine Sweep Excitation on a Large-scale Aircraft G. Hoople, K. Napolitano
1431
127 An Integrated Experimental and Computational Approach to Analyze Flexible Flapping Wings in Hover ̈ ̈ P. Wu, E. Sallstrom, L. Ukeiley, P. Ifju, S. Chimakurthi, H. Aono, C.E.S. Cesnik, W. Shyy
1441
128 Random Decrement Signal Processing of Modal Impact Test Data R. Brincker, A. Brandt
1453
129 Using Impulse Response Functions to Evaluate Baseball Bats D.L. Formenti, D. Ottman, M.H. Richardson
1461
130 FFT Integration of Time Series Using an Overlap-add Technique R. Brincker, A. Brandt
1467
xv
131 Requirements for a Long-term Viable, Archive Data Format A.W. Phillips, R.J. Allemang
1475
132 Modal Testing of Complex Hardened Structures J.C. Wolfson, J.R. Foley, L.M. Watkins, A.L. Beliveau, P.C. Gillespie
1481
133 Dynamic Force Characterization for an Industrial Process Using Response Measurements B. Pridham, B. Morava, B. Purnode, S. Mighton
1489
134 Operational Modal Analysis of a Rotating Tyre Subject to Cleat Excitation P. Kindt, A. delli Carri, B. Peeters, H. Van der Auweraer, P. Sas, W. Desmet
1501
135 Modal Identification of Flexible Structures with Applications in Robotic Manipulators Z.A. Rahman, A.A. Mat Isa
1513
136 Calibration of Very-low-frequency Accelerometers: A Challenging Task H. Nicklich, M. Mende
1521
137 Comparative Analysis of Triaxial Shock Accelerometer Output J.C. Dodson, L. Watkins, J.R. Foley, A. Beliveau
1529
138 Strain Sensors for High Field Pulsed Magnets C. Martinez, Y. Zheng, D. Easton, K. Farinholt, G. Park
1537
139 Wireless Noise and Vibration Management System for Construction Sites K. Veggeberg
1553
140 Estimation of Rigid Body Properties from the Results of Operational Modal Analysis A. Malekjafarian, M.R. Ashory, M.M. Khatibi
1559
141 Mass-stiffness Change Method for Scaling of Operational Mode Shapes: Experimental Results M.M. Khatibi, M.R. Ashory, A. Malekjafarian
1569
142 Sensitivity Analysis of Rigid Body Property Estimation from Modal Method M. Masoumi, S. Shahbazmohamadi, M.R. Ashory
1579
143 Vibration Absorber Design via Frequency Response Function Measurements N. Nematipoor, M.R. Ashory, E. Jamshidi
1587
144 Non Contact Eddy Current Dampers for Control Systems I. Shahini, M.R. Ashory, S. Shahbazmohamadi, A.A. Maddah, M.M. Khatibi
1595
145 Damping Augmentation of Nanocomposites Using Carbon Nanotube/Epoxy N. Kordani, A. Fereidoon, M. Ashoori
1605
146 Identification of Bolted Joints Under Repeatable Loads R. Khodadadberomy, M.R. Ashory, E. Jamshidi
1617
BookID 214574_ChapID 1_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
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T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_1, © The Society for Experimental Mechanics, Inc. 2011
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BookID 214574_ChapID 2_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII
February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc. &RPSDULVRQRI,PDJH%DVHG/DVHUDQG $FFHOHURPHWHU0HDVXUHPHQWV
Chris Warren, Pawan Pingle, Chris Niezrecki, Peter Avitabile Structural Dynamics and Acoustic Systems Laboratory University of Massachusetts Lowell One University Avenue Lowell, Massachusetts 01854 $%675$&7 Non-contacting measurements are extremely useful for many applications in which contacting measurements are not appropriate or impossible. Within this work, digital image correlation, dynamic photogrammetry, three dimensional (3D) laser vibrometry and accelerometer measurements are all used to measure the dynamics of a structure to compare each of the techniques. Each approach has its benefits and drawbacks, so comparative measurements are made using these devices to show some of the strengths and weaknesses of each technique, especially when measuring in a 3D environment. Comparisons are made in all cases to a well-studied finite element model as well as to each other. ,1752'8&7,21 Modal testing can be performed using a variety of different techniques. Accelerometer, laser vibrometer, and stereophotogrammetry measurement systems all have advantages and drawbacks, so each must be implemented where they will be most effectively employed, many times in conjunction with another technique. Accelerometers are by far the most traditional and widely-used sensors employed in modal testing. Their ease of use allows quick, broadband measurements to be made, but one must consider the effects of mass-loading, especially at higher frequency ranges. Laser Doppler vibrometers provide a non-contacting, broadband alternative to accelerometers, but large displacements and rigid body motion can contaminate the data dramatically. Conversely, digital image correlation (DIC) and dynamic photogrammetry are both displacement based approaches that analyze stereo image pairs to measure the three dimensional (3D) motion of surface patterns or specific points, respectively. Stereophotogrammetry has been used for many years in the field of solid mechanics to measure full field displacement and strain, but only very recently has the technique been exploited for dynamic applications to measure vibration [1-3]. As with laser vibrometry, line of sight with the measurement point must be maintained. Furthermore, surface preparation is required for both techniques; a speckle pattern is applied to a test object prior to imaging for DIC while dynamic photogrammetry tracks high-contrast, circular targets. To obtain mode shapes from accelerometer and vibrometer FRF measurements, modal parameter estimation must be performed. With both optical based systems studied in this paper, the mode shapes are measured directly using forced normal mode testing. Dynamic photogrammetry monitors the response at discrete targets, while DIC is capable of providing a relatively continuous measurement – on the order of tens of thousands of points – throughout a continuously patterned surface. A candidate structure used in previous studies was chosen to compare these 4 measurement approaches. Test setups and measurement considerations for each case are addressed. Each test is correlated to a well known and highly accurate finite element model and to the other test cases. Advantages and disadvantages observed are discussed.
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_2, © The Society for Experimental Mechanics, Inc. 2011
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6758&785$/'(6&5,37,21 7(67,1* A generic structure referred to as the BU (base – upright) was used as the test article for all tests presented in this paper. The base plate is 24x24 inches and rigidly bolted to the floor at four locations, while the upright is 24x36 inches. Both plates are 3/4” thick aluminum. A finite element model is available and has been shown to be very well correlated to other measured test data from previous studies [4-6]. Table 1 summarizes the Modal Assurance Criterion (MAC) and Pseudo-Orthogonalityy Check (POC) results for the accelerometer and laser vibrometer for the first 7 modes. In previous work, excellent correlation between both laser and accelerometer measurements and the BU finite element model was presented [7]. The average frequency difference is less than 2.5% and the minimum MAC is greater than 0.97 for the first 7 modes, confirming that the model is a very good representation of the true structure. The first four analytical mode shapes of the BU are shown for reference in Figure 1.
+]
+]
+]
+]
Figure 1. Structure model and its mode shapes for structure under study. Table 1. Previous accelerometer and laser Doppler vibrometer results (for the first 7 modes). Type of Correlation Term
Accelerometer Data
Laser Data
Average MAC Diagonal Average MAC Off-Diag. Average POC Diagonal Average POC Off-Diag.
0.989 0.002 0.996 0.021
0.992 0.055 0.995 0.027
Necessary changes in the test setup were made when transitioning from the more traditional modal approaches to DIC and dynamic photogrammetry testing. The two configurations are described in detail below. Accelerometer & Laser Doppler Vibrometer Test Setup When the accelerometer and laser data was acquired, shaker excitation was provided at an angle 45 degrees relative to all three principle axes so that all modes would be excited. The excitation was pseudo-random over a 400 Hz bandwidth. Thirty averages were taken with 1600 lines of spectral resolution to obtain adequate coherence over the frequency range of interest. Figure 2a depicts shaker mounted to the BU with the laser and accelerometer measurement points indicated by red dots and their numbers. Figure 2b displays an overlay of FRFs in the z-direction measured by an accelerometer and the laser Doppler vibrometer at point 3. A frequency domain, polynomial curvefitter was then used to extract the modal parameters and mode shapes. DIC & Dynamic Photogrammetry Test Setup Both optically-based measurement techniques monitored the response of the BU at the same 8 locations as in the accelerometer and laser tests. While DIC can be used over the whole surface [2], an alternative approach with patches of patterns was employed. This was done to illustrate that equivalent data could be extracted without having to pattern the entire structure. The surface treatment of the BU needed for the DIC test can be seen in Figure 3a. To create the patches, flat white spray paint coated the areas of interest. A black permanent marker was then used to create the speckle pattern. Circular targets with adhesive backs were subsequently applied over the speckle pattern prior to performing the dynamic photogrammetry test. An example of each can be seen in Figure 4.
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Figure 2. (a) Shaker orientation and measurement points. (b) Sample FRF from previous study [7]. Because these systems measure displacements (shapes) directly, forced normal mode testing was conducted to drive the structure at resonance. Two shakers were mounted near the base of the upright, as shown in Figure 3b. The camera pair then captured a series of images throughout 2 to 3 cycles using a phase-stepping approach. The amplitude and phase of vibration at 8 discrete points were monitored and used as feedback to tune the 2 shaker inputs such that the BU exhibited single mode behavior. Figure 3c shows an example of the motion at the same 8 points for the 26 Hz mode for one of the dynamic photogrammetry measurements. Note that each signal is in-phase, which is to be expected when measuring what is essentially the first bending mode of the upright. The 3 points along the top of the upright are moving at roughly the same amplitude while those along the midline move together, etc. The maximum values of displacement during a test were used when correlating the optical results to the finite element model, laser, and acceleration mode shapes because they naturally have the highest signal-to-noise ratio. In this example, the shape used was measured in Stage 18, indicated by the red cursor. When any forced normal mode (FNM) test is run, a variety of shaker configurations are typically needed to excite different mode shapes. By its nature, the stereophotogrammetry methods used here are essentially an extension of FNM testing, so not all modes of the BU could be measured with the dual shaker configuration. Only bending or torsion modes of the upright were targeted for this study.
Figure 3. (a) BU Prepped for DIC testing. (b) Shaker orientation. (c) Sample output from a dynamic photogrammetry test.
Also, a drawback to any displacement-base measurement technique is that relatively high frequency phenomena cannot be measured due to the low displacements associated with their vibrations. In these tests, which had a working distance of approximately 2 m between the cameras and BU, the out-of-plane noise floor was measured to be approximately 40 ȝm. These
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limitations only allowed the first and third modes of the structure to be captured using DIC and dynamic photogrammetry; for the sake of this study, only these two modes were used. &255(/$7,21 To evaluate the performance of the four measurement techniques, Modal Assurance Criterion (MAC) values were calculated using the FEMtools software package [4]. Overlay plots of the four experimental approaches and the finite element model can be seen in Figure 5. Tables 2 and 3 summarize the correlation results for modes 1 and 3, respectively, comparing all methods. Overall, the results are very good. The accelerometer and laser MAC values were all above 99.5 when compared to the FEM for Figure 4. Prepared measurement surface: speckled for DIC and circular targets for dynamic photogrammetry. the two modes evaluated. When dynamic photogrammetry (DP) was used, MACs of at least 99.7 were obtained. DIC yielded the slightly lower results for both modes 1 and 3 with values of 99.3 and 97.8, respectively. Originally, the value between the FEM and the DIC measurement for mode 3 was 96.8, but removing the center two patches improved the MAC by a full point. These patches lie along what is essentially the node line for the torsion mode of the upright. The structure exhibits little to no response in this area, so the measurement will have the lowest signal to noise ratio here. Removing the points reduced the variance between the measurement and the model. When comparing two experimental sets, the purity of the finite element model’s mode shapes is lost and variance on the measurements can compound to provide biased results. However, in this case good correlation was obtained for both modes (1 and 3); all MAC values were 95.6 and higher when the empirical results from two different approaches were correlated.
&DVH Laser Mode 1 Accel Mode 1 DIC Mode 1 DP Mode 1
Table 2. Comparison of MAC values for mode 1. Laser Mode 1 Accel Mode 1 DIC Mode 1 DP Mode 1 100 99.5 99.7 99.8 99.5 100 99.7 99.6 99.7 99.7 100 99.4 99.8 99.6 99.4 100
FE Model 99.9+ 99.8 99.3 99.9
&DVH Laser Mode 3 Accel Mode 3 DIC Mode 3 DP Mode 3
Table 3. Comparison of MAC values for mode 3. Laser Mode 3 Accel Mode 3 DIC Mode 3 DP Mode 3 100 98.4 99.6 99.5 98.4 100 97.3 98.5 99.6 97.3 100 95.6 99.5 98.5 95.6 100
FE Model 99.6 99.5 97.8 99.7
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Figure 5. FEM to measurement correlation results for the various methods, for modes 1 (26 Hz) and 3 (71 Hz).
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2%6(59$7,216$1',17(535(7$7,212)5(68/76 Each technique applied in this comparison has its advantages and drawbacks, but in the end all yielded good, consistent measurements. The biggest difference between the traditional and the optical techniques is the approach taken to measure multiple modes. Accelerometers and laser vibrometers measure multiple modes over a broad frequency range point by point. Conversely, the two optical approaches measure all points simultaneously, one mode at a time. Accelerometers provide an inertial reference frame, so establishing the calibration and orientation procedures (required by the scanning laser vibrometer and optical measurements) can be simpler. However, mass-loading has to be taken into consideration, especially when performing a test that requires numerous or roving accelerometers. The laser Doppler vibrometer has the widest dynamic capabilities as it can measure velocities from as low as a few Hertz up to 80 kHz. Low frequency measurements must be taken with care. When the laser is stationary, i.e. not tracking the global motion of the test article, the effective measurement point is constantly moving on the structure when high displacements or large rigid-body movements are present. The main advantages of digital image correlation are the immense number of effective measurement points and that the strain throughout these patches can be measured directly. With the analysis configuration (facet settings) used, each patch – about 3x3” – had on average 400 effective measurement points. Had the entire face been patterned while maintaining the same resolution, the number of points measured would be on the order of 30,000 to 40,000. It should be noted that the laser vibrometer has the ability to sequentially measure a comparable number of points, but the time needed to acquire the data would be quite large. In either case, the limiting factors become processing power and memory. As opposed to tracking facets of pixels from image to image, dynamic photogrammetry tracks the 3D motion of circular targets. The number of measurement points is on the order of the more traditional modal tests, so the computation time relative to DIC is greatly reduced. Because discrete targets are being tracked, local strain cannot be calculated as with DIC. A benefit to either optical technique is their ability to measure high amounts of rigid body motion. Cabling and tracking of the targets do not have to be considered as long as the test piece remains within the field of view. Conversely, the primary disadvantage to both optical approaches is that the low amplitude displacements associated with high frequency vibrations can fall below the noise floor of the optical measurement. When mode shapes are to be measured, these optical tests require at least a partial modal test in preparation of FNM testing, as well as feedback transducers (accelerometers or vibrometers). &21&/86,216 The results of this study show that digital image correlation and dynamic photogrammetry can be used to measure mode shapes which correlate very well to those obtained using accelerometers and laser Doppler vibrometers. All four measurement approaches were used to acquire the first and third mode of the structure studied and then were correlated to each other as well as a highly accurate finite element model. Excellent correlation between the measurements and the FEM was obtained; each MAC value was above 97.8. When the experimental results were compared, all MAC values were above 95.6. 5()(5(1&(6 1. 2. 3. 4. 5. 6.
Warren, C., C. Niezrecki, & P. Avitabile, “Applications of Digital Image Correlation and Dynamic Photogrammetry for Rotating and Non-rotating Structures.” Proceedings of the 7th International Workshop on Structural Health Monitoring, Stanford, CA, September, 2009. Helfrick, M., C. Niezrecki, & P. Avitabile, “3D Digital Image Correlation Methods for Full-Field Vibration Measurement.” Proceedings of the Twenty-Sixth International Modal Analysis Conference, Orlando, FL, Feb 2008. Helfrick, M., C. Niezrecki, & P. Avitabile, “Optical Non-Contacting Vibration Measurement of Rotating Turbine Blades.” sProceedings of the Twenty- Seventh International Modal Analysis Conference, Orlando, FL, Feb 2009. Butland, A., “A Reduced Order, Test Verified Component Mode Synthesis Approach for System Modeling Applications,” Master’s Thesis, University of Massachusetts Lowell, January 2008. Nicgorski, D. “Investigation on Experimental Issues Related to Frequency Response Function Measurements for Frequency Based Substructuring,” Master’s Thesis, University of Massachusetts Lowell, January 2008. Wirkkala, N.A., “Development of Impedance Based Reduced Order Models for Multi-Body Dynamic Simulations of Helicopter Wing Missile Configurations,” Master’s Thesis, University of Massachusetts Lowell, April 2007.
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7.
Pingle, P., J. Sailhamer, & P. Avitabile, “Comparison of 3D Laser Vibrometer and Accelerometer Frequency Measurements,” Proceedings of the IMAC-XXVII, February 9-12, 2009 Orlando, Florida USA. 8. FEMtools 3.0 – Dynamic Design Solutions, Leuven, Belgium. 9. LMS Test.Lab – Leuven Measurement Systems, Leuven, Belgium 10. Polytec Scanning Laser Doppler Vibrometer, Polytec Optical Measurement Systems
BookID 214574_ChapID FM_Proof# 1 - 23/04/2011
BookID 214574_ChapID 3_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Modal Parameter Estimation Using Acoustic Modal Analysis
W. Elwali, H. Satakopan, and V. Shauche R. Allemang, and A. Phillips University of Cincinnati, Department of Mechanical Engineering, Cincinnati, Ohio, 45221, USA
ABSTRACT Acoustic modal analysis (AMA) is of interest in cases where accelerometer measurements are limited by mounting techniques and where the mass of sensors affects the system dynamics. Major problems in performing AMA are time delay adjustment and the inability of obtaining true driving point measurements. For an impact test, the former problem causes difficulties because each measured acoustic frequency response function (AFRF) will have its own time delay as a function of the position of the reference microphone with respect to the structure. Thus, obtaining consistent modal parameters conventional multi-input, multi-output (MIMO) modal parameter estimation methods utilizing several microphones (MIMO AFRFs) becomes rather difficult. The latter problem complicates the computation of modal scaling which is frequently required in model validation. As an example, both microphone measurements and accelerometer measurements are utilized in an impact test for a heavy ringdisc structure. The results from each method are compared to study the effectiveness of estimating modal parameters from AFRFs compared to conventional FRFs. While some conventional modal parameter estimation tools such as the consistency diagram and the complex mode indicator function (CMIF) look slightly different, the frequencies, damping and mode shapes estimated using AFRFs are consistent with those of standard modal analysis. 1. INTRODUCTION Acoustic modal analysis is an interesting technique since it does not affect the system parameters such as mass, stiffness and damping. It is a useful technique especially when mounting accelerometers is difficult or in cases where accelerometers affect system parameters. Acoustic modal analysis is based on the assumption that emitted sound pressure levels vary linearly with vibration amplitudes at a certain frequency. Therefore for a linear structure, at a particular frequency, sound pressure level is directly proportional to the modal coefficient of the input point [1]. The proportionality is a function of the modal vibration pattern that is excited at each frequency in the structure and the coupled acoustic pattern that emanates from the structure. Based upon the impact point and the modal pattern, the input location may not result in a mode being excited (node at the input). Based upon the impact point and location, the coupled acoustic pattern may have a null at one of the reference microphones (node at the output). Finally, based upon the impact point and the modal pattern, if the modal pattern does not involve sufficient surface area to provide adequate vibro-acoustic coupling, the modal pattern cannot be observed (node at all outputs). There are mainly two problems with using acoustic modal analysis. The first one is the limitation of microphone to be placed in the near field since the distance affects the signal to noise ratio [1]. The other problem is the time delay problem. It is the time required for the sound wave to travel from impact point to microphone.
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_3, © The Society for Experimental Mechanics, Inc. 2011
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System dynamics
Vibrations
Sound pressure
Impact hammer
Accelerometer
Accelerometer measurement
Fluid structure interaction
Microphone
Microphone measurement
Figure 1. Accelerometer and microphone measurements flow charts In this paper, a circular disc is experimentally studied in order to find its modal parameters such as natural frequencies, mode shapes and modal damping. This research study was initiated by a need to identify vibration characteristic of a hypoid gear so the disc is just a simpler test structure that can be more accurately and easily modeled. The main objective of this work is to estimate modal parameters using two techniques: microphone measurements and conventional accelerometer measurements. Microphone measurements are to be examined in terms of their ability to estimate modal parameters compared with respect to conventional measurement. Moreover, the difficulties and limitations accompanied with acoustic modal analysis including the time delay problem are also discussed. 2. FINITE ELEMENT CHECK Before conducting the experiment, natural frequencies were found using the finite element method. The purpose of using the finite element method (FEM) is to check whether the in-plane (radial) modes are in the frequency range of interest (0-9000Hz). FEM is also used to detect the presence of repeated roots and thus decide on the minimum number of references to be used. It is necessary to check the presence of in-plane modes in the frequency range of interest, since exciting both sets of modes, in-plane modes and transverse modes, requires more FRF measurements, even though the inplane modes may not be well identified using acoustic modal analysis. Using the finite element method, it was found that a significant number of modes lie in the range 0-9000Hz with all being out-of-plane (transverse) modes. In-plane modes are above 9000Hz and will not be accounted for. 3. EXPERIMENTAL PROCEDURE The experiment was performed by exciting the circular disc using an impact hammer at 17 points perpendicular to the plane of the disc. For accelerometer measurements, three uni-axial accelerometers were mounted on the 2 disc. In this case frequency response functions [m/s /N] are measured and then processed to estimate modal parameters. In the case of acoustic modal analysis, three microphones are mounted in the nearfield of the disc perpendicular to the plane of the disc. Acoustic frequency response functions [Pa/N] are then measured. These acoustic frequency response functions can be used to estimate modal parameters under the assumption that sound pressure level varies linearly with vibration amplitudes.
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25 3.1. Experimental Setup There are a total of six fixed outputs (three accelerometers and three microphones). There are 17 input points on the disc. Impact hammer (Model E086C40) with metal tip is used for excitation since frequency range of interest is around 9 KHz. Since the response locations are fixed, the impact hammer is moved to various points for excitation. This is called a roving hammer method or a multiple reference impact test (MRIT). Uni-axial accelerometers (Model 352A56) are used to measure response (3 mounted in transverse direction). Three microphones (Model 130A10) are used for sound pressure measurements. The experimental setup is shown in Figure 2.
Figure 2. The experimental setup In setting the digital signal processing (DSP) parameters in the MRIT VXI test setup, the span frequency is set at 12.58 KHz and the number of spectral lines is set to 1600 lines for better frequency resolution. Force-Exponential windows are used for this test, since an impact type of excitation is used, and the pre-trigger value is set to 10%. All three accelerometers are calibrated using a hand held calibrator. The calibration is done at 159 Hz with 1g RMS acceleration level. The nominal sensitivity values (taken from the PCB website) are used for the microphones and load cell. The basic assumptions of experimental modal analysis like linearity and reciprocity are checked. 3.2. FRF Measurement Auto-ranging of the sensors is done for every impact point to prevent overloading of sensors. Five spectral averages are done. Force-exponential windows are applied to the data. The accelerometers are mounted using adhesive mounting, as it is effective up to 9000 Hz. 3.3. Correction for Time Delay in Microphone Signals Since the microphones are kept at a distance of around 5.5 inch from the test component, there is some time delay in the microphone signals. This time delay leads to the phase wrap in the AFRF over the entire frequency range. The phase plot of AFRF with time delay is shown in Figure 3(a). By impacting at the center of the component, the time delay values were obtained for each microphone. The microphone data in each channel were corrected based on the time delay. The phase wrap of AFRF reduced after the time delay correction. The phase plot of AFRF after time delay correction is shown in Figure 3(b).
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(b)
Figure 3. Phase plots of AFRF. (a) With time delay. (b) With time delay correction Once these values were fixed based on the impact at the center point, the same values were used when impacting at different points. Consistent poles were not obtained with this approach as shown in Figure 4.
Figure 4. Consistency diagram for acoustic modal analysis with constant time delay values This led to the speculation that time delay at different microphones varies significantly with impact point. So the time delays for the three microphones were calculated for each of the 17 impact points. These delay values were used for correcting each microphone channel while impacting at different points and thus consistent poles were obtained.
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27 If time delay is not corrected and the same time is used as that for the accelerometers, data will be lost out of the end of the data block contributing to more noise and poor coherence (not all data is observed).[2] The first correction mentioned in the above explanation reduces the coherence problem but is not sufficient to get optimum data. The second method in the above explanation is needed to be certain the best quality data is available, which is required by the multiple reference parameter estimation algorithms. 3.4. Creation of Wireframe Model and Trace Lines Using the Wireframe Editor option in X-Modal II software (developed by SDRL, University of Cincinnati), the wireframe model of the circular disc is created by entering the co-ordinates of all the measurement points and the creating trace lines connecting measurement points. This wireframe model is saved in universal file format (.ufb). These .ufb files are loaded into X-Modal II Data manager for parameter estimation. 4. PARAMETER ESTIMATION PROCEDURE There are several modal parameter estimation methods available to extract modal behavior of the system from the measured frequency response functions. These methods are implemented to estimate modal parameters from the measured data and classify the modes as consistent, non-realistic, etc. 4.1. Mode Indicator Function Mode indication functions (MIF) are normally real-valued, frequency domain functions that exhibit local minima or maxima at the natural frequencies of the modes [3]. Figures 5 and 6 show mode indicator functions from which the number of modes, including close or repeated roots, can be obtained. The number of peaks in the Complex Mode Indicator Function (CMIF) plot indicates the number of modes in the disc in the specified frequency range. Multiple peaks at the same frequency indicate repeated roots or close modes.
Figure 5. CMIF plot using accelerometer measurement From Figure 5 showing the CMIF based on accelerometer measurements, a total number of 13 peaks can be found in the 0 to 9000 Hz frequency range and thus number of modes is 13. The CMIF plot based on the microphone measurements (without correction) shows the presence of additional peaks (apparent repeated roots) which confuses the analysis of the data. The CMIF plot for microphone measurement is shown in Figure 6.
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Figure 6. CMIF plot for microphone measurement 4.2. MDOF Methods Time domain methods are usually limited to low damping case and may not perform accurately in the moderate or heavily damped cases. Knowing that the circular disc is a lightly damped structure, the Polyreference Time Domain (PTD) method, which is a higher order, matrix coefficient polynomial method, is used in the following analysis. 5. RESULTS AND DISCUSSION Modal order of 20 is used to get the 13 modes from the accelerometer measurements. The consistency diagram represents consistent poles and mode shapes; the poles with Mean Phase Correlation (MPC) values greater than 0.95 are selected. Figure 7 shows the consistency diagram for accelerometer measurements.
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Figure 7. Consistency diagram for accelerometer measurements Modal order of 30 is used to get 13 modes from the microphone measurements. The consistency diagram is shown in Figure 8.
Figure 8. Consistency diagram for microphone measurements
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30 Due to the geometrical symmetry of the plate, it is expected to observe close or repeated roots. However, the presence of repeated roots does not mean that they must be due to symmetry. Two roots are considered to be repeated if they are very close in frequency and in damping. The comparison of frequencies and damping values obtained using both the methods are given in Table 1. Table 1. Comparison of standard and acoustic modal analysis result
Mode
1 2 3 4 5 6 7 8 9 10 11 12 13
Accelerometer measurements Frequency (Hz) 1288.701 1298.466 2103.88 2987.763 2991.433 4637.023 4660.377 5163.566 5184.923 7745.799 7780.38 7796.185 8380.989
Damping % 0.502 0.505 0.38 0.591 0.283 0.205 0.204 0.173 0.157 0.37 0.191 0.145 0.154
Microphone Measurements Frequency (Hz) 1289.294 1298.981 2104.209 2988.226 2990.331 4637.171 4643.121 5164.086 5170.13 7738.605 7769.714 7788.736 8372.652
Damping % 0.437 0.482 0.394 0.396 0.326 0.209 0.223 0.164 0.166 0.448 0.255 0.159 0.171
% Difference in frequency
% Difference in damping
0.046 0.040 0.016 0.015 0.037 0.003 0.370 0.010 0.285 0.093 0.137 0.096 0.099
12.948 4.554 3.684 32.995 15.194 1.951 9.314 5.202 5.732 21.081 33.508 9.655 11.039
As seen from Table 1, the percentage difference in frequency values between the standard modal analysis and acoustic modal analysis is negligible. But there is considerable difference in some damping values, probably due to the vibro-acoustic nature of the test method involving the coupling of the vibration to the acoustic field and the characteristics of the acoustic pattern that develops. The close roots taken from standard modal analysis results are shown in Table 2. Table 2. Close roots from standard modal analysis Mode Frequency (Hz) Damping (%) 1, 2 1288.701, 1298.466 0.502 , 0.505 4, 5 2987.763, 2991.433 0.591, 0.283 6, 7 4637.023, 4660.377 0.205, 0.204 8, 9 5163.566, 5184.923 0.173, 0.157 11,12 7780.380, 7796.185 0.191, 0.145 The modal vectors derived from the residues are generated using the poles selected from the consistency diagram. For the accelerometer measurement, the FRF correlation coefficient (comparing the FRF measurement and the synthesis of the FRF measurement based upon the model) is close to unity for all cases checked. In order to measure the linear dependence of mode shapes, the Modal Assurance Criterion (MAC) is used. MAC can take values between zero and one. If the modal assurance criterion has a value near zero, this is an indication that the modes are linearly independent. If the modal assurance criterion has a value near unity, this is an indication that the modes are linearly dependent. Figures 9 and 10 show MAC plots of standard modal analysis and acoustic modal analysis respectively.
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Figure 9. MAC plot for Standard modal analysis
Figure 10. MAC plot for Acoustic modal analysis
The MAC plot of standard modal analysis (Figure 9) shows that most of the modes are linearly independent. Some small amount of linear dependence is observed in certain modes which is not unexpected. Note that there is good agreement (linear dependence) for the modal vectors and their complex conjugate pairs (lower diagonal of red squares). The MAC plot of acoustic modal analysis (Figure 10) shows that modal vectors associated with positive frequencies are again linearly independent of one another.
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32 However, note that there is not good agreement for the modal vectors and their complex conjugate pairs (lower diagonal) as in the previous case. The modal vectors for the conjugate frequencies appear to be more linearly dependent. This may be because the acoustic frequency response function is a ratio of pressure to force where the modal vectors for the conjugate frequencies may not be necessarily the same. 5.1. Mode Shapes Table 3. Mode shapes description Mode number Description 1, 2 Bending mode (Saddle mode) 3 Bending mode (Umbrella mode) 4, 5 Bending mode 6, 7 Bending mode (Inner ring see-saw) 8, 9 Bending mode 10, 11 Bending mode (Inner saddle) 12 Bending mode 13 Bending mode (Reverse umbrella mode) The mode shapes obtained using microphone and accelerometer measurement are similar as shown in Figure 11.
Accelerometer
Microphone
1288.701 Hz 0.5025 % zeta
1289.2945 Hz 0.4369% zeta
8380.989 Hz 0.154 % zeta
8372.652 Hz 0.171 % zeta
Figure 11. Comparison of mode shapes from standard and acoustic modal analysis 6. DISCUSSION OF ERRORS In conducting experiment, errors are usually generated by the following: • • • •
Leakage error: The leakage error will exist anytime the digitized time domain data does not match the requirements of the FFT (periodic in observation time T or totally observed transient in observation time T). This was minimized using exponential windowing. The FRF referred to as the driving point FRF is approximate, because it was not possible to hit at the exact same location where the accelerometers are mounted. In taking averages, some errors may occur due to not hitting the same point and in the same direction, average to average. Force with same magnitude not applied during every measurement which should not be a problem if the system is linear.
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33 •
• •
The time delay correction factor for microphone measurements is measured to within the nearest time difference based on the measured pressure and impact at a particular point. The measured pressure signal is the superposition of sound radiated from various points and each point can have its own time delay. The time delay used for the correction is the minimum time delay. The PTD method is a time domain method that applies an inverse FFT to get the impulse response, which may have a small frequency domain truncation error. The selected poles from the consistency diagram may not be at the centroid of the pole cluster leading to variation in damping estimation.
7. SUMMARY AND CONCLUSIONS Accelerometer measurements and microphone measurements were conducted on a circular disc and modal analysis results were compared in terms of modal parameters. For accelerometer measurements, the setup consists of three accelerometers (outputs) and 17 impact points (inputs). At each point, impact excitation is done in the transverse direction. Time delay correction is done for each microphone channel during each measurement. In order to minimize leakage and noise errors, force-exponential windowing was applied to the measured data. For microphone measurements, it is critical to account for the time delay associated with the reference microphone’s position relative to the part being tested. The PTD method was used to estimate modal parameters for both measurement techniques, from the measured FRFs and AFRFs independently. It was found that both techniques are able to find 13 modes in the frequency range 0 to 9000 Hz. In predicting natural frequencies, the acoustic modal analysis method is able to perform accurate estimation with an error of less than 0.096% relative to those found from accelerometer measurement. However, microphone and accelerometer measurements do not estimate the same value of damping with an average relative difference of 12.83% found. Some linear dependency was observed between modes 5 and 9 from the MAC plot of accelerometer measurements. More measurement points are required for better observability of these two modes. From the MAC plot of microphone measurements, it was observed that modal vectors associated with conjugate poles do not show a reasonable dependency. Modal scaling is not obtained from acoustic modal analysis since true driving point FRF measurements are not available. But, modal scaling can be obtained if a vibro-acoustic structure model with coupling is taken into consideration. It is probably easier to simply conduct a standard modal analysis test to get an estimate of the modal scaling even though the modal frequencies will be incorrectly identified. The two test cases can then be combined to give the complete modal model. REFERENCES [1] Allemang, R.J., Shapton, W.R., “Using modal techniques to guide acoustic signature analysis”, SAE, 780106, 1979. [2] Halvorsen, W.G., Brown, D.L., “Impulse Technique for Structural Frequency Response Testing”, Sound and Vibration Magazine,November,1977,pp. 8-21. [3] Allemang, R.J., Phillips, A.W., “The Unified Matrix Polynomial Approach to Understanding Modal Parameter Estimation: An Update”, ISMA 2004.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Mitigation of Vortex-Induced Vibrations in Long-Span Bridges
Atul Patil Research Assistant, Dept. of Civil and Environmental Eng., FAMU-FSU College of Engineering, 2525 Pottsdamer Street, Tallahassee, FL 32310 USA Sungmoon Jung Assistant Professor, Dept. of Civil and Environmental Eng., FAMU-FSU College of Engineering, 2525 Pottsdamer Street, Tallahassee, FL 32310 USA Soon-Duck Kwon Associate Professor, Dept. of Civil Eng., Chonbuk National University, 664-14 1Ga DeokjinDong, Jeonju-City, Jeonbuk, 561-756, South Korea
NOMENCLATURE ܽ݁ B ܾ௭ ǡ ۱ ܪ ܪଵ כǡ כܣଶ ܮ௫ ݉ ǡ ࡹ ௭ ܵ ߞ ߩ ߣ ߱ ı ~
= Aerodynamic quantities = Width of bridge deck = Non-dimensional load spectrum band width parameter = Damping coefficient, damping matrix = Frequency response function = Aerodynamic derivatives in lift, torsion = Wind exposed length = Equivalent mass, mass matrix = Non-dimensional rms lift coefficient = Cross spectral density = Damping ratio = Mode shape vector = Air density = Non-dimensional coherence length scale = Circular frequency = Variance = Indicates modal quantity
ABSTRACT Span lengths of bridges are ever increasing, leading them to be more slender. It is well known that the slender structures are more vulnerable to wind-induced forces and vibrations. Some bridges have shown significant windinduced vibrations, which were not anticipated during conceptualization and design phase. In order to improve the aerodynamic performance of a bridge after the construction, two approaches are commonly used. The first approach is to retrofit the cross-section of the bridge to be aerodynamically more favorable. The second approach is to add tuned mass dampers to dissipate the energy. In this paper, we propose a method to obtain an optimal solution when both approaches are used simultaneously. Multi-objective optimization technique is employed to
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_4, © The Society for Experimental Mechanics, Inc. 2011
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36 select the most significant solution form the set of possible solutions. The owner of the bridge may select the best solution based on the preferred performance and cost criteria. INTRODUCTION Wind is the one of the important factors that needs due consideration in conceptualization and design stages of long span suspension bridges. In the past few decades the maximum suspended span lengths have been increased to great extent. The Akashi- Kaikyo Bridge has central suspended span of 1991m. Even longer bridges like Messina Straits with 3300m and Gibraltar straits with 5000m are planned [1]. This increased suspended length has attracted one drawback in terms of slenderness of bridge decks, which are more susceptible to cross winds. Aerodynamic instability increases with increase in suspended span length. Wind direction and the behavior of the bridge deck under the wind forces set the bridge into four major aerodynamic responses such as flutter, galloping, buffeting and vortex shedding. A bridge deck may exhibit only one aerodynamic response exclusively, or multiple aerodynamic responses simultaneously. Trans-Tokyo Bay Crossing Bridge in Japan has exhibited the combined vortex and galloping during wind tunnel test on two dimensional sectional models [2]. Akashi-Kaikyo Bridge in Japan, Golden Gate Bridge across San Francisco Bay are the examples that have exhibited the wind induced flutter. Rio-Niterói Bridge [3] in Rio de Janeiro, has exhibited the vortex shedding oscillations even at low wind speed of 14m/s, which were not anticipated during the design and conceptualization phase. One solution to attenuate the aerodynamic response of an existing bridge lies in retrofitting the cross section and thereby improving aerodynamic response. In situations where structural modifications prove ineffective, energy absorbing units called dampers are used. The authors of [4] have shown that adding guide vanes to the main span of girder decreased the vortex induced oscillations in Great belt suspension Bridge. Wind tunnel test done on two and three dimensional sectional model, established that the oscillations in Trans-Tokyo Bay Crossing Bridge [2], due to first and second mode could be brought down to acceptable limit by using 16 TMDs. The vibrations due to higher modes were controlled by attaching the vertical plates. The focus of this paper is analytical investigation on mitigation of vortex induced vibrations using cross sectional retrofit along with tuned mass dampers. Multi-objective optimization technique is used to derive the possible solution set under conflicting objectives. The resulting solution set is termed as Pareto-optimal solutions. When a bridge deck is to be retrofitted to improve the aerodynamic performance, two objectives (cost and response of bridge such as deflection) conflict with each other. If the aerodynamic response of the bridge is required to keep lower, the cost involved is higher and vice versa. It is left to users’ discretion to choose a solution from the solution set. There is no single solution which can be termed as the best solution to a problem. This paper first presents a brief overview of vortex shedding analysis of a bridge deck. In the second section, conflicting objectives in retrofit of bridge deck is explained. In the third section brief overview of multi-objective optimization is discussed. In the last section, the combined approach of cross sectional retrofit and addition of tuned mass damper is discussed. VORTEX-INDUCED VIBRATIONS Vortex induced vibration is an aerodynamic response when a slender, flexible structure with low damping is subjected to cross wind. This type of aerodynamic vibrations resulting from the separation of air flow causes vortices to shed alternatively on either side of bridge deck. The vortices formed give rise to fluctuating across wind forces, cross sectional torsional moments accompanied by fluctuating vertical or rotational displacements. For analysis of vortex-induced vibrations, a two dimensional line-like structure is considered. Along wind speed effect is normally disregarded. The vortices generated give rise the fluctuating across wind forces qz, cross sectional torsional moment qș, and fluctuating displacements rz, rș. Experimental evidence shows that these fluctuating loads are narrow banded and centered at vortex shedding frequency fs. The vortex shedding frequency is the function of cross sectional depth D.
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fs =St
(1)
St is Strouhal number which has specific value for a typical cross section, V is cross wind velocity, D is Cross sectional depth. The relationship between the increment in velocity and frequency is linear till first frequency “lockin” state arrives. Vibrations are observed for various eigen frequencies, narrow banded about these frequencies. Vickery and Basu attributed this motion induced load effects at “lock-in” to the negative motion dependent aerodynamic damping Ƀୟୣ [5]. The total damping ratio associated with mode݅ is given by
ߞ௧௧ = ߞ െ ߞ ሚೌ
ߞ ൌ ۱ ൎ
(2) ೣ ۱ೌ ௗ௫
ൌ ෩
ଶఠ ெ
ఘ ଶ
(3)
ಽ ଶ ௗ௫
Ͳ ߱ ሺܸሻ Ͳ Ͳ
Ͳ ܪଵכ Ͳ
Ͳ Ͳ ଶ כ ܣ ܤଶ
(4)
From equations (3) and (4)
ߞ ൌ ݉ ൌ
మ כథమ ାమ כథమ ൯ ௗ௫ ఘಽ మ ഇ ೣ ൫ுభ
ಽ ࣐ ସ ࣐ ௗ௫ ෩ ெ
ಽ ௗ௫
ൌ
(5)
෩ ெ
(6)
మ ାథమ ାథ మ ൯ ௗ௫ ಽ ൫థ ഇ
Vickery and Basu, developed net motion independent cross sectional load spectra and incorporated in [5] is as follows. భష
ഘ మ
ഘೞ ۍ మ షቌ ್ ቍ ې ෝ ൯ ൫ಳ ۑ మێ ್ భ మ ܵ ሺ߱ሻ ቀ ఘ ቁ ێ ۑ మ ഘ భష ቈ ൌ మ గǤఠ ێ ഘ ೞ మ షቌ ξ ቍ ۑ ܵഇ ሺ߱ሻ ೞ ್ഇ ෝ ቁ ቀಳమ ഇ ێ ۑ ್ ێഇ ۑ ۏ ے
(7)
The variance contribution from any mode i, for resonant part of Aerodynamic response is given by ஶ
ଶ
ɐଶఎ ൌ หܪఎ ሺ߱ሻห ܵொ ሺ߱ሻ݀߱ In equation (8), load spectra value
(8)
ܵொ ሺ߱ሻ from equation (7) are used. Equation (7) gives the load spectra in
vertical (z) and rotational (ߠሻ direction.
CONFLICT OF OBJECTIVE FUNCTIONS Aerodynamic response of a bridge can be thought of as a function of cross sectional shape of bridge deck and can be mitigated by retrofitting the section to make it aerodynamically more favorable. Mitigation of the aerodynamic response results in conflicting objectives. For the sake of simplicity, this paper considers the
BookID 214574_ChapID 4_Proof# 1 - 23/04/2011
38 response in vertical direction only. If the displacement is required to be reduced, the corresponding cost will increase and vice versa. Mathematically these two objectives are expressed as two objective functions. ݂ଵ ሺࢄሻ ൌ ݐ݂݅ݎݐ݁ݎ݂ݐݏܥ ݂ଶ ሺࢄሻ ൌ ܸ݁ݐ݈݊݁݉݁ܿܽݏ݈݅݀ܽܿ݅ݐݎ
(9) (10)
The cost of retrofit can be framed as the ratio of length of retrofit to span length. The variable ࢄ describes the segments of the bridge to be retrofitted. ࢄ ൌ ሾ࢞ଵ ǡ ࢞ଶ ǡ ࢞ଷ ǥ ǥ ǥ࢞ ሿ ࢞ ൌ ሾܽ ǡ ܾ ሿ
(11) (12)
Where, ܽ indicates the start of retrofit section and ܾ is the length of retrofit. The two objectives conflict with each other and retrofit cannot be achieved based on only one of the objectives. Both the objectives are required to be considered simultaneously to arrive at a compromised solution. A simple way to address this problem is to reduce this multi-objective problem to a single objective problem by knowing the relative importance of the two objectives. It can be achieved by introducing the factors which determines the relative importance. Then the problem can be expressed as ݂ሺࢄሻ ൌ ߙଵ ݂ଵ ሺࢄሻ ߙଶ ݂ଶ ሺࢄሻ
(13)
Here, ߙଵ and ߙଶ are the factors which determines the relative importance of the two objectives functions. In practical situation it may not always be feasible to determine the importance of one object over the other. Secondly, using the arbitrary weights as explained may not always lead to the true optimal solution. The other way to address the problem using single objectives optimization is to run the optimization for various possible combination of ߙଵ and ߙଶ . This approach is computationally too expensive. Therefore the problem can be looked upon as the multi-objective optimization problem. BRIEF OVERVIEW OF MULTI-OBJECTIVE OPTIMIZATION Multi-objective optimization is an algorithm to find the multiple Pareto-optimal solutions. It is proved to be an effective tool for decision making under the conflicting objectives. Each objective involved has its own optimal solution. Due to the conflicting nature of the objectives no single solution can be termed as the best solution. The author of [6] has suggested the algorithm called as Non-dominated Sorting Genetic Algorithm (NSGA) to find the solution in Pareto-optimal region. In figure 1, the solution points on curve A,B,C are called as Pareto-optimal solutions or Pareto-optimal front. These solution points are termed as non-dominated solutions while the solution like D is called as a dominated solution. A solution set S1 is termed as dominated by another solution set S2, if S1 is not better than S2 in all objectives but is inferior to S2 in at least one objective. ϲ Cost
Pareto-Optimal Solution
ϰ
A
Ϯ
D B
C
Ϭ Ϭ
ϭ DeflectionϮ
ϯ
Figure 1: Pareto-optimal solutions In this paper, we propose the combined method of cross sectional retrofit and addition of tuned mass dampers to mitigate the vortex induced aerodynamic response. This makes the objectives to be achieved multifold and render the situation to objective conflict. Due to the inherent quality of multi-point, parallel searching capability, the multiobjective Non-dominated Sorting Genetic Algorithm (NSGA) is efficient tool to arrive at Pareto-Optimal solution
BookID 214574_ChapID 4_Proof# 1 - 23/04/2011
39 set under the conflicting objective situation. In our numerical example we have used NSGA-II [6] algorithm for multi-objective optimization. NUMERICAL EXAMPLE AND ILLUSTARION OF PROPSED APPROACH In our hypothetical situation we have considered a bridge of 850 m long. We have modeled this bridge on the basis of Rio-Niterói bridge [3]. For analysis three different modes are considered. Figure 2 shows the three normalized modes for this hypothetical bridge. Ϯ
ŵŽĚĞϭ
ŵŽĚĞϮ
ŵŽĚĞϯ
ϭ Ϭ Ͳϭ
Ϭ
ϮϬϬ
ϰϬϬ
ϲϬϬ
ϴϬϬ
ϭϬϬϬ
ͲϮ Figure 2: Mode shape for bridge The frequencies and the modal mass corresponding to the three modes are shown in Table 1 below. Table 1: Mode data for vortex analysis of bridge Mode No Frequency (Hz) Modal Mass 1 0.315 2865 2 0.624 2553 3 0.768 2189 To analyze the vortex induced response of the bridge deck, the procedure described in [5] is used. The coefficients in equation (7) given by Vickery and Basu assumed to have values are as follows given in Table 2. The assumed aerodynamic coefficients ܪଵ כare as shown in figure 3. Table 2: Coefficients used for analysis of vortex-induced vibrations Coefficient Original Section Retrofitted Section 0.145 0.1 ܵ௧ 0.4 0.1 ௭ 0.15 0.1 ܾ௭ 2 2 ߣ U/(nB) Ϭ Ͳϱ
Ϭ
ϱ
ϭϬ
H1*
ͲϭϬ Ͳϭϱ ͲϮϬ
KƌŝŐŝŶĂůƐĞĐƚŝŽŶ ZĞƚƌŽĨŝƚƚĞĚƐĞĐƚŝŽŶ
Figure 3: Assumed aerodynamic coefficients
ϭϱ
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40
Figure 4 and figure 5 compares the response before and after cross sectional retrofit. The response is found to be improved after cross sectional retrofit. Figure 5 is the response after retrofit of the entire sections. Although the reduction in displacement is satisfactory, this solution has high cost. Optimization analysis in which only part of the sections is retrofitted will be given in the next section. D = 425 m D = 345 m D = 170 m
0.25 0.3
0.2
0.3 SRSS displacement (m)
SRSS displacement (m) SRSS displacement (m)
0.3
D = 425 m D = 345 m D = 170 m
0.25
0.15 0.2
0.1
0.15
0.05 0.1
0.25 0.2 0.15 0.1 0.05
0.05 0
0
0 0
D = 425 m D = 345 m D = 170 m
10 10
20 30 Wind speed (m/s)
20 30 Wind speed (m/s)
40
40
0 0
50
10
50
Figure 4 : Aerodynamic response before retrofit
20 30 Wind speed (m/s)
40
50
Figure 5: Aerodynamic response after retrofit
MULTI-OBJECTIVE OPTIMIZATION In order to identify cost-effective retrofit, multi-objective optimization is performed with two objective functions shown in equations (9) and (10). Since the number of spans is three, n = 3 is used in equation (11) to obtain three groups of retrofitted segments. NSGA-II algorithm is used with 50 individuals 1000 generations. 900 Retrofit Non Retrofit
800 700
Cost
600 C
500
850
400 300 B
200
A B C A
100 0 0
0.05
0.1
0.15 0.2 Displacement (mm)
0.25
0.3
0.35
Figure 6: Optimized solution for cost vs. displacement for three segments of retrofit without TMD The Pareto-optimal solutions obtained from the optimization are shown in Figure 6. The Pareto-optimal solutions quantify the relationship between the cost and the performance, thereby enabling performance-based decision. A user can determine the best solution that meets the performance & cost need of the user. Three points A, B, and C show example solutions for the retrofit. For solution point A, the cost involved is 100, but the displacement is 0.16 mm. For solution point B, the cost is increased to 195 to restrict the displacement to 0.9 mm. For point C,
BookID 214574_ChapID 4_Proof# 1 - 23/04/2011
41 displacement is restricted to 0.045 mm m, with the cost involved is as high as 520. The e user can arrive at a particular solution based on the requirem ment and the need. 2500000
Retrofit Non Retrofit
2000000 D
850
1500000 Cost
A 1000000 B
A
ϯ dD
B
Ϯ dD ϭdD
C
500000
0 0
0.0 05
0.1
0.15
0.2
0.25
0.3
0.35
Displacement(mm) Figure 7: Optimized solution ffor cost vs. displacement for three segments of retro ofit with TMD In addition to the cross-sectional retrofitt, tuned mass dampers (TMD) can be used to reduc ce the displacement. In this paper, we increased effective damp ping to account for the effect of dampers. The dampers are assumed to be at the center of each span. Addition o of TMDs does not change the number of objective e functions, but rather, increases the design space shown in eq quation (11). In addition to the design variables for cross-sectional retrofit, the number of TMDs is also a design vvariable. The optimization problem becomes more complicated due to the c increased design space. Figure 7 depicts the results when the tuned mass dampers are added to the model, along with w the cross-sectional retrofit. All the points along D, A, B, C C, E curve are the optimized solutions under two conflicting objectives. Example solutions A, B, and C show the number of TMDs in addition to the cross-sec ctional retrofit. Point A indicates small displacement with high h cost and the point C indicates low cost with larg ge displacement. The sudden gap between the points D-A and C-E is due to the discontinuity in the design n space. Unlike crosssectional retrofit, the number of TMDs is a discrete variable, resulting in the discontinuity y in the Pareto-optimal solutions. CONCLUSION AND FUTURE WORK In this paper, we have proposed the a application of multi-objective optimization technique to find the solution to suppress the aerodynamic response ccaused due to vortex shedding of bridge deck. Th he approach has been demonstrated using a bridge similar to o the Rio-Niterói Bridge. Vortex shedding analysis was discussed briefly along with concept and application of m ulti-objective optimization technique. Further research is needed to improve e the formulation of the multi-objective optimizatio on. The current design space shown in equation (11) requiress pre-defined number of groups of retrofit. A more general formulation is desirable. Also, the proposed approacch can be extended to include other aerodynamic responses such as buffeting and flutter instability of a bridge e deck.
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42 ACKNOWLEDGMENTS This research was supported by the grant (09CCTI-A052603-02-000000) from the Ministry of Land, Transport and Maritime of Korean government through the Core Research Institute at Seoul National University for Core Engineering Technology Development of Super Long Span Bridge R&D Center. REFERENCES [1] Jones Nicholas P, Raggett Jon D, Ozkan Ender. Prediction of cable-supported bridge response to wind: coupled flutter assessment during retrofit. Journal of Wind Engineering and Industrial Aerodynamics 91(2003) 1445-1464. [2] Fujino Yozo, Yoshida Yositaka. Wind-induced vibration and control of Tran-Tokyo Bay crossing bridge. Journal of Structural Engineering 128(2002) 1012-1025. [3] Battista Ronaldo C, Pfeil Michࣉle S. Reduction of vortex-induced oscillations of Rio-Niterói bridge by dynamic control devices. Journal of Wind Engineering and Industrial Aerodynamics 84(2000) 273-288. [4] Larsen A, Esdahl S., Andersen J.E, Vejrum T. Vortex shedding excitation of Great Belt suspension bridge. Wind Engineering into the 21st century, Larsen, Larose & Livesey (eds) 1999 Balkema, Rotterdam. [5] Einar N. Str݊mmen. Theory of Bridge Aerodynamics. (2006) Springer. [6] Kalyanmoy Deb. Multi-Objective Evolutionary Algorithms: Introducing Bias Among Pareto-Optimal Solutions. Kanpur Genetic Algorithm Laboratory (KanGAL), Department of Mechanical Engineering, Indian Institute of Technology Kanpur, KanGAL report No. 99002.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
9LEUDWLRQDQGDFRXVWLFDQDO\VLVRIEUDNHSDGVIRUTXDOLW\FRQWURO Paolo Castellini, Paolo Chiariotti, Milena Martarelli and Enrico Primo Tomasini Department of Mechanical Engineering, Università Politecnica delle Marche, Via Brecce Bianche, 60131 Ancona, Italy
$%675$&7 Two procedures have been used for the damage detection of brake pads presenting two kind of defects, i.e. cracks and braking material detachments. Both the procedures are based on the monitoring of the pad structural resonances that changes in presence of defects. The first procedures employs a Laser Doppler Vibrometer (LDV) for the measurement of the vibration velocity of the pad forced into vibration by means of an impact hammer. The second one, instead, uses a system based on four microphone probe developed by SenSound and called SenQC system since is made for quality control. Both the techniques allowed to recognized damaged pads from the good ones from a sample of 37 elements. The sensitivity of the procedure to environment temperature has also been studied, since the structural resonance can be influenced by this parameter. The sensitivity analysis to temperature has been performed by using the LDV technique and by repeating the measurement on four different pads (two good and two damaged). The temperature has been varied from 15 to 30°C with a resolution of 2°C in order to monitor the complete range of the environmental temperature that can be reached in working conditions in the production line.
,QWURGXFWLRQ Quality control of brake pads is an important issue, being the pad a key component of the braking system. Up to now, the research has been addressed to the measurement and modelling of the whole system, consisting of disk, pad and calliper, all the parts, together, concurring to the formation of the annoying acoustic phenomenon known as squeal [1]. Most of the work has been focused on the improvement of the design of the brake system for the suppression or, at least, attenuation of the squeal [2]. The pad, therefore, has been studied only as part of the whole system, since its dynamic behaviour, in terms of natural frequencies that could be coupled with the disk ones and damping, influences the formation of the squeal. Larger is the brake damping higher is the capacity of the pad to damp out the vibration and to reduce the squeal. Several studies have been thus addressed to the definition of methods to accurately measure the pad damping in terms of Q factor or damping ratio. In particular in [3] a contactless excitation and measurement system has been designed, based on electromagnetic non contact shaker, to force into vibration the pad, and a microphone, to measure the pad response. Concerning the testing of the brake pad to perform quality assurance, i.e. to test structural integrity loss (like presence of cracks or detachments of the braking material from the backing steel plate), not many works are present in literature. Some of them [4] employ ultrasonic measurement systems to determine the mechanical properties of the pad, i.e. the in-plane, out-of-plane and shear elastic modulus. This paper shows the applicability of non contact measurement systems as Laser Doppler Vibrometer (LDV) and array of microphones for detecting damages in pads, in particular detachments of the adhesive layer connecting the pad with the backing plate. A statistical significant sample of undamaged and damaged pads has been tested with both methods, based on the pad’s dynamic characteristics variation due to the detachment. '\QDPLFFKDUDFWHUL]DWLRQRIEUDNHSDGV The dynamic behaviour of the pad has been first investigated by measuring on a grid of 63 points over the friction material. The measurement has been performed via a Scanning LDV (SLDV), a contactless sensor based on interferometry technique sensing the surface vibration velocity as a Doppler shift of the laser light incident on T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_5, © The Society for Experimental Mechanics, Inc. 2011
43
BookID 214574_ChapID 5_Proof# 1 - 23/04/2011
44 the surface itself. The pad has been forced into vibration by means of an impact hammer whose input force has been measured. The Frequency Response Functions (FRF) between the response velocity measured by the SLDV and input force, given by the hammer, have been acquired. An average FRF obtained by summing the FRF acquired all over the measurement points is given in Figure 1, where both the FRF attained for undamaged and damaged pad are reported. 10
-1
FRF Magnitude (m/s/N)
damaged undamaged
10
10
10
-2
-3
-4
0
1000
2000
3000
4000
5000 Frequency (Hz)
6000
7000
8000
9000
10000
Figure 1 Averaged FRFs for undamaged and damaged pad It is clear that the damage induces an evident shift on the first resonant frequency, indicating a stiffness reduction produced by the detachment. In Figure 2 some of the operational deflection shapes measured for the undamaged and damaged pads are given. With the exception of the first mode, the amplitude of the mode shapes decreases when the pad is damaged since the damping increases. Moreover, the damage provokes a phase scatter of the mode shape, thus it increases its complexity.
Undamaged 2769 Hz
Damaged 2150 Hz
Amplitude -15÷21 dB (dB ref 1.95 10 -3 m/s/N)
Amplitude -7÷22 dB (dB ref 1.95 10-3 m/s/N)
Phase -180÷180°
Phase -180÷180°
BookID 214574_ChapID 5_Proof# 1 - 23/04/2011
45 Undamaged 3538 Hz
Damaged 2450 Hz
Amplitude -10÷24 dB (dB ref 1.95 10 -3 m/s/N)
Amplitude -7÷21 dB (dB ref 1.95 10-3 m/s/N)
Phase -180÷180° Undamaged 5338 Hz
Phase -180÷180° Damaged 3544 Hz
Amplitude -12÷19 dB (dB ref 1.95 10 -3 m/s/N)
Amplitude -19÷15 dB (dB ref 1.95 10 -3 m/s/N)
Phase -180÷180°
Phase -180÷180°
BookID 214574_ChapID 5_Proof# 1 - 23/04/2011
46 Undamaged 5894 Hz
Damaged 5344 Hz
Amplitude -18÷12 dB (dB ref 1.95 10 -3 m/s/N)
Amplitude -14÷11 dB (dB ref 1.95 10 -3 m/s/N)
Phase -180÷180° Undamaged 7931 Hz
Phase -180÷180° Damaged 6581 Hz
Amplitude -28÷12 dB (dB ref 1.95 10 -3 m/s/N)
Amplitude -24÷7 dB (dB ref 1.95 10-3 m/s/N)
Phase -180÷180° Phase -180÷180° Figure 2 Operational deflection shapes for undamaged and damaged pad
4XDOLW\DVVHVVPHQWEDVHGRQG\QDPLFPHDVXUHPHQWV Being clear that the detachment influences mostly the first mode of the pad, it has been concluded that the parameters, useful for the discrimination between damaged and undamaged pads, are the first resonance
BookID 214574_ChapID 5_Proof# 1 - 23/04/2011
47 frequency and the related damping. For the quality assessment, thus, the pad has been supported along the two nodal lines of its first natural mode in order to not influence its dynamics, see Figure 3. The measurement set up remained as the one described in Section 3, with the only exception that a single point measurement has been performed.
hammer
LDV spot
support lines Figure 3 Experimental set-up for the pads quality control The time histories of the input force and the output vibration velocity have been acquired. The calculated FRFs for two damaged pads and for two undamaged ones are illustrated in Figure 4. The frequency shift due to the detachment is evident, see plots black and grey. 10
FRF magnitude (m/s/N)
10
10
10
10
10
0
Pad n.1 (undamaged) Pad n.2 (undamaged) Pad n.1 (damaged) Pad n.2 (damaged)
-1
-2
-3
-4
-5
0
1000
2000
3000
4000
5000 Frequency (Hz)
6000
7000
8000
9000
10000
Figure 4 FRFs measured on two undamaged pads (black) and two damaged ones (gray) Having measured 18 good pads and 19 damaged, a statistical analysis has been performed in terms of both first natural frequency and related damping. The damping has been measured from the response time histories as the decay ratio (σ) that, for light damped structures, is related to the damping ratio ξ by the natural frequency ω n ( . From the bar plots shown in Figure 5 an 6, it can be noticed that the first natural frequency is always below 2500 Hz when the pad is damaged and there is a large spread between different pads. In fact, the standard deviation on the damaged pads is 177 Hz while in the good ones is 31 Hz. It is not the same for the damping of the first natural mode which cannot be used to identify the damage. The analysis has been performed also at different temperatures of the pad between 15 and 30°C. Four pads have been measured: N.7 e 17 good and N.11 e 17 damaged. The variation of the first natural frequency with the temperature has been studied. In the bar plots in Figure 7 and 8 the natural frequency is plotted against the pad temperature.
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48
3500
0.07 undamaged damaged
3000
0.06 0.05 Damping ratio
Frequency (Hz)
2500
2000 1500
0.04 0.03
1000
0.02
500
0.01
0
0
2
4
6
8
10 Pad n.
12
14
16
18
0 0
20
Figure 5 First natural frequency measured for 18 good pads and 19 damaged
2
3000
3000
2500
2500
2000
2000
1500
1000
500
500
1
2
3
4
5 6 7 Temperature (°C)
8
Figure 7 Undamaged pad (N. 7)
9
10
6
8
10 Pad n.
12
14
16
18
20
1500
1000
0
4
Figure 6 Damping ratio related to the first natural mode measured for 18 good pads and 19 damaged
Frequency (Hz)
Frequency (Hz)
undamaged damaged
0
1
2
3
4
5 6 7 Temperature (°C)
8
9
10
Figure 8 Damaged pad (N. 17)
The variation of the temperature within 15-30°C cannot be considered an important interfering input, since the natural frequency remains always above 2500 Hz for the good pad and below 2500 Hz for the damaged one. A marked separation is kept between good and damaged pads also when the temperature oscillates between 15 and 30°C. This is demonstrated by the first natural frequency distribution measured with variable temperature, see Figure 10, even if the probability distributions of both the undamaged and damaged pad are enlarged, see Figure 9.
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49
4 3.5
N. of occurences
3
undamaged histogram damaged histogram undamaged distribution damaged distribution
2.5 2 1.5 1 0.5 0 1600
1800
2000
2200
2400 Frequency (Hz)
2600
2800
3000
3200
Figure 9First natural frequencies histogram and probability distribution measured at constant temperature (white: damaged pads, red: good pads). 6
N. of occurrences
5
undamaged histogram damaged hisogram undamaged distribution damaged distribution
4 3 2 1 0 1600
1800
2000
2200 2400 Frequency (Hz)
2600
2800
3000
Figure 10First natural frequencies histogram and probability distribution measured at temperature varying between 15 and 30°C (white: damaged pads, red: good pads). The distribution shapes are larger but the separation is still evident. 4XDOLW\DVVHVVPHQWEDVHGRQDFRXVWLFPHDVXUHPHQWV The analysis has been also performed with the SenQC system, made of a four microphone probe to be placed at a short distance from the pad. This system enables the user to clean acquired signals from background noise: this is obviously very usefull when tests are performed in the production line, where high level of background noise can lead to misleading results, causing bad products to pass and good ones to be rejected. As a consequence of the denoising procedeure no acoustic enclosure is needed. SenQC is also able to show SPL of direct sound from the target and represent an all-rounded non-destructive/non-contact defect detection technique. The denoising algorithm needs a train phase with respect to the target and the environment, while the defect identification requires the definition of pass/fail metrics on denoised signals.
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50
Figure 11 Set-up for SenQC test (left) and close-up on probe measuring distance (right) The product has been tested in real operating environment (Figure 11), i.e. with background noise reproducing the floor noise present in production line. The effect of the decreasing first natural frequency is appreciated also by using the acoustic measurement. Therefore the same kind of analysis can be used with the SenQC system.
Damaged
Good
Figure 12 Frequency response between the output, i.e. the acoustic response of the pad (combination of the acoustic pressures measured by the microphones) and the input force. The learning phase of the system has been performed by measuring 3 good pads and 3 damaged ones. By fixing the discrimination region (the so-called metric) in the range between 2500 and 3000 Hz the good pads can be recognized from the damaged. In this range, in fact, the level of the acoustic emission of the good pad is higher than the damaged one. On this basis the green and red regions have been set. If the measured pad will lie in the green region it will pass, if it will lie in the red it will fail.
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51
Figure 13 Pass/Fail metric definition The results are given below (Table 1). All the pads have been correctly recognized. An example of SenQC way to show test results is reported in Figure 14. Brake Pads Good Damaged
% Recognized 100 100
Table 1 Pass/Fail test results with SenQC system
Figure 14 Results table example
BookID 214574_ChapID 5_Proof# 1 - 23/04/2011
52 &RQFOXVLRQV Two measurement techniques for quality control of brake pads have been tested: the first one based on vibration measurement by a non contact LDV and the second one using a 4 microphones probe and a dedicated signal processing developed by SenSound Acoustic Imaging. Both the systems are non contact and require excitation of the pad by means of an impact hammer. The damaged pads are completely recognized from the good ones by both the techniques. The influence on the quality control system, in particular the vibrational one, of the pad and environment temperature has been investigated, it evidencing that the feature characterizing the damage (i.e. the pad first natural frequency) has got a probability distribution distinctly separated from the same feature distribution relative to good pads. 5HIHUHQFHV [1] R. A. Ibrahim, “Friction-induced vibration, chatter, squeal and chaos. Part II: Dynamics and modelling“, ASME Applied Mechanics Reviews, 47:227-259, 1994. [2] U. von Wagner, D. Hochlenert, T. Jearsiripongkul, P. Hagedorn, “Active control of brake squeal via ‘smart pads‘“, SAE Technical Paper Series, 2004-01-2773. [3] R.P. Uhlig, ”Brake pad assembly damping and frequency measurement methodology” United States Patent No. US 6,382,027 B1, 7 May 2002 [4] IMS inc. Internal Report, ”Quality Assurance Measurements on 4 As-manufatured Brake Pads”
BookID 214574_ChapID 6_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Vibration Monitoring of a Small Span Composite Bridge 2 ¨ D. Siegert1 , M. Dohler , O. Ben Mekki3 , L. Mevel2 , M. Goursat4 , F. Toutlemonde1 1 2
Universite´ Paris-Est, LCPC, 58 boulevard Lefebvre, 75732 Paris, France
INRIA, Centre Rennes - Bretagne Atlantique, Campus de Beaulieu, 35042 Rennes, France 3 4
´ ENIT, Laboratoire de Genie Civil, Tunisia
INRIA, Domaine de Voluceau-Rocquencourt, 78153 Le Chesnay, France
Abstract Small localized damages are hardly detected by global monitoring methods. The effectiveness of vibration based detection depends on the accuracy of the modal parameter estimates and is limited by the low sensitivity of the modal parameters to a local stiffness reduction. A local reduction of stiffness related to frequency changes less than 1 % was successfully detected on a 10 meter span composite UHPFRC - FRP reinforced timber beam bridge loaded in laboratory conditions up to the serviceability limit state (SLS). Such a small decrease in the stiffness was not detected by the monitoring of the static load-deflection measurements but was confirmed by non-linear local strain measurements. Statistical subspace-based damage detection successfully detected the change of the modal parameters of the investigated structure. Further analysis with a finite element model was conducted for assessing the consistency of the expected location and extent of the damaged elements. 1 INTRODUCTION In many cases, the effect of damage on the dynamic response of structures is hardly detectable [1] [2]. The performance of the vibration-based damage detection method depends on the accuracy of the experimental modal parameter estimates and is limited by the relatively low sensitivity of the modal parameters to local stiffness reductions. This paper focuses on the application of two global detection techniques: the monitoring of the first resonant frequencies and a damage detection test derived from the null-space of the Hankel matrix whose components are the correlation estimates of the measured output responses. Among the different vibration-based techniques, the monitoring of the resonant frequencies remains still promising for automatic structural health monitoring applications. Recent advances related to effective implementations of the covariance driven subspace identification method were proposed for meeting the requirements of automatic modal identification procedure [3]. The damage detection test proposed by some of the authors offers an alternative which is derived from the subspace formulation of time invariant linear system models [4]. An extended non-parametric version of this damage detection algorithm which rejects temperature effects was successfully experimented on several test cases [5]. In this paper, another example of damage detection on a real structure tested in laboratory conditions is presented. The test of an innovative composite bridge deck concept made of an ultra-high performance fiber-reinforced concrete (UHPFRC) slab and fiber-reinforced polymer (FRP) at the bottom of longitudinal timber beams, was carried out at the LCPC structural laboratory within the framework of the NR2C project. The two vibration based monitoring techniques were applied for detecting structural changes in the bridge mockup loaded up to the serviceability limit state (SLS). The modal tests were complementary to an extensive instrumentation of the structure involving strain and displacement measurements. The design, test procedure and results are described in details in the final report of the project [6]. 2 DESCRIPTION OF THE TEST STRUCTURE The structural design and tests were part of a program on short span bridges easy to assemble on site, and likely to be used alone for small spans (typically 10m spans over a two-lane-road) or themselves supported by structural elements, when used transversally, for most important spans. The element of the bridge deck considered here is formed by 4 wooden beams, a top slab made of ultra-high performance fiber-reinforced concrete (UHPFRC) and
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_6, © The Society for Experimental Mechanics, Inc. 2011
53
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54 fiber-reinforced polymer (FRP) 3mm thick at the bottom of the timber beams. The transverse cross-section of the deck is shown in Figure 1 where the main dimensions of the structural elements are indicated. The beam extremities are supported by neoprene bearing pads.
Figure 1: Transverse cross-section. This concept derives from preliminary design considerations developed in [7]. Important preliminary researches were carried out for assessing the respective advantages and drawbacks of different types of connections, especially between wooden members and concrete, and their influence on the design issues of composite structures [8] [9]. The 10 m-span, 2.5 m-wide structural element, constructed and tested by the LCPC, has been studied and designed by JMI, Greisch, LCPC and LCPC-LAMI.
Figure 2: SLS loading configuration. The design hypotheses mainly relied on existing codes [10] [11] complemented by recommendations for UHPFRC and FRP reinforcement [12] [13]. The only design situation considered in this paper concerns the local and transverse bending. This loading case appears as critical for the UHPFRC slab. According to EN 1991-2 [10] the traffic load to be considered for serviceability limit state (SLS) verification corresponds to 150 KN on each 0.4 m x 0.4 m spot. The SLS loading configuration is shown in Figure 2. The loads are applied 0.75 m apart from the mid-span section, and their transverse distance is 1.5 m, which corresponds to the most severe situation relative to the timber beams position. 3 VIBRATION-BASED DAMAGE MONITORING 3.1 Modal test The deck of the tested bridge was excited by an impact hammer. The vertical vibrating response of the structure was measured with eight uniaxial inductive accelerometers fixed on the top slab as shown in Figure 3. The transient responses were filtered by a low pass filter with a cut-off frequency of 75 Hz and recorded with 3000 samples at a sampling frequency of 600 Hz. Since the vibrating structure was strongly damped, the duration of the free vibrations
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55 produced by the transient excitation lasted less than 2s. Ten transient responses were recorded for each subsequent extraction of the modal characteristics from the correlations of the measured responses.
Figure 3: Accelerometer setup.
3.2 Subspace identification The considered modal identifications algorithm use the covariance driven subspace methods which are described in full detail in [14] [15]. Here, we just present some of the definitions and basic computational steps, useful to set the parameters of the method for processing the measurement data of free vibrations to identify the linear time invariant system. The linear state-space difference equations of the corresponding model are given by Xk+1
=
F Xk + Vk+1
Yk
=
HXk + Uk
where Xk denotes the state vector of the system sampled with a frequency Fs = τ1 and Yk the vector of the measured output responses. Vk and Uk are respectively the white noise excitation process and the white noise in the measurement. The identification procedure of a linear time invariant system is based on the correlation estimates Ri which are defined as N 1 T Ri = Yk Yk−i (N − i) k=i+1
T
where the upper script is the transpose operator, N the number of recorded samples and i the time lag of the correlation. The Hankel matrix of size (p + 1)r × (p + 1)r is then written as ⎡ ⎤ .. R R . R 0 1 p ⎢ ⎥ ⎢ ⎥ .. ⎢ ⎥ R2 . Rp+1 ⎥ ⎢ R1 Hp+1,p+1 = ⎢ ⎥ .. .. .. ⎢ .. ⎥ ⎢ . ⎥ . . . ⎣ ⎦ .. Rp Rp+1 . R2p where r is the number of sensors or measurement channels selected for the identification data processing. The identification of the state transition matrix F of the state-space model of the system relies on the factorization of the Hankel matrix which is derived from the state-space equations Hp+1,p+1 = Op+1 (H, F )Cp+1 (F, G) where O is the observability matrix and C is the controllability matrix defined as ⎛ ⎞ H ⎜ HF ⎟ ⎜ ⎟ Op+1 (H, F ) = ⎜ . ⎟ , Cp+1 (F, G) = G F G · · · ⎝ .. ⎠ HF p
Fp
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56 in which G = E(Xk YkT ) is the cross correlation between the state vector and the vector of the measured output responses. Only the observability matrix needs to be identified using the singular value decomposition of the Hankel matrix. Then the observation matrix H is obtained from the first block row of Op+1 and the state transition matrix F is obtained from the least squares solution of the over determinated linear system Op+1 (r + 1 : r(p + 1), :) = Op F where the symbol : denotes a vector of integer subscripts. In order to improve the modal identification in the stabilization diagrams for automatic process, the algorithms were modified leading to clearer and more stable stabilization diagrams [3]. This is also called “crystal clear”. 3.3 Damage detection Two different versions of the subspace-based damage detection test are described in this section. 3.3.1 Parametric approach The eigenanalysis of the state transition matrix leads to the definition of the reference vector θ0 whose components correspond to the eigenvalues and the observed components of the eigenvectors stacked in a single column vector. A change in this signature indicates damage of the monitored structure. We want to compare this modal signature p+1,q computed on new data. with a Hankel matrix H We assume that the eigenvectors of F are chosen as the basis for the state space, in which the observability matrix Op+1 writes: ⎛ ⎞ Φ ⎜ ΦΔ ⎟ ⎜ ⎟ Op+1 (θ0 ) = ⎜ . ⎟ ⎝ .. ⎠ ΦΔp
where the diagonal matrix Δ = diag(λ) contains the eigenvalues and Φ the eigenvectors. Thus, Op+1 (θ0 ) can be filled with the information of the modal signature θ0 . We now compute the left null space S(θ0 ) of this matrix which is also the left null space of the Hankel matrix of the reference state. Then the characteristic property of a system in the reference state writes: S T (θ0 ) Hp+1,q = 0. Now, the damage detection problem is to decide whether a new data sample (Yk )k=1,...,n from the (possibly damaged) system is still well described by the reference parameter θ0 (identified on data recorded on the undam p+1,p+1 on the new data. For damage aged reference system) or not. For this we compute the Hankel matrix H detection, the following residual is introduced in [17]: def √ p+1,p+1 ζn (θ0 ) = n vec ST (θ0 ) H Under convenient assumptions, this residual is asymptotically Gaussian, and manifests itself to the damage by a change in its own mean value. Hence the χ2 -test statistics writes as −1 def −1 (θ0 )J(θ0 ) JT (θ0 )Σ −1 (θ0 )J(θ0 ) −1 (θ0 )ζn (θ0 ) χ2n (θ0 ) = ζnT (θ0 )Σ JT (θ0 )Σ (1) 0 ) are estimates of the sensitivity and covariance of ζn (θ0 ), see [17], [18] for details. where J(θ0 ) and Σ(θ 3.3.2 Non-parametric approach For the non-parametric damage detection we do not use the modal signature θ0 . The left kernel matrix S is directly computed as the left null space of the Hankel matrix in the reference state and it holds S T Hp+1,p+1 = 0. p+1,p+1 to the reference state S is Then the residual for comparing a new Hankel matrix H √ p+1,p+1 ζn = nvec ST H The value of the corresponding χ2 test writes as ˆ −1 ζn χ2n = ζnT Σ ˆ = E(ζn ζnT ) is the covariance matrix of the residual. The monitoring of the system consists in calculating where Σ the value of the test with the Hankel matrices estimated from the newly recorded output data. A significant increase
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57 in the χ2 value indicates that the system is no more in the reference state. Since the test may be sensitive to modal changes induced by environmental temperature variations, a robust version of the null-space based method has been introduced in [5] for rejecting the temperature effects. It consists in determining the left null-space of the average of Hankel matrices over various temperature conditions of the undamaged structure. This version is the so-called non-parametric test since it does not rely on the modal identification of a reference signal recorded in the undamaged state of the monitored structure. 4 RESULTS Now results are presented for both the subspace identification and detection approaches. 4.1 Modal identification The signals were processed with the COSMAD toolbox developed under the Scilab software to extract the modal parameters of the vibrating beam [16]. Figures 4 and 5 show the frequency stabilization diagrams where the frequencies in Hertz are plotted against the model order of the subspace method. The stabilization diagram in Figure 5 is the result of the “crystal clear” implementation of the covariance-driven Stochastic Subspace Identification. Ten successive transient responses recorded were concatenated to improve the accuracy of the correlation estimates in the Hankel matrix. The mode shapes are plotted in Figure 6.
Figure 4: Frequency stabilization diagram without “crystal clear”.
Figure 5: Frequency stabilization diagram with “crystal clear”.
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1.0 1.0
0.8
0.8
0.6 0.4 Z
Z
0.6 0.4
0.2 0.0 −0.2
0.2
−0.4
0.0
−0.6
0
−0.8 1
−1.0 0
2
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Y
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4
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6
5
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6 X7
8 9 10
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8
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9 10
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Z
0.40 −0.20 −0.80 0 1 2 3 4 5 6 7 Y
8 9 10
2.5
2.0
1.5
1.0
0.5
0.0
X
Figure 6: Mode shapes of the longitudinal flexural mode (13.66 Hz), torsion mode (16.63 Hz), transverse flexural mode (37.84 Hz). ".1-*56%&
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Figure 7: Mode shapes of the FE model
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59 A finite element model of the structure has been developed, using the free software CAST3M. This model consists in a shell-beam model i.e. modeling the slab with shell elements and the girders with beam elements and rigid offsets for the plate-beam connection. The two first calculated bending and torsion modes match well the experimental modes. Without updating the design values of the model parameters, the calculated frequencies are only 2 % higher than the experimental ones. The second bending mode of the model was not identified in the stabilization diagram. The third measured mode of vibration corresponds to the mode 6 of the model. Further analysis is needed for accomodating the experimental modal analysis results to the updated model. The three first calculated mode shapes are displayed in Figure 7. 4.2
Damage detection Repeated estimations of the modal parameters were performed before and after the loading test up to the SLS load value (520 kN). Experimental modal tests were not carried out during the application of the loads because the system was too sensitive to the details of the loading conditions which were not held constant during the test. The result of the frequency monitoring of the first bending mode is shown in Figure 8. The graph indicates a drop in the resonant frequency of 1 %. The same relative variation of the frequency was observed for the torsion mode and a shift of 1.3 % was measured for the third mode. Since the coefficient of variation of the frequencies was below 0.15 %, the measured frequency shifts are significant and indicate a modification of
13,16 13,15 13,14 13,13 13,12 13,11
frequency (Hz)
13,1 13,09 13,08 13,07 13,06 13,05 13,04 13,03 13,02 13,01 13 12,99 0
1
2
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6
7
8
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Series
Figure 8: Frequency monitoring of the first bending mode.
6,00
18 ref. 3
ref. 3-6-13-19 ref. 3-6-13-15-19
16 5,00 14
12
4,00
Chi2
Chi2
10 3,00
8
6
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0
0,00 0
1
2
3
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Figure 9: Parametric test series for the reference parameter identified in series 3.
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 Series
Figure 10: Non-parametric test series for reference data sets over different mass distributions prior to the modification of the stiffness.
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60 the structure. This observation is in agreement with the local measurements of the strains of the slab at the load locations since the elastic limit of concrete in tensile bending has been reached. This modification was not detected in the load-displacement records where no significant decrease of stiffness was noticed. The damage was simulated as a reduction in the stiffness of the shell elements located at the four load pads shown in Figure 7. The measured drop in the frequency of the first bending mode is induced by a reduction of 40 % of the Young modulus of the UHPFR concrete. However, the frequency of the first torsion mode is almost not sensitive to this damage case. A deeper analysis taking into account the strain history recorded during the load test is required before selecting relevant candidate damage configurations. Figure 9 shows the parametric χ2 -test values that were computed using the modal parameters of the reference state identified prior to the loading of the structure. The modification induced by the loads is successfully detected in the graph of the series. Subsequent local modifications were made by adding or removing concentrated masses in the range of 2 % of the overall mass of the bridge. Using the average of the Hankel matrices over the different non damaged configurations as a reference restricts us to process the test in its non-parametric formulation only. This test was then applied on time series recorded on the former modified configurations of the structure. Figure 10 shows that the non-parametric test does not detect the modifications of the mass distribution as it was designed to, but still remains sensitive to the stiffness reduction induced by the SLS loading test. Furthermore, a data set, collected when one of the eight neoprene bearing pads was heated, was included in the average reference state for smoothing the test series. This gave us an even more robust damage test. 5 CONCLUSIONS The frequency monitoring of the first vibration modes has exhibited a small decrease of stiffness because the variability of the modal parameter estimates was low enough. Since in our experimental conditions, the temperature was fairly constant and the vibration data was collected in free vibration conditions, the first resonant frequencies were determined from repeated transient responses with an uncertainty below than 0.15 %. The first vibration modes were quite precisely described by a finite element model using the design values of material parameters. However further investigations are needed to validate the model predictions over the frequency range of the experimental modal analysis. In addition the extent of the stiffness reduction remains to be estimated. The effectiveness of the non-parametric test for discriminating different structural modifications has been shown in the test case presented. Further applications for on-line monitoring are now considered for assessing the numerical efficiency and robustness of the method. ACKNOWLEDGEMENTS The present research was made possible by the experimental program performed in the frame of NR2C R&D European Project funded by the European Union within 6th FRDP. The authors are grateful to the LCPC technicians of the Structure Laboratory for their valuable contribution in the realization of the experimental project. They gratefully acknowledge the ENPC-LAMI partners for their contribution and advice for the design of the test structure. References [1] F. Galanti, A. van Doormaal Feasibility of dynamic test methods in classification of damaged bridges, paper 313 in proceedings of the 25th International Modal Analysis Conference (IMAC), Orlando Fl., February 2007. [2] D. Siegert, S. Staquet, G. Cumunel, M. Goursat, F. Toutlemonde, Vibration based structural health monitoring of prebended steel-vhpc beams, paper 302 in Proceedings of the SEM annual conference, St Louis Mo, June 2006. [3] M. Goursat, L. Mevel, Algorithms for covariance subspace identification: a choice of effective implementations, Proceedings of IMAC-XXVII, February 9-12, 2009 Orlando, Florida USA. [4] L. Mevel, L. Hermans, H. van der Auweraer, Application of a subspace-based fault detection method to industrial structures, Mechanical Systems and Signal Processing 13, 6, November 1999, pp.823-838. [5] E. Balmes, M. Basseville, F. Bourquin, L. Mevel, H. Nasser, F. Treyssede, Merging sensor data from multiple temperature scenarios for vibration-based monitoring of civil engineering structures, Structural Health Monitoring, Vol.7, 2, June 2008, pp. 129-142. [6] F. Toutlemonde, O. Ben Mekki, J.F. Caron, F. Gens, Detailed design and experimental validation of innovative aspects of a 10 m-span composite UHPFRC – carbon fibres – timber bridge, NR2C final report, Proceedings of the final workshop organised by FEHRL, Brussels, November 2007. ´ ´ ´ Projet de fin d’etude, ´ [7] R. Delfino, Conception d’un pont en materiaux composite de faible portee, Ecole Nationale ´ des Ponts et Chaussees, juin 2006.
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61 ´ ´ ` ´ des [8] H. Son Phan and R. Le Roy, ponts mixtes Bois-Betons– Etude des systemes de connexion, Journes ´ ´ 5-6 decembre ´ Sciences de l’Ingenieur, Marne La Vallee, 2006. [9] H. Son Phan et et al., Nouveaux tabliers composites Bois-BHP-Carbone pour les ouvrages d’art, GC’2007, AFGC, Paris, 21-22 mars 2007. [10] NF EN 1991-2, Eurocode 1 – Partie 2, Actions sur les ponts, dues au trafic, 2004. [11] EN 1995-1-1, Eurocode 5, Conception et calcul des structures en bois, 2005. ´ ´ a` ultra-hautes performances, recommandations provisoires, documents scientifiques et [12] AFGC, Beton fibres ´ techniques, Association Franc¸aise de genie civil, 2002. ´ ´ ´ [13] AFGC, Reparation et renforcement des structures en beton au moyen des materiaux composites, recomman´ dations provisoires, documents scientifiques et techniques, Association Franc¸aise de genie civil, 2003. [14] M. Basseville, A. Benveniste, M. Goursat, L. Hermans, L. Mevel, H. van der Auweraer output-only subspacebased structural identification: from theory to industrial testing practice, ASME Journal of Dynamic Systems, Measurement, and Control, Special issue on Identification of Mechanical Systems, Vol. 123,no 4, Dec. 2001, pp. 668-676. [15] B. Peeters, G. De Roeck Reference-based stochastic subspace identificationfor output-only modal analysis, Mechanical Systems and Signal Processing 13, 6 , 1999, pp. 655-878. [16] L. Mevel, M. Goursat, M. Basseville, A. Benveniste Subspace-based modal identification and monitoring of large structures, a Scilab toolbox, in Proceedings of the 13th Symposium on System Identification, IFAC / IFORS, Rotterdam, The Netherlands, August 2003, pp. 1405-1410. [17] M. Basseville, M. Abdelghani, and A. Benveniste, Subspace-based fault detection algorithms for vibration monitoring. Automatica, 36(1):101-109, Janvier 2000. [18] M. Basseville, L. Mevel, and M. Goursat, Statistical model-based damage detection and localization : subspacebased residuals and damage-to-noise sensitivity ratios, Journal of Sound and Vibration, 275(3), pp. 769–794, 2004.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
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:'=KX Professor and Corresponding Author Email:
[email protected]; Tel: 410-455-3394
Department of Mechanical Engineering University of Maryland, Baltimore County 1000 Hilltop Circle Baltimore, MD 21250
$%675$&7 As a global structural damage detection method, a vibration-based method that uses changes in natural frequencies of a structure to detect the locations and extent of damage can in principle detect various types of structural damage with minimum test data, including damage that occurs at the joints and boundaries of the structure. However, it is a challenging task to practically detect damage in engineering structures such as space frames using the vibration-based method. The major challenges can arise from the forward problem, i.e., the creation of an accurate physics-based model for both the undamaged and damaged states of a structure, and the inverse problem, i.e., the development of a robust iterative algorithm to search for the locations and extent of damage. With the recent development of the modeling techniques for fillets and bolted joints by the authors, space frames with filleted thin-walled beams and bolted joints can be accurately modeled with a reasonable model size for both the undamaged and damaged states. The damage detection algorithm developed earlier for under-determined systems is improved here by introducing a logistic function transformation to convert the constrained optimization problem into an unconstrained one and a trust-region search method to solve the
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_7, © The Society for Experimental Mechanics, Inc. 2011
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nonlinear least-square problem. The new iterative algorithm is ensured to converge to a stationary point of the objective function, the convergence speed is significantly increased, and the robustness of the algorithm is improved. A real lightning mast at an electric substation and a scale mast with machined sections in the laboratory can be accurately modeled using mainly shell and beam elements, respectively, and numerical and experimental damage detection is performed using the new damage detection algorithm. The modeling techniques for fillets and bolted joints and the new damage detection algorithm are used to numerically and experimentally detect loosening of bolted connections in a component of a space frame, which consists of two Lshaped beams with a bolted joint. The exact locations and extent of damage can be detected in the numerical simulation for both types of structures. With the errors between the measured and calculated natural frequencies within 2%, the locations and extent of both types of damage can also be successfully detected in the experiment
,1752'8&7,21 It is known that the presence of damage in a structure, which usually manifests itself as structural stiffness reduction, will change the natural frequencies of the structure. Different locations and extent of damage will lead to different patterns in the changes of the natural frequencies. Hence it is possible in principle to identify the locations and extent of damage in a structure using its natural frequency changes, which is the theoretical basis of the vibration-based damage detection method discussed herein. Since natural frequencies are global vibration characteristics of a structure, the vibration-based method is a global damage detection method. It can be used to detect a broad range of damage where stiffness reduction occurs, including damage at the joints and boundaries of a structure and material degradation due to fatigue, which are difficult to detect using the conventional nondestructive testing methods. An iterative algorithm that uses changes in the natural frequencies and mode shapes of a structure to determine the stiffness parameters of the structure was developed earlier [1]. Since it is much easier to measure the natural frequencies of a structure than the mode shapes, using only the changes in the first several natural frequencies to detect the locations and extent of damage was numerically and experimentally investigated on simple structures such as cantilever beams [2]. The algorithm in Ref. [1] along with the Moore-Penrose inverse was used for the under-determined systems in Ref. [2] where the number of the stiffness parameters to be determined exceeds that of the natural frequencies used. The method in Ref. [2] is essentially the Gauss-Newton line search method [3] and can converge slowly when the system equations are severely under-determined. To apply the vibrationbased method to more complex engineering structures such as space fames, two major challenges need to be addressed. Since the vibration-based method is a model-based method, the first challenge arises from the forward problem, i.e., the creation of an accurate physics-based model for both the undamaged and damaged states of a structure. While accurately modeling simple structures such as beams is not an issue, modeling assembled structures with filleted thin-walled beams and bolted joints can be difficult. The model used for damage detection should be physics-based so that both the undamaged and damaged states of a structure can be represented by the same model. Since the smallest detectable damage in a structure is determined by the accuracy of the model, the model should have sufficient accuracy to ensure that the critical damage in the structure can be detected. In addition, the model should have a reasonable size so that it can be easily processed by the damage detection algorithm without worrying about the computational cost. Due to the presence of fillets in thin-walled beams and bolted joints, developing accurate physics-based models of space frames, while maintaining the model sizes at a relatively small level, is difficult. Considerable physical insight is required to develop the models to be used for damage detection. With the recent developments of the modeling techniques for fillets [4] and bolted joints [5] by the authors, the challenge associated with the forward problem can be successfully resolved. The second challenge arises from the inverse problem: with the known changes of the natural frequencies of a structure, the locations and extent of damage can be identified by the damage detection algorithm. The inverse problem can be considered as a nonlinear least-square problem [6]. The stiffnesses of a structure at potential damage locations can be represented by a set of non-dimensional stiffness parameters; the natural frequencies of the structure are a nonlinear function of these parameters. The nonlinear least-square problem is to minimize the least-square error between the calculated and measured natural frequencies of the damaged structure in the
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stiffness parameter domain. The nonlinear least-square problem for damage detection is usually severely underdetermined and has an infinite number of solutions at each iteration. While a minimum norm solution can be obtained at each iteration using the Gauss-Newton line search strategy [3,6], the method does not ensure that the iterations will converge to a stationary point of the objective function of an under-determined system [6,7]. In addition, the Gauss-Newton method can amplify the effects of modeling error and measurement noise in experimental damage detection [3]. A trust-region search strategy, called the Levenberg–Marquardt (LM) method [6] is introduced here, which ensures that the iterations will converge to a stationary point of the objective function of an underdetermined system and improves the robustness of the iterative algorithm in experimental damage detection. A new logistic function transformation is employed to convert the constrained optimization problem into an unconstrained one, since the Gauss-Newton method and the LM method are formulated here for unconstrained systems, and to increase the convergence speed of the iterative algorithm. With the challenges associated with the forward and inverse problems being resolved, the vibration-based method is used to detect damage in a scale lightning mast with machined sections and loosening of bolted connections in a component of a space frame, which consists of two L-shaped beams with a bolted joint. The exact locations and extent of damage can be detected in the numerical simulation when there is no modeling error and measurement noise. Both types of damage can also be successfully detected in the experiment. 7+()25:$5'352%/(0 Modeling of Lightning Masts. Detecting damage in lightning masts at an early stage can avoid their premature failure. Shown in Fig. 1(a) is a typical lightning mast at an electric substation, which consists of two steel pipes connected by bolted flanges, with a spike bolted to the upper pipe at the top of the mast. The length, radius, and thickness of the lower pipe are 6.845 m, 0.1055 m, and 0.0081 m, respectively, and those of the upper pipe are 6.8358 m, 0.0806 m, and 0.0071 m, respectively. The length of the spike is 2.1336 m, and its radius is 0.0127 m. The radius and thickness of the flanges are 0.1762 m and 0.0222 m, respectively. The elastic modulus of the lightning mast is 200 GPa , the Poisson's ratio is 0.29, and the mass density is 7800 kg/m3 . The natural frequencies of the lightning mast were measured at the ground level. The lower pipe of the mast was excited using a random impact hammer test [8] to reduce the effect of noise excitation due to wind, and the dynamic response of the mast was measured by two accelerometers placed in two perpendicular, radial directions of the lower pipe. A spectrum analyzer was used to obtain the measured frequency response functions of the mast and its measured natural frequencies. While the geometry of a lightning mast is relatively simple, modeling it using a relatively simple model cannot provide sufficient accuracy for damage detection. The lightning mast in Fig. 1(a) was first modeled using beam elements using SDTools [9] (Fig. 1 (b)). Different cross-sectional properties associated with the lower pipe, the flanges, the upper pipe, and the spike of the mast were assigned to the corresponding beam elements in the finite element (FE) model. The maximum error between the first eight measured and calculated natural frequencies is 5.83% (Table 1). While the modeling error can arise from the upper joint that connects the spike and the upper pipe as well as the lower joint that connects the two pipes since the major uncertainties are associated with the two joints, from the simulation it was observed that the modes with the largest modeling errors are those with relatively large local vibrations of the spike, and the natural frequencies of the lightning mast are not sensitive to the change of the stiffness of the bolted flanges because the thickness of the flanges is relatively small compared with the length of the mast. In the beam element model, the stiffness of the upper joint is overestimated, since the beam elements representing the spike are connected to those representing the upper pipe through rigid links, and the shell-like deformations of the upper pipe wall adjacent to the spike are neglected. To resolve this problem, a more intensive model of the lightning mast, which models the lower and upper pipes and the flanges using shell elements and includes more details of the mast, such as the steel cap at the top of the upper pipe and the mass of the brackets at the upper joint (Fig. 1(a)), was created using SDTools [9] (Fig. 2). With the shell element model, the maximum error between the first eight measured and calculated natural frequencies is reduced to 1.69% (Table 1).
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Fig. 1 (a) A lightning mast at an electric substation, along with expanded views of the two joints, and (b) its beam element model. Table 1 The first eight measured and calculated natural frequencies of the lightning mast in Fig. 1(a) 0HDVXUHG +] %HDP(OHPHQW0RGHO +] (UURU 6KHOO(OHPHQW0RGHO +] (UURU 1.17
1.21
3.42%
1.15
-1.30%
5.09
5.36
5.30%
5.08
-0.23%
7.81
7.78
-0.38%
7.87
0.71%
16.56
17.00
2.66%
16.33
-1.41%
29.31
31.02
5.83%
29.10
-0.71%
45.22
45.49
0.60%
45.98
1.69%
47.65
46.09
-3.27%
46.85
-1.69%
53.98
55.83
3.43%
53.81
-0.31%
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Fig. 2 A more intensive shell element model of the lightning mast in Fig. 1 (a) with 40,122 degrees of freedom (DOFs) Since it is difficult to introduce damage to a real operating lightning mast, a scale model of the lightning mast in Fig. 1(a), with machined sections, was built in the laboratory to experimentally validate the vibration-based damage detection method (Fig. 3(a)). The length, radius, and thickness of the lower pipe are 0.776 m, 0.0124 m, and 0.000889 m, respectively, and those of the upper pipe are 0.776 m, 0.0093 m, and 0.000889 m, respectively. The length and radius of the spike are 0.2530 m and 0.0016 m, respectively. The radius and thickness of the flanges are 0.0254 m and 0.0051 m, respectively. The length, width, and thickness of the base plate are 0.0762 m, 0.0762 m, and 0.0062 m, respectively. The elastic modulus of the scale mast is 200 GPa , the Poisson's ratio is 0.29, and the mass density is 7,833 kg/m3 . A section from 0.0923 m to 0.1348 m from the fixed end and 0.000285 m deep was machined from the surface of the lower pipe, and another section from 0.8959 m to 0.9471 m from the fixed end and 0.000355 m deep was machined from the surface of the upper pipe. They represent a 30% and a 55% stiffness reduction, respectively. Unlike in the real lightning mast, the spike of the scale mast is welded to the upper pipe. Since the shell-like deformations of the upper pipe wall adjacent to the spike can be neglected in the scale mast, there is no problem to model the pipes using beam elements. The flanges and the base plate are modeled using shell elements, and the base plate is assumed to be fixed at the corresponding bolt locations. The welds between the pipes and the flanges and those between the lower pipe and the base plate need to be modeled, and each weld is modeled by a rigid link at its corresponding location. The maximum error between the first 15 measured and calculated natural frequencies is 1.88%, and the machined sections introduce a maximum change of 3.57% in the first 15 calculated natural frequencies (Table 2).
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Fig. 3 (a) A scale mast with two machined sections, along with the expanded views of the two joints and the base plate, and (b) its beam and shell element model with 5,892 DOFs. Table 2 The first 15 measured and calculated natural frequencies of the scale mast in Fig. 3(a) and the differences between the calculated natural frequencies of the undamaged and damaged masts 0RGH0HDVXUHG +] &DOFXODWHG +] (UURU &DOFXODWHG +] 'LIIHUHQFH GDPDJHG GDPDJHG XQGDPDJHG 1
9.780
9.698
-0.85%
10.026
-3.27%
2
9.812
9.817
0.05%
10.074
-2.55%
3
42.341
42.785
1.05%
44.371
-3.57%
4
43.151
43.152
0.00%
44.427
-2.87%
5
74.019
75.152
1.53%
75.433
-0.37%
6
76.714
75.271
-1.88%
75.535
-0.35%
7
147.976
146.212
-1.19%
148.146
-1.31%
8
149.430
147.555
-1.25%
148.851
-0.87%
9
249.081
252.208
1.26%
257.696
-2.13%
10
254.010
254.395
0.15%
258.158
-1.46%
11
425.630
428.982
0.79%
429.910
-0.22%
12
435.639
431.942
-0.85%
432.578
-0.15%
13
495.239
498.128
0.58%
500.235
-0.42%
14
505.554
500.641
-0.97%
501.165
-0.10%
15
543.332
549.123
1.07%
557.286
-1.46%
Modeling of Fillets and Bolted Joints. Detecting structural damage in a space frame is a common application of a vibration-based damage detection method. A space frame, as shown in Fig. 4, is usually assembled by filleted thin-walled beams connected by bolted joints; each bolted joint can have multiple bolted connections. The
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presence of the fillets in thin-walled beams and the bolted joints can significantly affect the free vibration characteristics of the assembled structure, and they need to be accurately modeled. Otherwise the modeling errors associated with the fillets and bolted joints can accumulate in the model of the assembled structure [5].
Fig. 4 An aluminum three-bay space frame with filleted L-shaped beams and bolted joints Modeling a fillet in a thin-walled beam usually requires an intensive solid element model with a large number of DOFs due to stress concentration at the fillet, which will increase the model size of the assembled structure and the computational cost for damage detection. The model size of a fillet must be reduced, but its effect on the free vibration characteristics of a thin-walled beam, which is mainly the stiffness effect, should be accurately represented in the model. The equivalent stiffness of a fillet in a thin-walled beam, e.g., an L-shaped beam, can be decomposed into the in-plane stiffness and the out-of-plane stiffness, which can be assumed to be uncoupled and calculated separately [4]. The in-plane stiffness is associated with the cross-sectional deformation and the out-of-plane stiffness is associated with the bending and the torsional stiffness of the thin-walled beam. The equivalent in-plane stiffness can be calculated by modeling half of the fillet as a curved cantilever beam subjected to a shear force and a bending moment at the tangent section of the half fillet (Fig. 5), and the equivalent out-ofplane stiffness can be obtained by calculating the area moments of inertia and the torsional stiffness factor associated with the fillet area [4]. In the FE model, the two stiffnesses can be modeled using shell and beam elements. The shell elements are used to mainly capture the in-plane stiffness although they also provide some out-of-plane stiffness; the beam elements are used to compensate for the out-of-plane stiffness that is not fully represented by the shell elements (Fig. 5) [4]. The length and thickness of the shell elements are determined by matching the rotational displacement of the tangent section with that calculated from the curved beam model under the same shear force and bending moment. The area moments of inertia of the beam elements are calculated by subtracting the area moments of inertia associated with the shell elements from that of the actual fillet area. The torsional stiffness factor of the beam elements is obtained by matching the torsional displacement of the beam elements with that of the fillet area subjected to the same torques at the tangent sections. The method was used to model an L-shaped beam of the space frame in Fig. 4 with free boundary conditions (Fig. 5) [4]. The errors between the measured and calculated natural frequencies of 28 elastic modes are within 2% and the associated Modal Assurance Criteria (MAC) values are all above 95%.
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Fig. 5 A fillet in an L-shaped beam in the space frame in Fig. 4, its analytical model, and its shell and beam element model. Modeling the bolted joints in the space frame in Fig. 4 is difficult due to their complicated nonlinear behaviors and the uncertain parameters such as the clamping forces, the contact interface properties, and the assembling prestresses. Direct simulation of the nonlinear behavior of a bolted joint will encounter some challenges in experimental parameter identification and the numerical calculation for structural dynamic analysis [5,10]. Since only the linear, free vibration characteristics of a structure are used in the vibration-based damage detection method, it is possible to develop a relatively simple model to accurately represent the mass and stiffness effects of a bolted joint. Modeling the mass effect of a bolted joint is relatively easy if the material density and the geometry of the bolted joint can be measured, and the main challenge lies in modeling its stiffness effect. Many parameters associated with a bolted joint, including the uncertain parameters mentioned above, can contribute to its stiffness effect; modeling all of them can lead to an intensive model involving nonlinear contact problems [5]. Since the contact problems are mainly related to the damping effect of a bolted joint rather than the stiffness effect, the task of modeling a bolted joint is to determine the critical parameters, to which the natural frequencies of the assembled structure are sensitive, and represent them in the FE model. A relatively simple FE model is created using SDTools [9] for a component of the space frame in Fig. 4, which consists of two L-shaped beams and a bolted joint with ten bolted connections (Fig. 6(a)), with free boundary conditions. The length and thickness of the bracket are 0.13 m and 0.0064262 m, respectively; steel hex head bolts ( M 10 u 35 8.8 ) are used for the bolted connections. The walls of the L-shaped beams and the bracket are modeled using shell elements, and each bolted connection is modeled by a solid cylinder [5]. The length of the cylinder is the sum of the length of the gap between the shell elements that represent the L-shaped beam and the bracket and that of the extruded part of the bolt outside the bracket surface. The radius and the material properties of the cylinder are the parameters that need to be determined. Figure 7 shows the sensitivities of the natural frequencies of the first nine elastic modes of the component in Fig. 6(a) to the radius and the elastic and shear moduli of the cylinders. It is seen that the natural frequencies are more sensitive to the changes in the radius of the cylinders than the elastic and shear moduli. The radius of a cylinder, which is associated with the effective area of the corresponding bolted connection, can be approximated by the radius of the static contact area of the two clamped components under a sufficiently large clamping force, which was calculated, from an intensive contact model using ABAQUS 6.7-1, to be 0.01043 m [5]. From the numerical simulation, it was observed that the contact area is not sensitive to the changes in the contact interface properties and the material properties of the clamped components, and remains a constant once the clamping force is greater than a threshold value [5]; hence the FE model is valid as long as the bolted connection is tightened and the clamping force is sufficiently large compared to the external loading. The elastic modulus of the cylinders can be obtained by matching the axial stiffness of a cylinder with that of the corresponding bolted connection calculated using the Rotsher's pressure cone method [11]. The calculated elastic modulus of the cylinders is 1.17 times the original material elastic modulus. For a tightened bolted connection, the shear modulus of the cylinders is set to be that of the material shear modulus of the clamped components since the slip between them is neglected in the linear
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model of the bolted joint [5]. In addition, rigid-link constraints that can restrict the relative translational motion between the corners of the L-shaped beams and the bracket, and prevent the penetration between the clamped components, are included in the FE model [5]. With the parameters of each cylinder being determined, the mass and stiffness effects of the bolted joint in Fig. 6(a) can be accurately modeled. The maximum error between the measured and calculated natural frequencies of the first 19 elastic modes of the component in Fig. 6(a) is 1.99% (Table 3), and the associated MAC values are all over 95% [5].
Fig. 6 (a) A component of the space frame in Fig. 4 and (b) its FE model with 97,866 DOFs
Fig. 7 Sensitivities of the least-square errors between the measured and calculated natural frequencies of the first nine elastic modes of the component in Fig. 6(a) to changes in the radius of the cylinders (solid) and the nondimensional elastic (dashed) and shear (dash-dotted) moduli of the cylinders [5] Table 3 The measured and calculated natural frequencies of the first 19 elastic modes of the component in Fig. 6(a) 0RGH0HDVXUHG +] &DOFXODWHG +] (UURU 0RGH0HDVXUHG +] &DOFXODWHG +] (UURU 1
105.83
104.71
-1.05% 11
927.06
940.04
1.93%
2
182.21
182.14
-0.04% 12
1057.26
1036.2
-1.99%
3
254.02
250.61
-1.34% 13
1306.62
1306.6
0.00%
4
313.75
318.15
1.74%
14
1328.52
1331
0.19%
5
371.75
373.57
0.49%
15
1381.74
1362.6
-1.39%
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6
470.45
469.03
-0.30% 16
1533.87
1511.4
-1.46%
7
555.46
553.57
-0.34% 17
1682.24
1664.9
-1.62%
8
650.46
641
-1.45% 18
1740.3
1729.9
-0.60%
9
903.25
910.18
0.77%
1998.16
1999.9
0.09%
10
917.8
921.67
0.42%
19
7+(,19(56(352%/(0 As mentioned earlier, the inverse problem for damage detection is usually an under-determined nonlinear leastsquare problem, which minimizes the objective function
1 m 1 m 2 1 7 2 (1) ( f ( [ ) y ) ¦ j ¦ rj 2 U U j 2 j1 2 j1 where [ [ xi ] in which 0 xi 1 ( i 1, 2!n ) is the i -th non-dimensional stiffness parameter at a potential Q ([)
damage location of a structure, with the value 1 being the undamaged stiffness parameter and the stiffness parameters;
n
the number of
f j ([) and y j are the j -th calculated and measured natural frequencies of the structure,
respectively; m is the number of the natural frequencies used in damage detection; and
U [rj ] in which rj is
the difference of the j -th calculated and measured natural frequencies. Solving the nonlinear least-square problem is to iteratively search the
[
domain for a stationary point [ of Q ( [ ) , with Q ([ )
0 . For a solution
of the nonlinear least-square problem, [ , its entries whose values are between 0 and 1 indicate that there are some stiffness reductions occurring at the corresponding locations of the structure, with the extent of damage indicated by the values of the entries. At a selected initial point [ 0 , Q ( [) can be approximated by a quadratic function M ( [ ) , which is the secondorder Taylor series expansion of Q ( [ ) around [ 0 :
1 M ([) Q([ 0 ) Q([0 )([ [0 ) ([ [ 0 )T 2Q([ 0 )([ [ 0 ) (2) 2 A stationary point of M ( [ ) in the neighborhood of [ 0 can be found by setting the first derivative of M ( [ ) with respect to [ to zero: M ([) Q([0 ) 2Q([0 )([ [0 ) 0 (3) Note that (Nocedal and Wright, 1999)
Q ( [ ) where
- [ [
wrj wxi
- T [ U [
(4)
] is the Jacobian matrix, the superscript T denotes the transpose of a matrix, and 2Q ( [ )
m
- T ( [) - ( [) ¦ rj 2 rj
(5)
j 1
In the Gauss-Newton method, the second term on the right-hand side of Eq. (5) is neglected for small residual problems, and Q ( [ ) is approximated by - ( [) - ( [) [3,6,7]. Equation (3) becomes 2
T
- T [0 U [0 - T ([0 )- ([0 )([ [0 ) 0
(6)
Equation (6) happens to be the normal equation of a liner least-square problem [3,7]:
min - ([ [ 0 ) U [
2 2
The solution of the linear least-square problem, as well as that of Eq. (6), can be directly obtained by letting
(7)
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- ([ [0 ) U
(8)
For an under-determined system, Eq. (8) has an infinite number of solutions. A minimum-norm solution of Eq. (8) can be obtained using the Moore-Penrose inverse [3,6,7]:
[
[0 - U
(9)
where the superscript + denotes the Moore-Penrose inverse. If only the stiffness reduction is considered, the Jacobian matrix can be written as [1,3]
w. Ij] (10) wxi where . is the stiffness matrix of the structure and I j is its j -th mode shapes. Due to the difference between - [ [I j T
M ( [ ) and Q ( [) and the approximation of 2Q ( [ ) by - T ( [) - ( [) , [ in Eq. (9) may not be a stationary point of Q ( [ ) , but it is closer to a stationary point of Q ( [ ) than the initial point [6]. If one replaces [ 0 in Eq. (3) by [ in Eq. (9) and repeats the same procedure, an iterative algorithm, and
[ k 1
[ k G[ , can be established, where [ k
[ k 1 are the current point and the next search point, respectively, k is the iteration number, and G[ - U
is the search step of each iteration. While the Gauss-Newton method provides a unique minimum norm solution for Eq. (3) when it is an underT determined system [6], it can amplify the errors associated with U ( [ ) when the condition number of - - is T
relatively large [3]. Since - - is rank-deficient for an under-determined system, some of its zero singular values can become non-zero and have small values due to the presence of modeling error and measurement noise in T experimental damage detection, which will significantly increase the condition number of - - . If the modeling error and measurement noise are relatively large, the Gauss-Newton method may not converge to the right
[ 2Q ( [)]T Q ( [)
0 for an under determined system [7], which is a weak convergence condition, rather than a stationary point [ satisfying
Q([ ) 0 [6]. solution. In addition, the Gauss-Newton method converges to a solution of
To avoid the drawback of the Gauss-Newton method for an under-determined system and enhance the robustness of the iterative algorithm, the LM method [6], which is a trust-region search method, is used here. Instead of directly calculating the magnitude and the direction of the search step using a line search strategy as in the Gauss-Newton method, the LM method restricts the magnitude of each search step to a region called the trust region, within which approximating Q ( [ ) by M ( [ ) is acceptable, and then calculates the search direction under the magnitude restriction. The radius of the trust region is updated at each iteration to ensure that sufficient descent of Q can be obtained [6]. By restricting the magnitudes of the search steps, the trust-region search T
strategy can avoid relatively large search steps caused by the small singular values of - - , which result from the modeling error and measurement noise; hence it is more robust than the Gauss-Newton line search strategy in experimental damage detection. In addition, the LM method is ensured to converge to a stationary point of the objective function of an under-determined system [6]. Since [ is bounded between 0 and 1, the nonlinear least-square problem is a constrained optimization problem. Since the convergence of the Gauss-Newton method and the LM method discussed above is based on the formulation for an unconstrained system, a logistic function transformation is introduced to convert the constrained optimization problem into an unconstrained one by letting
xi
1 1 e O si
(11)
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O
is a positive constant, which is selected to be 1 in this work, and si ( f , f ) are the new search variables. With the above transformation, the convergence speed of the iterative algorithm can also be increased. While increasing O can increase the convergence speed, numerical overflow can occur when O is too large. where
'$0$*('(7(&7,215(68/76 With the forward and inverse problems being resolved, the vibration-based method can be used to detect damage in different structures. The first application is to detect possible damage in the real lightning mast in Fig. 1(a) using the first five measured natural frequencies. In the FE model, the lightning mast, including the spike, is evenly divided into 20 element groups in the vertical direction, and the elastic modulus of each element group is represented by a non-dimensional stiffness parameter. When damage detection is performed on the lightning mast using the beam element model, some stiffness reductions are detected around the two joints of the mast (Fig. 8), because the relatively large modeling error is treated as the natural frequency changes caused by the possible damage there. When the model of the lightning mast is improved by the more intensive shell element model, there is no significant stiffness reduction in the damage detection results in Fig. 8, and the lightning mast is essentially healthy.
Fig. 8 Damage detection results for the real lightning mast in Fig. 1(a) using the beam element model and the improved shell element model The second application is to detect the stiffness reductions associated with the machined sections in the scale mast in Fig. 3(a). In the FE model, the lower pipe, the upper pipe, and the spike are evenly divided into 20, 14, and 4 element groups, respectively, in the vertical direction, and the elastic modulus of each element group is represented by a non-dimensional stiffness parameter. The first six measured natural frequencies are used to detect damage, and both the locations and extent of damage can be successfully detected in the experiment (Fig. 9); the maximum error between the first six measured and calculated natural frequencies of the damaged mast is 1.73%. The exact locations and extent of damage are detected in the numerical simulation in Fig. 9, when there is no modeling error and measurement noise.
Figure 10 shows the numerical results that illustrate the effect of the logistic function transformation on the convergence speed of the Gauss-Newton method. Consider a scale mast similar to that in Fig. 9 with a 50%
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is sstiffness reduction from 0.5 5044 m to 0.5432 m from the fixed end d (groups 13 and 14 in Fig. 11). The damage d detected using a meth f first hod similar to that in Refs. [1,2], withoutt using the log gistic function n transformation, and it takes 1300 ite erations to co onverge to a solution that is close to th he exact solu ution (Fig. 10(a)). When th he logistic function transsformation is used, it takess only 13 iterrations to con nverge to the e exact solutio on (Fig. 10(b)). Similar re c comparison esults can be obtained for the LM metho od.
Fig. 9 Experimental and numerical dam mage detectio on results for the scale ma ast in Fig. 3(a a) with a 30% % and a 55% stiffness redu pipe, re changess in the first six natural uction at a secction of the lo ower and the upper u espectively, using u fre equencies.
Fig. 10 Num merical damag ge detection re esults and the e numbers of iterations forr a scale mastt similar to tha at in Fig. 9 with a 50% % stiffness re eduction at a section of the lower pipe using changess in the first six natural freq quencies: (a) the logistic function transforma ation is not ussed, and (b) the t logistic fun nction transfo ormation is ussed. The last appliccation is to de etect loosenin ng of bolted connections in n the compon nent in Fig. 6((a) using the measured n natural freque encies of the e first 16 ela astic modes. Four of the ten bolted connections c in the bolted joint are lo oosened to hand-tight, as shown in Fig g. 11, which causes a maxximum change e of 4.31% in n the measure ed natural f frequencies o the first 16 elastic mode of es (Table 4). A loosened bolt connection is modele ed by a reducced shear
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modulus of the solid cylinder [5]. In the FE model, the portion of each L-shape beam that is not in contact with the bracket is evenly divided into 10 element groups (groups 1 to 10 for one L-shaped beam and groups 18 to 27 for the other L-shaped beam), the elements representing the bracket are grouped together as one element group (group 11), and those representing the L-shaped beams in contact with the bracket are grouped together as one element group (group 17). The elastic modulus of each element group above is represented by a non-dimensional stiffness parameter. Each pair of the bolted connections that are symmetric about the corner of the bracket, which is assumed to have the same tightness, is grouped together as one element group (groups 12 to 16), and the corresponding shear modulus of the cylinders is represented by a non-dimensional stiffness parameter. The loosened bolted connections can be successfully identified, and the maximum error between the first 16 measured and calculated natural frequencies of the damaged component is 1.50% (Table 4). In the numerical simulation, the shear modulus of the cylinder for a loosened bolted connection is assumed to be reduced by 60%; the exact locations and extent of damage are detected using changes in the natural frequencies of the first 16 elastic modes (Fig. 11).
Fig. 11 Experimental and numerical damage detection results for the component in Fig. 6(a) with four of the ten bolted connections loosened Table 4 The measured natural frequencies of the first 16 elastic modes of the component in Fig. 6(a) with all the bolted connections tightened (undamaged) and four of the ten bolted connections loosened (damaged) and the calculated natural frequencies of the first 16 elastic modes of the damaged component 0RGH8QGDPDJHG +] 'DPDJHG +] 'LIIHUHQFH'DPDJHG +] (UURU PHDVXUHG PHDVXUHG FDOFXODWHG 1 105.83 104.118 -1.62% 102.765 -1.30% 2 182.21 174.351 -4.31% 174.9026 0.32% 3 254.02 248.216 -2.28% 246.6243 -0.64% 4 313.75 310.17 -1.14% 314.8317 1.50% 5 371.75 366.39 -1.44% 367.86 0.40% 6 470.45 464.933 -1.17% 461.3521 -0.77% 7 555.46 550.085 -0.97% 543.7143 -1.16% 8 650.46 634.573 -2.44% 628.7344 -0.92% 9 903.25 893.243 -1.11% 890.2972 -0.33% 10 917.8 907.601 -1.11% 905.5672 -0.22% 11 927.06 916.04 -1.19% 922.6876 0.73% 12 1057.26 1022.72 -3.27% 1024.462 0.17% 13
1306.62
1281.12
-1.95%
1272.863
-0.64%
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77
14 15 16
1328.52 1381.74 1533.87
1319.41 1332.49 1492.72
-0.69% -3.56% -2.68%
1306.332 1338.277 1484.867
-0.99% 0.43% -0.53%
&21&/86,216 The creation of an accurate physics-based model for both the undamaged and damaged states of a structure and the development of a robust iterative algorithm for an under-determined system are the keys to implementing the vibration-based damage detection method using changes in the natural frequencies of the structure. The real lightning mast and the scale mast can be accurately modeled using mainly shell and beam elements, respectively. The new modeling techniques for fillets and bolted joints can be used to accurately model the component of the space frame with tight and loosened bolted connections. The trust-region search method can resolve the erroramplifying problem associated with the Gauss-Newton method in solving the under-determined nonlinear leastsquare problem, and improve the robustness of the iterative damage detection algorithm. With the logistic function transformation, which converts the constrained optimization problem into an unconstrained one, the damage detection algorithm is ensured to converge to a stationary point of the objective function. The logistic function transformation can also significantly increase the convergence speed of the iterative algorithm. The new methodology can successfully detect damage in the scale mast and loosening of bolted connections in the component of the space frame. The exact locations and extent of damage can be detected in the numerical simulation when there is no modeling error and measurement noise.
$&.12:/('*(0(17 This work is supported by the National Science Foundation through Grant No. CMS-0600559 and the American Society for Nondestructive Testing (ASNT) through the 2007 ASNT Fellowship Award. The authors would also like to thank the Baltimore Gas and Electric Company and the Maryland Technology Development Corporation for their previous support, and Benjamin Emory for measuring the natural frequencies of the real lightning mast and building the scale mast in the laboratory.
5()(5(1&(6 [1] Wong, C.N., Zhu, W.D., and Xu, G.Y., “On an Iterative General-Order Perturbation Method for Multiple Structural Damage Detection,” Journal of Sound and Vibration, Vol. 273, pp. 363-386, 2004. [2] Xu, G.Y., Zhu, W.D., and Emory, B.H., “Experimental and Numerical Investigation of Structural Damage Detection Using Changes in Natural Frequencies,” Journal of Vibration and Acoustics, Vol. 129, Dec. pp. 686-700, 2007. [3] Friswell, M. I. and Mottershead, J. E., Finite Element Model Updating in Structure Dynamics, Netherlands, Kluwer Academic Publishers, 1995. [4] He, K. and Zhu, W. D., “Modeling of Fillets in Thin-walled Beams Using Shell/Plate and Beam Finite Elements,” Journal of Vibration and Acoustics, Vol. 131, 051002 (16 pages), 2009. [5] He, K. and Zhu, W.D. , “Finite Element Modeling of Structures with L-shaped Beams and Bolted Joints,” Proceedings of the ASME Biennial Conference on Mechanical Vibration and Noise, San Diego, CA, Sept. 2009; also Journal of Vibration and Acoustics, submitted. [6] Nocedal, J. and Wright, S. J., Numerical Optimization, New York, Springer-Verlag, pp. 252-270,1999. [7] Galantai, A., "The Theory of Newton's Method", Journal of Computational and Applied Mathematics, Vol. 124, pp. 25-44, 2000.
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[8] Zhu, W.D., Zheng, N.A., and Wong, C.N., "A Stochastic Model for the Random Impact Series Method in Modal Testing", Journal of Vibration and Acoustics, Vol. 129, pp. 265-275, 2007. [9] Bálmes, E., Bianchi, J. P., and Leclère, J. M., Structural Dynamics Toolbox, Users Guide, Version 6.1, Paris, France, Scientific Software Group, 2009. [10] Segalman, D. J. and Starr, M. J., "Modeling of Threaded Joints Using Anisotropic Elastic Continua," Journal of Applied Mechanics, Vol. 74, pp. 575-585, 2007. [11] Shigley, J. E. and Mischke, C .R., Mechanical Engineering Design, 5th Edition, New York, McGraw-Hill Inc., 1989.
BookID 214574_ChapID 8_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Order Tracking with Multi-Sine Excitation on a Ground Vibration Test Gordon Hoople Kevin Napolitano ATA Engineering, Inc. 11995 El Camino Real, Suite 200 San Diego, California 92130
Abstract This paper presents a new method of postprocessing multi-reference sine sweeps using order tracking methods. In combination with conventional multi-reference signal processing using load cells as independent references, this method of postprocessing can be used to extract the equivalent of multi-reference, symmetric sine, and antisymmetric sine sweeps from a single data set. By capturing all of this information in a single run, as opposed to up to seven runs that might be required on a typical six-shaker test, testing time can be dramatically reduced. With this method, a complete set of mode shapes can be more easily extracted from a single data set with a consistent input force level. This provides consistent results for use in model correlation.
1. Introduction Multi-sine excitation is a new method of excitation that can be used on ground vibration tests (GVTs). GVTs are an important step in the preflight certification of new and newly modified aircraft. Different implementations of multi-sine excitation offer equivalent or improved results while reducing testing time when compared to traditional testing methods [1]. With advances in computing power and data storage, it has become possible to manipulate and store vast amounts of time history data rather than having to immediately transform the data to the frequency domain during testing. This has opened the door to new, more computationally intensive postprocessing methods. This paper will build on the work presented by Hoople and Napolitano in “Implementation of Multi-Sine Sweep Excitation on a Large-Scale Aircraft [1].” This paper will focus on a single type of multi-sine testing presented in that paper: full multi-sine. The paper will show that a single multi-sine run can be postprocessed using order tracking to extract the equivalent of burst random excitation, symmetric sine sweeps, and antisymmetric sine sweeps. Full multi-sine is defined as having the same number of independent reference signals as shakers, similar to multi-reference random where all shaker signals are independent. The implementation of full multi-sine presented here involved passing a signal composed of two sine waves to each pair of shakers on the test article. These sine waves covered the same frequency range; however, the two sine sweeps were slightly offset in the time domain so that at each instant in time the structure was excited at two different frequencies. In order to simultaneously excite all of the modes of the aircraft, the sine waves were added with the appropriate differences in phase to simultaneously excite both symmetric and antisymmetric modes. Figure 1 shows both time and frequency domain plots T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_8, © The Society for Experimental Mechanics, Inc. 2011
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for shakers positioned on the wingtips of a test article at a particular instant in time. Notice that each signal is composed of the addition of two frequencies; however, for the lower frequency these signals are in phase, and for the higher frequency these signals are out of phase. Figure 2 shows the frequency pairs sent to each of three pairs of shakers at a particular instant in time. Notice that these are also offset so that no independent input signal frequency matches another. For example, the wing shakers do not contain the same frequencies as the horizontal stabilizer or the engine shakers. Full Multi-Sine Time Domain - Wing Shakers Only
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2. Data Collection Data was collected during the GVT on Lockheed Martin’s Advanced Composite Cargo Aircraft (ACCA). To build the ACCA, Lockheed Martin replaced the mid/aft fuselage and vertical tail of a Dornier 328J aircraft with an advanced composite structure. A more detailed description of the test and data collection processes is presented in a previous
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paper [1]. Signal processing techniques that can be used on full multi-sine data are described next, so only a brief summary of the relevant data sets is provided. The aircraft was excited at six locations: the engines, wing tips, and horizontal stabilizer tips. The aircraft was tested in a simulated free-free boundary condition. For all of the test runs, data was acquired using a sampling frequency of 128 Hz. Data was processed with a blocksize of 2048 samples, which corresponds to a frame length of 16 seconds. In order to help validate the multi-sine method, a complete set of traditional data was taken. Burst random data was collected using six shakers with uncorrelated inputs. No windows were applied, and a total of thirty frames of data were measured with no overlap. Traditional sine data was collected using a single pair of shakers at a time. Symmetric or antisymmetric sweeps were used to separately excite shakers at the wings, the horizontal stabilizers, and the engines, for a total of six different runs. The data was processed using a single reference with an overlap of 90% and a Hanning window. The sine sweep began at 50 Hz and swept down to 1 Hz at a rate of 0.2 decades per minute. Full multi-sine data was collected using six shakers with six independent signals. For each pair of shakers, two independent signals were injected into the shakers to simultaneously provide symmetric and antisymmetric excitation. The sweep started at 50 Hz and swept down to 1 Hz at a rate of 0.2 decades per minute. During the sweep, a ratio of 1.1 was kept between the frequencies of all six signals in order to ensure independence. Signals for the shakers were created using ATA’s Multi-Sine Sweep Creator [2]. The data was processed using all six references with an overlap of 90% and a Hanning window. Order tracking methods, described next, were also used to process the data.
3. Order Tracking Order tracking is a method of signal processing developed to primarily aid in the understanding of mechanical systems under periodic loading. Order tracking methods are generally applied to rotating machinery, where one or more shafts rotate and the fundamental order is the speed, or frequency, of rotation. The system responds at this frequency as well as at higher harmonics (orders) of this frequency. Generally, for operating machinery, many different rotating components produce multiple sets of orders and rich frequency content. Order tracking filters are used to focus on those responses associated with a particular frequency, or a set of harmonics in a given data set [3]. Multi-sine data from a single shaker can be viewed as a simple rotating machine which contains only two frequency components of interest: the symmetric sine excitation and the antisymmetric sine excitation. (This was shown in Figure 1.) For multi-sine testing, the frequency of the excitation changes throughout the test. This is analogous to the frequency changes during a run-up or run-down test of a piece of rotating equipment. The goal of using an order tracking filter on the multi-sine data is to break apart the shaker input signal into its two components: the symmetric piece and the antisymmetric piece.
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These filtered input loads to the structure can then be used, when combined with acceleration measurements, to calculate frequency response functions (FRF). One important point to note is that by using the H1 estimator to compute the FRF, it is not necessary to filter the acceleration measurements. While the acceleration measurements contain responses from both the symmetric and antisymmetric cases, when using the H1 estimator the undesired response will be averaged out as noise. If it is desired to use another estimation technique for computing FRF, then the order tracking filter should be applied to the acceleration responses as well. The Vold-Kalman order tracking filter [3] was used to extract symmetric and antisymmetric sweeps from the full multi-sine data. This filter is specifically designed to decouple close orders, which is ideal for the full multi-sine sweeps that are relatively close in frequency. The filter is also designed to handle the constantly changing frequency of the sweep. ATA Engineering's Rotate software was used to apply the VoldKalman filter to the load cell data from each of the shakers. In this case, the specific filter used on the data was a 2-pole filter designed to track a single order. Using Rotate, six different time history signals were extracted from a single full multisine data set. Next, six sets of frequency response functions, each using only a single filtered time history as the reference, were computed. This data was processed in the traditional manner that would be used for a single sine sweep data set: a blocksize of 2048, a Hanning window, and an overlap of 90%. The next section compares the results of the order-tracked multi-sine data to the traditional single sine data.
4. Presentation of Results The following is a comparison of traditional testing results to those obtained from a single multi-sine test performed on the same test article. It is important to remember that all of the multi-sine data presented in this section were collected from a single run and that order tracking methods were used to help process the data. Using postprocessing techniques, the multi-sine data can be compared to seven data sets collected using traditional random and sine methods. A comparison of burst random and unfiltered full multi-sine data is presented first. A plot of the complex mode indicator function (CMIF) is shown in Figure 3. The CMIF is a type of summary plot of all the collected FRF and is similar to FRF in that peaks indicate modes of the structure [4]. The concentrated energy associated with the multi-sine testing, particularly at the low frequencies, helped avoid the signal-to-noise problems sometimes associated with burst random. The full multi-sine exhibited more clearly defined peaks in the CMIF, and the curve-fitting algorithm was able to solve for potential mode shapes with a finer tolerance for frequency and damping.
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The next sets of data were produced using the order tracking methods previously discussed. Overlay plots of the CMIF for single sine data and filtered multi-sine data are presented in Figure 4, Figure 5, and Figure 6. Results for symmetric and antisymmetric excitations at the wing tips, horizontal stabilizer tips, and engines are provided. For each of these data sets, the primary modes of the structure have been tagged in the figures. The target modes of the structure were successfully extracted using the filtered multi-sine data. For all of the cases shown, the CMIF for the multi-sine and single sine are nearly identical up to 40 Hz. Both frequency and damping match extremely well for the primary modes, and the degree of similarity between the CMIF functions demonstrates that there is good agreement between the two different testing methods.
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Figure 6 - Engines – symmetric and antisymmetric.
There are a few small discrepancies between the CMIF plots; one example can be seen around 17 Hz in Figure 6. The slight discrepancy here is most likely due to small nonlinearities in the structure caused by the differences in force levels between the two runs. While the peak levels of excitation for the single sine and multi-sine tests were matched, as previously mentioned the multi-sine is the addition of the symmetric and antisymmetric cases. This means that when the data was filtered using order tracking, the effective force level for that case was only half of the total force of the shaker for that test. This is shown graphically in Figure 7. In the future, for a true one-to-one comparison, the peak multi-sine level should be twice the level of the single sine data.
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5. Further Beat Frequency Analysis Multi-sine excitation can be susceptible to beating effects on the shaker input signals [2]. Figure 8 shows an example of the beat frequency which occurs when two source signals are added together. Notice that at a particular point in time, which corresponds to a particular frequency, there is effectively no input to the test article. As previously presented by Napolitano and Linehan [2], there are three ways to deal with this phenomenon: (1) decrease block size, (2) decrease sweep rate, or (3) maximize the frequency difference between the source signals. Beat Frequency Phenomenon shown for Engine Shakers 10 8
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For the majority of the data collected, the effects of the beating phenomenon were not observed when following the guidelines presented by Napolitano and Linehan. The only area where the beating phenomenon was observed to be an issue was at high frequencies on accelerometers that showed low responses due to a particular shaker input. For this test, that condition was particularly prominent for the shakers on the engines of the air-
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craft. The large mass of the engines acted as a mechanical filter and prevented much of the engine shakers' energy from exciting accelerometers located on other parts of the test article. To understand this phenomenon, consider Figure 6 and Figure 9. Figure 6 is the CMIF calculated from the engine shakers to the engine accelerometers – the accelerometers which are expected to have the highest response to these shakers. Notice that the multisine CMIF shows no noisy characteristics. Next, consider Figure 9; here, the CMIF has been computed between the engine shakers and the accelerometers on the horizontal stabilizer. These accelerometers have a response four to six times lower than those on the engine. Consequently, the CMIF for the multi-sine case shows noise effects associated with the beating phenomenon, particularly at higher frequencies. With careful consideration and frequency spacing, these beating effects can be avoided. When there is an indication of this problem, the sweeps can be repeated with different frequency spacing and sweep rates.
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6. Conclusions This paper has explained that by using order tracking filtering techniques, the equivalent of seven different data sets can be extracted from a single multi-sine run. As such, this method can potentially reduce a testing program by half of a day or more. Furthermore, it has compared the data from these two methods and found that they produce equivalent modal information. A major advantage of this technique is that it enables the test engineer to more easily extract all of the mode shapes for a particular test article from a single set of data. This is advantageous for the correlation process where consistency between sets of modal information is highly desired. The potential issues associated with the beat frequency phenomenon have been addressed and, while present in a few select cases, do not render the testing method unusable. While further work needs to be performed to minimize the beating effect of the low responding accelerometer signals, this paper has shown that order tracking can be a viable alternative to traditional testing methods.
References [1] Hoople, G. and Napolitano, N., “Implementation of Multi-Sine Sweep Excitation on a Large-Scale Aircraft.” IMAC XXVIII, Jacksonville, Florida, Feb. 2010. [2] Napolitano, K and Linehan, D., “Multiple Sine Sweep Excitation for Ground Vibration Tests,” IMAC XXVII, Orlando, Florida, Feb. 2009. [3] Gade, H., Herlufsen, H., Konstantin-Hansen, H., and Vold, H., ”Characteristics of the Vold-Kalman Order Tracking Filter.” Brüel&Kjær Technical Review No. 1 – 1999. [4] Shih, C.Y. and Brown, D.L., “The Complex Mode Indicator Function Approach to Modal Analysis,” Pre-IMAC 8 Symposium, UC-Irvine, 1989.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
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$%675$&7 Impact excitation is the most common excitation type for measurements of frequency response functions for modal analysis and other purposes. The method used is almost always based on setting the data acquisition system up with triggering, fixed FFT analysis settings, and then using an accept/reject step where each impact is either accepted if the impact seems good, or rejected if it contained some error such as double impacts or overload. This method has several drawbacks that often lead to non-optimal frequency responses. In this paper, an improved method based on time recording of all signals and subsequent post processing is proposed. The data acquisition part is made easier with the proposed method, while at the same time the importance of a skilled operator is reduced. It is shown on a real test structure that the quality of the resulting frequency responses can be significantly improved (measured by the coherence function) compared to the traditional method, and at the same time the total acquisition time can be shortened. An automatic optimization procedure which allows for fully automated post processing is proposed.
,1752'8&7,21 The way most vibration data acquisition systems are designed is based on the premises in the 1970’s when the first FFT analyzers became available. One restriction in those days was the price of memory, and thus the way the data processing was implemented was to reduce data as soon as possible after acquisition. The result became the frequency block averaging that we use today and which is illustrated in Figure 1. The process waits for a trigger event and, when this is fulfilled, acquires a block of N samples into the buffer, which is then sent off to the FFT processor as soon as the buffer is full. Once the FFT process is completed, the data is sent to the averaging process where each frequency value of the latest FFT results is averaged into auto and cross spectra. Usually, for impact testing, there is an interrupt after the FFT process, so that, prior to including the new FFT results in the averaging process, the user can decide to include the new
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_9, © The Society for Experimental Mechanics, Inc. 2011
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90 frequency results in the averaging process, or, if it was a bad impact, the user can decide to discard the data and make a new impact. Several drawbacks of this technique are well known, for example that a relatively experienced operator is required for best results. The main reason for this, is that as the impacts are acquired and accepted into the averaging process, the end result is often dependent on the fact that all impacts are approximately equally large, and impacting the same location on the structure accurately. During the data acquisition, it is often also required to have one operator running the measurement system, accepting and rejecting each impact, and one operator working the impact hammer, although in some measurement systems there is an “auto reject” feature. From practical experience, though, we know that most of the time it is best to be two people for a good test.
)LJXUH Illustration of the data acquisition and FFT process involved in traditional impact testing.
We assume that impact testing is rather well known. Details about the signal processing involved in the spectrum, frequency response (FRF) and coherence estimates can be found in e.g. [1].
,03529('0(7+2' The improvement to the traditional way of performing an impact test which we propose in this paper is to replace the online averaging process by a separate recording of time data, followed by offline post processing to obtain frequency response and coherence. To evaluate the proposed method data were acquired on a slalom ski under free-free boundary conditions. The ski was instrumented with an accelerometer and an impact hammer was used to excite the ski in several locations, although only one measurement point will be presented here. The method is very easy to implement and has been implemented in MATLAB. The main steps in the process are 1. Record a long time record with 5 to 10 impacts with enough time in between each impact 2. Find the impacts using a trigger level and apply a suitable number of samples pretrigger to make sure each impact is well defined in the time block 3. Apply a force window to the force, and exponential window to both force and acceleration 4. Select impacts to add for the averaging process by evaluating the coherence function as a quality estimator, only adding the impacts that give a good quality frequency response, i.e. one with a coherence function near unity. The proposed method offers several advantages over the traditional way of performing impact testing. First, the measurement phase is considerably easier, since there is no accept/reject procedure involved, and no
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91 trigger conditions to set up. This is particularly important when the impact test is done for trouble shooting purposes in an operating environment. Second, the actual analysis process in steps 2. to 4. can be fully automated, and we will propose a simple yet powerful optimization algorithm to obtain high-quality frequency responses. In addition to these advantages there are several “side effects” which can prove to be important. The essential advantage is to have the entire recording as one, long time series. This allows for efficient signal processing such as removal of line power noise, and highpass filtering to remove rigid body vibrations, which can sometimes cause leakage effects when the structure is slowly oscillating on free-free supporting soft springs. Impact testing involves a number of steps normally implemented in the measurement system. They include triggering, application of force and exponential window on each triggered time block, an FFT of each time block, and appropriate averaging to auto and cross spectra which are finally combined into either an H1 or an H2 estimate of the frequency response. As the noise in the force sensor can usually be almost entirely removed by the force window, the H1 estimator which minimizes noise on the output (accelerometer) is generally the best estimator. [0 PD[ DEV [ 0D[SHDNLQ[ IRUFHVLJQDO 7/ 7ULJ3HUF [0 7ULJJHUOHYHOLQXQLWVRI[ [ HQG [0 7RPDNHGLIIZRUNZLWKRXWORVLQJODVWLPSDFW G[ GLII VLJQ [7/ 'LVWDQFHEHWZHHQWULJJHUSRLQWV 7ULJ,G[ ILQG G[! G[KDVDYDOXHRIZKHUHWKHUHLVDWULJJHU
)LJXUH MATLAB code to find the trigger locations on the force signal in vector x. The code works for positive peaks and positive slope.
)LJXUH Plot showing the force signal (upper) and response signal (lower). The entire signal is plotted in blue, and each defined block from the trigger point and N consecutive samples is plotted in green.
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92 The first step in the post processing is to find the impact occasions. This is done in MATLAB very simply by a few short lines of code as shown in Figure 2. The data is in the vector denoted x, and the trigger level is given as a percentage of the maximum impact peak. After the trigger events are found, the indices should be subtracted by the number of pretrigger samples one wants, so that the beginning of each force impact is well inside the data block. In Figure 3 a plot is shown where each block of data (using a preselected blocksize, see more below) is marked in green, plotted on top of the entire time data series in blue. It is easy to implement a double impact detection and for example plotting the blocks which contain a double impact in red in a plot similar to the one in Figure 3. This has been implemented but we omit it here for simplicity. After each impact and response block are thus defined, appropriate frequency analysis settings should be evaluated. This is another great advantage with the proposed method, as we can now very easily analyze the data in Figure 3 over and over again with different blocksize, force window parameters, and exponential window parameters. The only limitation is that we can only use blocksizes that make each block well defined, i.e. the blocks may not be so long as to go into the next impact. When estimating frequency responses, the frequency increment must be sufficiently small so that the bias error that comes from the discrete approximation of the continuous frequency response becomes negligible. This should be investigated by calculating the frequency response with successively decreasing frequency increment (i.e. increasing blocksize) until two consecutive blocksizes show approximately the same peak height at the first resonance (assuming same damping for all modes). The blocksize can then be set to the lower of the two.
)LJXUH Plot showing the force signal (upper) overlaid with the force window, and the spectra of the unwindowed and windowed force signals (middle), and a response signal and exponential window (lower). Note that the upper two plots are highly zoomed in time and frequency, respectively. Also, the force windowed has a scaling to fit the plot, its actual values are of course 1 over the flat part. A scaling has also been applied to the exponential window, which in reality starts at one for time zero.
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93 Once an appropriate block size is selected, the force window should be optimized. A good force window can be defined by a certain length with unity values, followed by a half period of a cosine with approximately the same duration. In Figure 4 (top) one impact force is plotted together with the force window chosen for the example later in this paper. The force window removes the noise in the force spectrum (middle plot) almost entirely. The final step in the frequency settings stage is to optimize the exponential window. This can be done by applying a successively stronger exponential window until no apparent improvement is seen in the coherence. It should be noted especially that it is important not to forget to add the exponential window also to the force signal, as the increased damping effect can otherwise not be properly compensated for, [2] if the frequency response is to be analyzed for damping (for example through modal analysis curve fitting). 237,0,=,1*)5)48$/,7< As can be seen in Figure 3 there is some discrepancy between the force levels of each impact. This illustrates how difficult it is to perform identical impacts with a common impulse hammer. On a slightly nonlinear structure (as many structures, if not all, are) this will result in a badly estimated frequency response. However, as we are post processing data, we can very easily decide to choose only impacts with approximately the same force level. In Figure 5 a plot is shown of frequency response and coherence where the first five impacts have been used. As is seen in the plot the quality is good (coherence of unity) at most frequencies, with an area between approximately 70 and 90 Hz where, at the deep antiresonance region of the FRF, the coherence drops, indicating a low signal-to-noise ratio. At approximately 75 Hz there is a resonance which coincides with a dip in the coherence, a most unwanted situation.
)LJXUH Plot showing a frequency response (upper) and the corresponding coherence function. The first five impacts of the time series in Figure 3 have been used in the averaging process to compute both functions.The result corresponds with what one would expect from a measurement using the traditional impact testing process. Note that the results in Figure 5 is what one would expect as the result from a traditional averaging process for impact testing, as none of the five impacts used showed any apparent signs of problems. In order to
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94 possibly find a better estimate of the FRF, a simple optimization algorithm was developed. To limit the computational time, first all pair combinations of two out of the (in our example) 10 impacts are used to compute the coherence. The pair is chosen which minimize the error function H , defined by
H
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k
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f
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of N samples. A frequency region of interest is chosen and used for the error calculation. After the first two impacts have been chosen, each of the remaining impacts is added individually, the coherence function is calculated, and for each impact the error function defined in Equation 1 is calculated. Those impacts that reduce the error function are added to the averaging process. The result of applying the optimization algorithm is shown in Figure 6. As is seen in the figure, the coherence function is substantially improved, and thus the FRF quality. Only a narrow dip remains at the strong antiresonance of the FRF.
)LJXUH Plot showing a frequency response (upper) and the corresponding coherence function for the optimally chosen impacts. The problem area in the frequency range 70 to 90 Hz is considerably improved compared to using the five impacts that produces the FRF and coherence in Figure 5.
&21&/86,216 We have proposed a modified method for impact hammer excitation and shown on example data that the method can improve the quality of estimated frequency responses. It is based on time recording of a number of impacts and responses into a file, and subsequent post processing. An optimization algorithm to automatically select impacts for best FRF estimate was presented. The method has a number of advantages, such as
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Easier data acquisition because the operator only needs to make impacts, and not have any interaction with the measurement system during the data acquisition. Less importance of an experienced impact hammer operator, as impacts of equal force level can easily be selected in the post processing stage. Improved means of optimizing frequency analysis settings as no new measurement is needed after each change. Better coherence can be more easily and reliably obtained by choosing only impacts that give a better coherence function (FRF quality). Easier to apply signal processing to remove 50/60 Hz line frequency noise and highpass filtering to remove effects of low frequency oscillations on soft springs support of the structure.
5()(5(1&(6 [1] Bendat, J. and Piersol, A. (2000), Random Data: Analysis and Measurement Procedures, Wiley Interscience. [2]
Fladung, W. & Rost, R. “Application and correction of the exponential window for frequency response functions,” Mechanical Systems And Signal Processing, 1997, (11), p. 23-36.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
5HYLHZRI5HFHQW'HYHORSPHQWVLQ0XOWLSOH5HIHUHQFH,PSDFW7HVWLQJ 05,7 '/%URZQDQG0&:LWWHU Structural Dynamics Research laboratory University of Cincinnati $EVWUDFW Impact testing was one of the first practical applications of the Fast Fourier Transform (FFT) technique in the late 60s. Prior to the development of the FFT, measuring the Frequency Response Function (FRF) was limited to sinusoidal testing procedures. The sine testing methods were slow, and required elaborate fixtures for excitation (electro-mechanical or hydraulic exciters). Impact testing had an order of magnitude faster test time and minimal fixtures. As a result, it became a very good field testing and trouble shooting method, as well as a pretesting method for controlled laboratory testing. In this paper a general review of the evolution of impact testing from its development in the 60s and 70s to the present time, with a more extensive review of recent developments in testing procedures and parameter estimation for Multiple Reference Impact Testing (MRIT).
$FURQ\PV 3D ADC ARMA CMIF EFRF EMIF ESSV DFRF DOF DOT DSIT DSP DSS ERA GVT FEM FRF FFT HP
Three Dimensional Analog Digital Converter Auto Regressive Moving Average Complex Mode Indicator Function Enhanced Frequency Response Function Enhanced Mode Indicator Function
Extended State Space Vector Directional Frequency Response Function Degree-of-Freedom Department of Transportation
Digital System Interface Transmitter Digital Signal Processing Digital Sensor System Eigenvalue Realization Algorithm Ground Vibration Test Finite Element Model Frequency Response Function Fast Fourier Transform Hewlett Packard
IRF LSCE MAC MDOF MIMO MRIT NASA PTD SDA SDOF SDRL SDRC SDOF SST SVD UC UMPA UIF
Impulse Response Function Least Squares Complex Exponential Modal Assurance Criteria Multiple Degree-of-Freedom Multiple-Input Multiple-Output Multiple Reference Impact Testing National Aeronautics and Space Administration Poly Reference Time Domain System Dynamic Analysis Single Degree-of-Freedom Structural Dynamics Research Laboratory Structural Dynamics Research Corporation Single Degree of Freedom Spatial Sine Testing Singular Value Decomposition University of Cincinnati Unified Matrix Polynomial Algorithm Unit Impulse Function
%DFNJURXQG In the mid 60s, the mathematical properties of the Fourier Transform were well known, but its applications were limited and it was the development of the FFT which made the numerical computations of the Fourier Transform practical. The FFT was a revolutionary breakthrough which led to many developments in digital signal processing which were applied in many disciplines, including acoustics, controls, and structure dynamics. In 1966/67 a project to develop a transient testing procedure for measuring Frequency Response Functions was initiated for a master’s thesis. In this initial effort an impact hammer was used to excite a machine tool structure, with measurement of the transient input and response on an FM tape recorder. Tape loops of the T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_10, © The Society for Experimental Mechanics, Inc. 2011
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transient responses were played into a Transfer Function Analyzer (TFA). The input and response signals were processed by using the tracking filters in the TFA to filter and ratio the response to the input signal, thereby estimating the frequency response between the input signal and the response signal. This method proved to be impractical due to signal-to-noise problems. However, it triggered the investigation of other methodologies including Fourier analysis. A prototype Real Time Analyzer was made available by Spectral Dynamics Research Corporation in the late 60s, and this single channel spectrum analyzer was used to estimate the response spectrum to an impact on a machine tool base. This spectrum measurement had good agreement with the response spectrum estimated from the measured FRF using sinusoidal testing. Based upon this result, a serious effort was initiated to develop a measurement process which would use the newly developed FFT algorithm to estimate an FRF from the FFTs of the digitized input and responses signals. In 1968/69 the large Applied Dynamics computer located in Department of Electrical Engineering at the University of Cincinnati was used to develop a software program which used the analog part of the hybrid computer to digitize the force and the response (accelerometer) signal measured by testing a machine tool base. The IBM 1130 computer (Digital part of Hybrid computer) was used to compute FRFs and Coherence Functions. These measurements were compared to FRF function measured with a Spectral Dynamics Transfer Analyzer and the comparisons were good. The hybrid computer filled a complete room. As a result, only small test objects could be taken into the computer room to be tested in real time. For large test articles, the measurement data had to be recorded on an AM and/or FM tape recorder and this recorded data were processed with the hybrid computer. In general, this was the same in other organizations where data was recorded and processed in their computer centers. In the late 60s and early 70s, small minicomputers systems manufactured by Hewlett Packard and the Time Data Corporation (DEC Computer) became available which were portable and utilized Fourier analysis. These systems could be located next to the test article, making laboratory and field testing practical. These systems were an important step in the evolution of measurements from the analog to digital arena. 7UDQVLHQW7HVWLQJ'HYHORSPHQWV The advent of the portable Fourier Analyzer system totally revolutionized the experiment measurement arena in the early 70s. The revolution was the ability to measure power spectrum and frequency response functions using a wide variety of difference signal types (sinusoidal, random and transient). It should be noted that a portable two channel system in the early 70s was the size of a large TV or small refrigerator instead of a room full of equipment. This paper will concentrate on the evolution of testing methods utilizing transient signal types. It will briefly itemize important developments of the 1970s and 1980s, and will concentrate mainly on the developments in the late 1980s thru the 2000s.
V±'HYHORSPHQWVLQ7HVWLQJ3URFHGXUHVDQG'LJLWDO6LJQDO3URFHVV '63 The University of Cincinnati Structural Dynamics Research Laboratory (UC/SDRL), was loaned a prototype HP 5450 Fourier Analyzer System in 1969/70. The HP 5450 was based upon the HP 2114 8K mini- computer system. This system was used initially to evaluate the potential of using Fourier analysis with transient testing. A graduate student performed an impact test on a small milling machine which had been previously evaluated for its chatter characteristics. The modal characteristics and important Directional Frequency Response Functions (DFRF) had been measured using standard analog techniques of the 60s and these characteristics were measured with by impact testing. The results were very encouraging. 1
2
This testing was discussed in a technical note for Hewlett Packard and a paper presented at a conference at the University of Birmingham in England, including a live demonstration of the impact testing as part of the conference. After the conference, a seminar and a demonstration of impact testing was presented at the
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University of Leuven in Belgium. This was the start of a very high energy effort to develop an understanding of the digital signal process and the measurement aspects of Fourier analysis. This effort spanned the decade of the 70s. Based upon these results, the HP system was updated to a HP 5451A system and a more extensive set of tests were performed by the undergraduate advanced vibration class (class of 71 and 72). Groups of 2 to 3 students were given a project to use impact or step relaxation for testing of a variety of test objects. There were approximately 10 groups per class. Fourier analysis was in its infancy and the student projects actual contributed significantly to the transient testing state-of-art. Some of the contributions of the student projects to transient testing: x Impact Testing R An impact hammer with a built-in load cell was developed. Hammer could easily be roved. R A load cell with an impact surface was mounted on the structure. The load cell measured the transient input force as well as the transient force due the mass added by the load cell. Thus, the mass additive effect was taken into consideration. This testing condition can be used when the accelerometer is roved. R A ratio calibration process was developed for calibrating instrumented hammers. R Tips and mass of impact hammers can be changed to control frequency content of the input. R Moving systems can be impacted. A student group wrote a technical paper describing the use of impact testing to measure the FRF for a rotating spindle with a hydrodynamic bearing where the 3 stiffness of the bearing depended upon the rotation of the spindle . x Unit Step Function (step relaxation testing) R A large machine tool isolation foundation was tested with a step relaxation method and a response ratio method for measuring the mode shapes. The force was not measured and an accelerometer mounted at the input point was used to trigger the data acquisition and to serve as the reference sensor. R A small shear model of a high-rise building was tested using a load cell to measure the input force. The FRFs were computed using the step input. The input force and response channels were AC-coupled to reduce the influence of the DC component. The major activity in the 70s was the development of signal processing techniques to make good measurements with different excitation methods. The secondary effort was the development of parameter estimation methods which could be used to extract characteristic functions (Modal Parameters and Impedance Functions) which could be used to characterize the systems being tested. Some of the important developments related to transient testing in the 1970s are summarized as follows: x
Special Transient Testing Windows R In the early 70’s, the most important signal processing development for impact testing was the development of force and exponential windows. This development was the fallout from work being done by Ron Potter, who was the person at HP primarily responsible for the development of the HP 5450 series Fourier Analyzer System. He was the real GURU of Fourier analysis in the late 60’s and 70s. As part of one of his activities, he was developing a parameter estimation algorithm for extracting modal parameters. He was trying to get starting values for the modal parameters, using the shift theorem of the Fourier Transform to reduce the apparent damping in his FRF measurements. The shift theorem states that multiplying a time function by an exponential function will shift the damping and/or frequency axis in the transformed domain. In other words, the apparent damping of system can be changed in a predictable way, simply by multiplying the unit impulse response measurement of a system by a damped exponential. It became apparent that this could be used as a window to eliminate “leakage errors” and to improve the signal-to-noise of impact measurements. As result,
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an exponential window could be applied to both the input force and the output response signal. An additional force window could be applied to the force signal to eliminate noise on the force channel after the impact. These windows have been documented in a 4,5 number of references . Animated Display R Impact testing was often used for trouble-shooting vibration problems in the field. The ability to visualize mode shapes quickly and conveniently is especially important in the field. Initially, the mode shapes were plotted by hand which was time-consuming. Several plotting routines were developed in which the mode shapes could be plotted in the field. Several of these plotting programs would generated 3D images that could be viewed with special training.(Crossed-eye 3D images). In general, this was not completely satisfactory since many people had difficultly viewing these images; as a result, this technique was quickly abandoned as a functional method. Later, there was development of an animated display for the HP 5451B system, in which the mode shapes could be viewed directly on the 5451 display. This was revolutionary. It used: Completion Algorithms: In testing, there were often only a few points defined on a given component and measurement at a given point in only a few directions. These conditions led to confusing displays. These findings led to the development of methods where the measured DOFs could be used to estimate missing DOFs. One method was to estimate rigid body properties for a rigid component or for a section of the component which behaved in a rigid fashion. The rigid body properties were estimated from the measured data and the resulting rigid body characteristics could be used to interpolate and predict the response for missing Degrees-of-Freedom (DOFs). A second method used a Slave DOF (a point whose motion is the same as a measured point) were used to estimate missing DOFs. Completion algorithms were particularly important for impact testing since it is difficult and/or impossible to make measurements at certain points and in certain directions. Reducing Periodic Noise R One of the negative aspects of the exponential impact window is that periodic noise components are smeared by the window over a frequency range band centered around the frequency of the periodic noise. In an initial effort in the 70’s, the DC component and frequency components that were periodic in the window were filtered by taking the FFT of both input and response channels and setting the Fourier coefficient of the noise components to the mean values of the adjacent Fourier coefficients. The data were then transformed back into the time domain. The force/exponential window could then be applied to the force and response signals. Impact Hammers R A modally-tuned hammer was developed in which the influence of the modes of the hammer was controlled to reduce artifacts in the FRF measurements due to the hammer dynamics. A wide range of impactors were developed, ranging from very small hammers to large masses whose weight could exceed hundreds of pounds and could be used as a pendulum to impact large test objects. Parameter Estimation for transient testing methods, particularly trouble-shooting applications were normally restricted to simple Single Degree-of-Freedom (SDOF) methods, with quadrature being the most popular. However, Multiple Degree-of-Freedom (MDOF) algorithms were developed during the 70’s, including: 6 R Complex Exponential Algorithm (CEA) 7 R Ibriham Time Domain Algorithm(ITD) 8 R The Least Squares Complex Exponential Algorithm (LSCE)
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V±'HYHORSPHQWV±0XOWLSOH&KDQQHO'DWD$FTXLVLWLRQDQG3DUDPHWHU (VWLPDWLRQ The major innovations in transient testing methodology were accomplished in the 60s and 70s. In the 80s, developments in data acquisition and parameter estimation led to the significant advancements in Multiple Reference Impact Testing (MRIT) of the 90’s. 'DWD$FTXLVLWLRQ'HYHORSPHQWV V It became clear in the 70s that in order to improve modal parameter estimation, it would be necessary to develop multiple reference parameter estimation algorithms and affordable multiple channel data acquisition systems. One of the major problems with parameter estimation was the consistency of the measurement database. Measurements taken at difference times from different reference points were inconsistent. As a result, the estimate modal parameters were inconsistent, even when the fit to individual measurement appeared to be excellent. In order to address this problem, a large multiple channel affordable data acquisition system would be required. In the late 70s the groundwork for the application of a multiple channel acquisition system was developed with the formulation of the Multiple-Input-MultipleOutput (MIMO) FRF technique. Initially, a four channel system with two inputs and two outputs was used, followed by testing with an eight channel system with two inputs and six roving response channels. Several vehicles were tested in this manner with encouraging results. The dream was a system where with two to four inputs and hundreds of responses could be measure simultaneously and multiple reference parameter estimation algorithms could extract modal data from this set of measurements. 3DUDPHWHU(VWLPDWLRQ'HYHORSPHQWV V In the early 1980’s, with the breakthrough development of the Poly Reference Time Domain Algorithm 9 (PTD) the parameter estimation part of the dream came true. This was followed by the development of the 10 Eigenvalue Realization Algorithm (ERA) a few years later. The PTD algorithm was a multiple reference version of the Least Square Complex Exponential Algorithm and the ERA algorithm was effectively a multiple reference version of the ITD method. The PTD method could run effectively in a small mini computer system, however, in the early 80s the ERA required a larger mainframe computer. As a result, two difference groups of users were employing the two methods: NASA was the primary user of ERA; industrial users (machine tool, auto companies, etc) used the PTD method. By the mid 80’s, a more general unifying approach to the parameter estimation was being developed. The Unified Matrix Polynomial Approach (UMPA) concept was being developed in the late 80s and early 90s. Using the UMPA concept all of the important parameter algorithms could be rederived from a common starting point. Important mathematical techniques like Singular Value Decomposition (SVD) became a significant part of these developments. In fact, a parameter estimation procedure based on SVD, the Complex Mode Indication Function (CMIF), was developed in the late 80s and perfected in the early 90s. It became a standard parameter estimation tool used with Multiple Reference Impact Testing (MRIT) in the 90’s 'DWD$FTXLVLWLRQ'HYHORSPHQWV V The 2nd part of the dream, the ability to measure hundreds of channels simultaneously, took a little bit longer to develop. In 1981 Boeing Aircraft Company conducted the 1st large scale modal test of the Boeing 767, where up to 128 channels of responses could be measured simultaneously. The raw time data was recorded to a large disk file and was post-processed into FRF measurements. This was the ideal case; by recording the raw data, it was available for post-processing after the test object was released. Different
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signal processing could be used to enhance certain aspects of the analysis, for example, zooming into a certain frequency band to enhance an important target mode. Unfortunately, this type of data acquisition was too expensive in the early 80’s for most users. The development of the Struccel system11 in the mid 80’s significantly reduced the cost of the multiple channel sensor systems. The widespread demand for CD players and further developments in digital music led to mass production of delta sigma ADCs which, by the early 90s, were available and very inexpensive. This made it possible to design a very inexpensive multiple channel data acquisition system. The possibility of conducting a test with hundred of channels became practical in the 90s for groups outside of the Aerospace and Auto industries.
7UDQVLHQW7HVWLQJ VWR3UHVHQW Advances in state-of-the-art data acquisition and modal testing can be clearly illustrated with a comparison of impact testing between the 1970’s and 2000s. In the 1970s, a two-channel Fourier Analyzer System cost approximately seventy five to one hundred thousand dollars, and senior undergraduate engineering students used this system to measure the modal parameters of a guitar. In 2002 a 13 year old middle school student competing in a science fair was given a small instrumented hammer and an inexpensive accelerometer. He programmed the sound board in his six hundred dollar PC to collect data from a guitar using impact testing with a pseudo random sequence of impacts. Given access to MATLAB, minor help from his father, and a MATLAB animation program he duplicated the effort of the students of the 70’s and was a major winner in his science fair. This comparison demonstrates the clear advancements in the cost of data acquisition and the availability of powerful computational tools. An ordinary PC of the 2000’s is much more powerful than the most powerful main frame of the 70’s; data acquisition costs have been reduced by orders of magnitude; new parameter estimation with powerful new mathematical tools and concepts were available by the early 2000s. 7UDQVLHQW3DUDPHWHU(VWLPDWLRQ3URFHGXUHV VWR3UHVHQW Obviously, any parameter estimation procedure developed to extract modal parameters from measured FRF can be used with FRF measured with transient testing. With the development of rather inexpensive and portable data acquisition systems in the early 90s, Multiple Reference Impact Testing (MRIT) became practical. In MRIT a large number of excitation points are used in the testing. This significantly increases the amount of redundant spatial information available in the parameter estimation process, which improves the possibility of uncoupling closely-coupled modes. The measurement of this enhanced multiple reference database led to the development of specialized multiple reference parameter estimation algorithms for transient testing in the early 90s. In the late 80s, an inexpensive multiple channel sine testing system was developed that took advantage of the additional spatial information. This testing method was identified as the Spatial Sine Testing Method (SST), where a large number of electro-mechanical exciters were distributed on a structure and, at a given frequency, a number of forcing patterns were used to excite the structure. This process was repeated at a number of frequencies, and the forcing vectors and resulting response vectors were processed with a 1st order UMPA model to estimate the modal parameters. Details of this method can be found in references 12. In the early 90’s, a major infrastructure testing program was initiated by the Department of Transportation (DOT) in the State of Ohio, USA. This program was initiated to develop a procedure for testing mediumsized bridges typical of those found on Interstate Highways for accidental damage due to seismic, flood, and/or accidental impact from large vehicles. The standard inspection method includes visual inspection and measurement of static deflection due to large loads. The initial effort used changes in the bridge’s modal parameters as a measure of its health. This study had limited success. It was discovered that significant measured changes in discrete modal parameters were not sensitive enough to reliably predict the health of a bridge. In fact, the deformation of the bridge to static loads was a better indicator of significant bridge damage. The static deformation is a measure of the
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simultaneous contribution of a number of bridge modes. Unfortunately, it was easy to load the bridge but was difficult at the time to measure its static deformation. Today, with some of the newer laser systems available, it may become more practical. The proposed solution was to experimentally measure a modal model of the bridge and predict its flexibility to a variety of diagnostic loading conditions. This required the ability not only to get estimates of the bridges eigenvalues and eigenvectors but to generate a scaled modal model. A bridge was located which was scheduled to be demolished. The bridge was in reasonably good shape, but was being demolished because the road it serviced was being retired. Figure 1 is a picture of the bridge. The first phase of the project was to determine the best testing procedure. The Civil Engineering Department of the University of Cincinnati was responsible for the project and they had access to both large hydraulic, electromechanically and impact exciters systems. An initial study indicated that a MRIT test was the best testing method based upon the requirement for a fast testing cycle. The bridge testing process used is described in a following section of this paper. See 13,14 reference for several technical references on this project.
)LJXUH± Seymour Street Bridge
A MRIT testing procedure was used to generate multiple reference FRF data sets and the commercial parameter estimation of the early 90s were used to process these data sets with little success. The criterion for success was to be able to predict the static deflection of the bridge to statically applied truck loads. A special load frame was built to measure the static deflection of the bridge using an array of potentiometers connected between bridge and the load frame. The initial expectation was that the static deformation would only require getting good estimates of the lowest ten or so modes of the bridge and was assumed to be possible. Unfortunately even for the lower modes the problem of trying to sort out good estimates of the modal parameters from computational parameters was determined to be impractical using the MDOF parameter estimation available at the time. It was not possible to generate a modal model which gave results that correlated well to the measure static deformations. Fortunately, a technique developed in the mid to late 80s as a mode indicator function (CMIF) was modified to give estimates of the eigenvectors; then, using the estimates of eigenvectors as modal filters, to estimate an Enhanced Frequency Response Function (EFRF). The eigenvalues and the modal scale factor could be extracted from the EFRF.
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)LJXUH± CMIF Plot Seymour St. Bridge
The resulting eigenvalues, eigenvectors and modal scale factors defined a modal model of the system. The modal model generated with this method successfully predicted the static deflection of the bridge to truck 15 loads. A complete description of this method is documented in the following thesis . Figure 2 shows a
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typical CMIF plot for the bridge, and Figure 3 shows a comparison between predicted static deflection of the bridge due to truck loads and measured deflection. It should be noted that the static deflection was not measured completely across the bridge, using potentiometers, because of the difficulty in constructing a load frame the length of bridge with traffic flow under the bridge.
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)LJXUH – Typical Comparison between static deformation due to truck loads and predicted deformation from modal model For cases in which the modal density makes it difficult to use the CMIF method to extract the modal parameters, a method developed in the late 80s as part of the Spatial Sine Testing development was modified for MRIT testing. This Enhanced Mode Indicator Function (EMIF) method utilized a 1st order frequency domain UMPA parameter estimation model. The mathematical formulation of the model is shown in the following equation:
ª¬( j Z ) ª¬ A º¼ ª¬ A ºº ^ X (Z )`k 1 0 ¼¼ Where
[Ai] [Bj] {X(Z)} {F(Z)} M K
ª M¦ ( j Z )m ª B º º ^F (Z )` k ¬ m ¼ »¼ «¬ m 0
= Response Coefficient Matrix = Input Coefficient matrix = Response Vector = Input Vector = Order of Input polynominal = Index for kth equation
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The number of eigenvalues estimated by the 16 1st order UMPA (see references ) was fixed based upon the number of eigenvalues observed to be active in the narrow frequency band of interest, using the CMIF plot of the quadrature FRF responses. The FRF matrix is a 3D matrix of the (inputs times the response times the frequency) for the structure being tested. The EMIF eigenvalues are an average value of the estimates of the eigenvalues where the A0 and A1 were used to normalize the UMPA model in a least squares sense. This is similar to the normalization used in the (H1 or H2) FRF estimation process. For a particularly difficult region of the CMIF plot (see Figure 4), the CMIF plot for the fit region of the measured FRF matrix and the synthesized FRFs that were estimated using the EMIF algorithm is shown for comparison. The correlation is very good.
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)LJXUH – EMIF reconstruction of band 1 shown in Figure 1 for the Seymour St. Bridge
In the 2000s, better formulations of the over-specified UMPA models and automated methods for sorting of the computational and real eigenvalues makes it possible for novices to use commercial software to obtain acceptable modal parameters from MRIT testing of complex structure such as the bridge testing of the early 17 90s. The general UMPA model formulation is given below for the time and frequency domains
UMPA Models -- General Formulation for kth equation
n
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¦ > A @^x(t )` ¦ >B @^f (t )` k i
i
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= time = scaled frequency (if s=jZ then Z is the unscaled frequency) = response Coefficient Matrix where responses can be virtual = input Coefficient Matrix = response vector – time response = input vector – time response = response vector – generalize responses can include states = input vector – generalize response = order of input polynomial = order of response polynomial = number
In the 2000s, there were several significant advancements in the implementation of the parameter estimation processes which made it easier for the user. One of the biggest problems for the users involved selection of a realistic set of eigenvalues from an over-determined set of estimates. In general, the UMPA
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model is drastically over-specified in order to help reduce the influence of noise. The process increases the order of the UMPA model which likewise increases the estimated number of eigenvalues. The real eigenvalues of the system is a subset of the estimated eigenvalues. The solution process involves: 1. selecting a particular UMPA model a. Time Domain—Equations are very well conditioned b. Generalize Frequencies Domain i. Unscaled frequency -- very poorly conditioned ii. There has been much research on how to scale the frequency in order to improve the conditioning of the solution. 1. Normalized Frequency – Frequency is scaled by maximum frequency so frequency values are between plus and minus one 2. Orthogonal polynomials -- effectively reduce the order of the polynomial. 3. Reduce the order of the polynomial by state-space-expansion or by measuring more response points. 4. Complex z mapping – effectively maps the data to the unit circle, or uses the equivalent of the inverse Fourier Transform, which effectively maps the frequency domain data to the scaling of the time domain. As a result, the conditioning is similar to the time domain. c. High order model d. Low order model 2. solving a set of linear equations for the coefficients of UMPA model a. There are many more equations than unknowns, so that a pseudo inverse solution is required i. Selection of objective function (function for minimizing the error) 1. Normalizing equations so that [An] equals the identity matrix. In this case the damping is over-estimated, in other words, the estimated eigenvalues appear to be more heavily damped. In general, this makes it more difficult to sort out the computational modes. 2. Normalizing equations so that [A0] equals the identity matrix. The damping of the eigenvalues is under-estimated. This makes it easier to sort out the computational modes. In fact, many of the computational modes show negative damping, which is unrealistic, making it easy to reject these modes. As a result, this is often the objective function used. One of the negative aspects is that very lightly damped system modes will often have negative damping. 3. A total least squares objective function can be used. ii. Least squares Solutions 1. normal equation solution 2. total least squares – eigenvalue solution 3. etc. iii. Using transformations 1. SVD 2. LU, etc 3. solving for the eigenvalues/eigenvectors of the UMPA model 4. filtering the estimated eigenvalues into system eigenvalues and computational eigenvalues due to the over specification of the model. This was an area which had the biggest impact in the 2000’s in terms of the usability. This is also the step where there is a little black magic or art involved in the process. Every vender of commercial parameter estimation software has made an effort to make this step more intuitive or possibly more autonomic. It should be noted that steps 1-3 in the solution process and the characteristics of the FRF database have big influence on the filtering process. In this paper we will not go into detail on the many methods for filtering the data but will concentrate on methods that work well for MRIT testing.
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With MRIT testing, a large number of excitation points are typically used during the test. For a typical MRIT test, ten to thirty impacting locations are often used. For the previously-described bridge, 15 excitation points were used and up to 128 response points were measured. As a result, there is significantly more spatial information available from a MRIT test. This spatial information can be used to help filter out computational information. The methods used in the early 90s for the bridge test were the CMIF and/or the EMIF method. The CMIF method is a purely spatial domain method where the eigenvectors are estimated from the SVD of the FRF matrix. The EMIF uses a reduced first order frequency domain UMPA model which can solve for a fixed number of eigenvalues in the frequency range of interest. In the early 2000s an alternate method using a more conventional UMPA model approach can be used with the MRIT database. In general, a first order UMPA model in either the time or frequency domain can be used. In the time domain the UMPA model is the equivalent of the ERA algorithm. The first order ERA algorithm is generated by a state expansion of the second order equations of motion. For the bridge example with 128 response points, the UMPA model would have a solution with 256 eigenvalues and the eigenvector would be a state space vector with length of 256. The state space vector is the system’s eigenvector augmented by the system’s eigenvector multiplied by its eigenvalues. It should be noted that the UMPA can be further expanded by adding additional state space expansions where each expansion would expand the solution for this example by 128 eigenvalues. This state space vector is referred to as the Extended State Space Vector (ESSV). (See Reference 18) Using the ESSV will generate a model with hundreds of computational modes. In order to filter the computational modes from the systems modes, the correlation between the state space vectors estimated by using the A0 and A1 solutions for the same model can be used. The system modes should be highly correlated and computational modes poorly correlated. The correlation value can be used as the filter cutoff. It should be noted that the correlation computation is equivalent to the Modal Assurance Criteria (MAC) often used to compare modal vectors. The authors of this paper have used this procedure successfully with MRIT testing for trouble-shooting applications since the early 2000s. Specialized MATLAB programs and the X-MODAL program developed by UC/SDRL have been used in the data processing for the MRIT testing. In order, to demonstrate the application of using an ESSV as a spatial filter the following MRIT data set will be used to extract the eigenvectors and eigenvalues with a specialize MATLAB program. It should be noted that this same type of analysis can be done using X-MODAL The data set which will be analyzed was taken from a multiple reference data set taken in the late 70s on a circular plate and has been used as an example in many papers and in many modal courses over the past thirty or more years. The data were taken with a two channel analyzer by mounting a transducer at a point on the structure and impacting at 36 points, moving the transducer to 6 other points and repeating the process. By using reciprocity, this generated a FRF 3D matrix (7 inputs by 7 responses by 512 spectral lines). The CMIF plot for this data set is shown in Figure 5. This data set is a historical data set but the C-Plate structure is a standard test article. C-Plate )LJXUH – CMIF Plot C-Plate type structures have been tested over the past thirty years with every new version of data acquisition and sensors and they have been analyzed with nearly every type of parameter estimation algorithm. This original data set has been used as a test case for evaluating many software packages. It has repeated modes, is lightly damped and the data has a number of inconsistencies. This data set is typical of a
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MRIT dataset. It should be noted that in a more modern test, all the response data would be taken simultaneously, which would generate a more consistent data set. The data were analyzed with a standard UMPA model emulating the ERA Time domain algorithm with an extended state space vector with 7 state space shifts or extensions. This extended model will generate 7x36 or 252 eigenvalues while the CMIF plot indicates that there are 28 eigenvalues. The eigenvalues are filtered by comparing the MAC value between the A0 and A1 solutions estimated eigenvectors. The filter cut-off was set to accept any MAC value greater than 0.95. It should be noted that for this data set, all the selected data have a MAC values greater than 0.99. In Figure 6, a MATLAB pcolor map (252x252) of the MAC values is shown. In Figure 7, a zoomed region is shown so that highly-correlated eigenvalues are visible. The small dark brown spots are highly correlated solutions. The correlated poles (eigenvalues) are plotted in Figure 8 and a zoom of the region around the three highest frequencies modes is shown in Figure 9. For each pole the estimated pole from the A0 and A1 solution is plotted and the average value of the two solutions is plotted. The average value is the value used as the estimate for the pole. From experience, using this method with analytical data sets, the average values are, in general, a better estimate than either the A0 or A1 solutions.
)LJXUH –Pcolor Plot of MAC between A0 and A1 eigenvector estimates.
)LJXUH – Zoom of Figure 6 showing filter poles which appear as small dark brown spots
)LJXUH – Cplate filtered Pole plot – Zoomed area will be shown in Figure 9
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)LJXUH – Zoom of region in Figure 8 showing three closely coupled modes, a repeated mode around 2322 Hz. and a single mode around 2338Hz. The legend shows the pole location for the A0, A1 and the average estimates for each pole. There are two parameters used in the process: 1. 2.
The number of state space extensions: In ERA, a state space extension is often described as adding a virtual measurement set to the data set, where the virtual measurement set is simply a time shift of the previous measurement set. The MAC value for the filter cut-off: In order to select the cut-off, a stability type of plot can be generated where the filtered cut-off is plotted versus pole frequency. As the MAC cut-off is reduced, more modes will become visible. The CMIF plot can be used as a guide to how many modes are active in a given frequency range. However, in the end, it requires engineering judgment to make a good decision as to what is a reasonable cut-off. In other words, like most things associated with modal parameter estimation, experience is important.
For narrow frequency ranges, the generalized frequency equivalent of the ERA algorithm can be used with either unscaled or scaled frequency in the same manner as in the example given above. In the future, a more complete paper on this type of spatial filtering will be developed with several more detailed case histories.
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7UDQVLHQW7HVWLQJ3URFHGXUHV VWR3UHVHQW Two historical transient testing procedures were developed in the 70’s. In one testing procedure the transient inputs were input at one or more points and roving response measurements were taken to estimate the FRFs of interest. In the second method, reference response sensors were mounted at the input points and a roving transient force (normally an impact, but occasionally a step relaxation) was used to excite the points of interest, and reciprocity was used to determine the FRFs of interest. In both cases, transient testing makes it is easy to measure data from a large number of reference points, since there is minimal fixturing and data acquisition involved. As a result, transient testing has become a powerful field testing and trouble shooting tool. It should be noted that historically impact testing has been the transient method of choice, probably greater than ninety percent of the time. In the 1990s/2000s a large MIMO modal test using exciters involves the use of two to four simultaneous exciters and hundreds of response sensors (normally accelerometers). A number of exciter configurations may be used in the overall modal testing program. A single data set consists of the data taken from one of the MIMO configurations. A large MRIT modal test involves mounting potentially tens or hundreds of sensors and impacting at a large number of fixed-input points. A different impacting device can be used at the various input points. One data set consists of a single input and all of the response points. For the MRIT testing, it is not necessary to use MIMO signal processing. Since the MIMO testing procedure is a more controlled test, it has been the method of choice for large laboratory modal testing. However, in the pretest phase, impact or transient testing has often been used to get an initial set of data: x to determine or check exciter and sensors locations; x to get a initial estimate of frequencies, damping and modal density; x to identify potential local modes which can present a problem during the testing; x to identify rattles, clearances, or other local noise sources or non-linearities. In 2003, “A New Concept: Ground Vibration Testing (GVT)” was conducted at Boeing Aircraft (Figure 10) to evaluate a number of GVT concepts: x Evaluation of using a reduced suite of sensors with the existing modeling technology for modal model verification. x Evaluation of transient testing methods as a primary modal testing method. R MRIT Impact Testing R Step Relaxation testing – Transient Force was not measured; in most cases, a known static force was applied and then released by suddenly cutting the static restraining line. R Evaluation of amplitude dependant modal parameters )LJXUH New Concepts GVT (Non-linear ID) x Evaluation of a new digital sensor system (the DSS system), which drastically reduced cabling requirements. x Evaluation of testing the aircraft’s on it’s landing gear with reduce tire pressure acting as a soft support. x Evaluation of the noise floor of current generation sensors.
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The results of this program were encouraging concerning the use of a reduced sensor set and the possibility 19 of using transient testing procedures. A summary of the results are summarized in a 2004 IMAC paper . It should be noted that impact testing has been used historically not only as a pretest procedure during a GVT but also for documenting aircraft components such as: x Control surfaces which have been tested using impact testing by several aircraft vendors. x Stores configuration have been tested both by the Air Force and by vendors.
6HQVRUDQG'DWD$FTXLVLWLRQ'HYHORSPHQWV WR3UHVHQW Data acquisition improved significantly in the late 80s and early 90s with incorporation of inexpensive 24 bit delta sigma ADCs in data acquisition. Fairly portable multiple channel acquisition systems and powerful notebook computers which are well suited to MRIT testing also became available. Eight to thirty two channels systems can conveniently be transported and used into the field. These systems make trouble shooting or testing of infrastructure in the field practical. Depending upon the application, either a roving input or response could be used. In 2000 a prototype data acquisition system, the Digital Sensor System (DSS), was developed. In it, a number of sensors could be mounted along a single wire. This system was demonstrated in the early 2000s at IMAC, JMAC, ISMA and several other conferences, and was demonstrated at a number of organizations internationally. It significantly reduced the cabling problems for conducting a high channel count modal test. It was small and portable and worked well for the MRIT applications. The digital components in the prototype system would not fit into a small modal sensor so it required a small patch panel (Digital System Interface Transmitter -- DSIT) to be mounted along the cable to interface with the sensor. A prototype seismic digital sensor which could be mounted on the cable was developed; however, in order to build smaller sensors, it would require the development of an Application Specific Integrated Circuit (ASIC). To develop the ASIC and/or to further commercialize the existing system was judged to be too expensive by the developers. As a result, the DSS system development was stopped in the mid 2000s. A steady improvement in the cost, sensitivity and size of sensors has improved from the 70s with low cost, high sensitivity and small size being desirable. The fantasy of every test engineer has been wireless sensors. Wireless systems with a limited number of channels are commercially available. However, for the large channel count applications with hundreds of sensors, the technology is still not practical. A wireless hammer channel would certainly be desirable for the roving hammer applications and several hammers of this type have been developed but have not been widely accepted by the marketplace. /DWHVW7UDQVLHQW7HVWLQJ0HWKRGRORJLHV The impact of the current state-of-art in sensors, data acquisition, parameter estimation and testing techniques on several of the prominent historical applications of transient testing will be examined in this section. 7URXEOH6KRRWLQJDQG)LHOG7HVWLQJ One of the first applications in which transient testing techniques were used was the trouble shooting of vibrations and acoustics problems (forced and self excited) in the field. The testing techniques have not changed significantly from those used in the 70s. The difference is in the number of channels of data acquisition and the improvements in sensors. There are more channels of acquisition and the ADCs have 24 bits of dynamic range. Historically, auto ranging of the data acquisition was important for obtaining good measurements. This was a very time consuming process, particularly when a number of acquisition channels were used. The 24 bit ADC tremendously reduces the magnitude of this problem. The newer generations of IPC sensors also have an improved dynamics range which improves the process. Data acquisition software has been modified such that if one of the response channels is overloaded during an impact, the data from the channel is rejected and the gain of the overloaded channel is reduced for the
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next impact. Channels that are not overloaded are averaged. As a result, the number average taken per channel is different, but good data is not rejected for the channels that are not overloaded. If the input is overloaded then all the channels are rejected and the input channel gain is reduced for the next impact. Some software packages have the capability of processing the data in real time, but also saving all the impact data for post processing. If necessary, the saved raw data can be reprocessed at a later time for data enhancement if there are data analysis questions. For testing a large piece of infrastructure like the bridges described in the previous section hundreds of sensors can be mounted on the structure and the structure can be impacted at ten to thirty input points. For the bridge testing where testing time is important, the sensors are premounted on the cabling and can be unwound and located on the bridge in a very short time. The bridge is impacted at 10 to 20 points to take a complete data set and the cables are rewound and the testing is move to the next bridge. For large trouble shooting projects where it is difficult to find good impacting points, a small array of tri-axial accelerometers are roved over the structure. The tri-axial sensors are located on different regions of the structure so that it is fast and easy to relocate the tri-axial sensors for the next measurement cycle. The tri-axial sensors are used in order to get a good three dimensional animate display of the mode shapes during the testing process. Visualization is important for trouble shooting, since it is a very interactive process where new points are often selected on-the-fly. A portable data acquisition system with 8 to 32 channels is typically used for these applications. A 32 channel system can be configured with 10 roving tri-axial accelerometers and one fixed reference accelerometer located at an important point on the structure. For each measurement configuration, 5 to 10 common input points are impacted. Standard MRIT signal processing for 12 each measurement cycle is used . The modal information is also processed on-the-fly using the CMIF parameter estimation process, where mode shapes are quickly animated to be used as feedback, for selecting new response locations for the next pass. The raw impact data should be save for post-processing after the test is completed to )LJXUH—Trouble shooting test to solve self enhance the understanding of the problem. excitation problem. The problem was due to thermal baring of the rubber roll in a paper Calender For trouble shooting, an iteration process is often Stack. performed where a fix is proposed and quickly implemented, and a second test performed. A picture of a typical trouble shooting project is shown in Figure 11 to illustrate an industrial testing environment. In these environments roving the hammer is often difficult and dangerous.
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/DERUDWRU\7HVWLQJ9HULI\LQJDQGRU*HQHUDWLQJ6\VWHP0RGHOV In the past 10 to 15 years, an increasing number of MRIT tests have been performed in the laboratory on system components (frames, bodies-in-white, engine blocks, etc.) with some success. As was the case with the trouble shooting discussion, the impact testing techniques and signal processing have not changed significantly since the 70’s, only the number of simultaneously measured channels. The vast amount of spatial information that is available using the MRIT testing procedures significantly improves the ability of the modal parameter estimation procedures to extract a good modal model of these components. In the laboratory testing, most of the components have been tested by roving the input, mounting a number of reference accelerometers, and using reciprocity. Since with impact testing, it is not possible to impact tangential to the surface of the test article, completion algorithms are often used to determine missing DOFs and/or to interpolate and estimate the modal data at unmeasured points. An example of this is taken from the System Dynamics Analysis (SDA) Course sequence at the University of Cincinnati. A modal test of an HFrame structure is performed by the students in the course using MRIT and a completion algorithm to generate a modal model of the HFrame structure with six-DOF at important connection points. A typical mode is shown with the mode shape completed at the measurement points in Figure 12. In Figure 13, the same measured mode is shown where the data are interpolated to points common to some of the nodes in the FEM model using the completion algorithm. In this figure the modal coefficients at 90 percent of the points are interpolated from a small subset of measured DOF. Six-DOF information is estimated by the completion algorithms at each of the displayed points on the end masses.
)LJXUH Competed mode shape at measured points.
The students then use the experimentally )LJXUH Completed mode shape at estimated modal model to predict modifications interpolated points made to the HFrame. They also use the modal model to verify a Finite Element Model (FEM) of the HFrame, which is built by a subset of students in the SDA class who are also taking the FEM course.
&RQFOXVLRQV Impact testing was one of the first applications of using Fourier analysis in the area of Structural Dynamics and has a long history as a method well suited for field testing and trouble shooting. In recent, years it has been used more frequently as a method for developing and validating modal models. It has advantages both in the laboratory and in the field due to simplified test set-up and the ease of measuring a data set with significant amounts of spatial information. The additional spatial information has benefits in developing
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and/or validating a modal model. It may not replace a well controlled MIMO laboratory test but it may certainly augment the capabilities and speed up the testing in the laboratory. It is clear that transient testing has many historical application and the advancement in sensor, data acquisition and data processing is expanding its horizons.
5HIHUHQFHV 1. “Dynamic Testing of Mechanical Systems Using Impulse Testing Techniques”, Application Note 140-3, Hewlett Packard Corporation, 1972 2. Morse, I. E., Shapton W. R., Brown, D. L., Kuljanic, E. “Application of Pulse Testing For Determining Dynamic Characteristics of Machine Tools”., 13th International Machine Tool Design and Research Conference, Universoty of Biringham, England, 1972 3. Allemang, R.J., Powell, C., Graef, T., “Dynamic Characteristics of Rotating and Non-Rotating Machine Tool Spindles”, ASME Paper No. 73-DET-29 4. Brown D. L, "Grinding Dynamics", PhD Thesis, Department of Engineering, University of Cincinnati, 1976 5. Brown, D.L., Halvorsen, W.G.,"Impulse Techniques for Structural Frequency Response Testing", Sound and Vibration, pp. 8-21, Nov. 1977. 6. Spitznogle, F.R., "Representation and Analysis of Sonar Signals, Volume 1: Improvements in the Complex Exponential Signal Analysis Computational Algorithm", et al,Texas Instruments, Inc. Report Number U1-829401-5, Office of Naval Research Contract Number N00014-69-C-0315, 1971, 37 pp. 7. Ibrahim, S. R., Mikulcik, E. C.,"A Method for the Direct Identification of Vibration Parameters from the Free Response," Shock and Vibration Bulletin, Vol. 47, Part 4, 1977, pp. 183-198. 8. Allemang, R.J., Zimmerman, R.D., Mergeay, M., "Parameter Estimation Techniques for Modal Analysis", Brown, D.L., SAE Paper Number 790221, SAE Transactions, Volume 88, pp. 828-846, 1979. 9. Vold, H., Rocklin, T., "The Numerical Implementation of a Multi-Input Modal Estimation Algorithm for Mini-Computers," Proceedings, International Modal Analysis Conference, pp. 542-548, 1982. 10. Juang, J.N., Pappa, R.S., "An Eigensystem Realization Algorithm for Modal Parameter Identification and Model Reduction", AIAA Journal of Guidance, Control, and Dynamics, Vol. 8, No. 4, 1985, pp. 620-627. 11. Poland J. B., ”An Evaluation of a Low Cost Accelerometer Array System – Advantages and Disadvantag es”, Master of Science Thesis, University of Cincinnati, Dept of Mechanical and Industrial Engineering, 1986 12. Fladung, W.F., "Multiple Reference Impact Testing", MS Thesis, University of Cincinnati, 1994. 13. Catbas, F.N., Lenett, M., Brown, D.L., Doebling, C.R., Farrar, C.R., Tuner,A., “Modal Analysis of Multi-reference Impact Test Data for Steel Stringer Bridges”, Proceedings IMAC, 1997 14. Lenett, M.,Catbas, F.N., Hunt, V., Aktan, A.E., Helmicke, A., Brown, D.L., ”Issues in MultiReference Impact Testing of Steel Stringer Bridges, Proceedings, IMAC Conference, 1997 15. Catbas, F.N., “Investigation of Global Condition Assessment and Structural Damage Identification of Bridges with Dynamic Testing and Modal Analysis”, PhD. Dissertation University of Cincinnati, Civil and Env. Engineering Department, 1997 16. Fladung, W.F., "A Generalized Residuals Model for the Unified Matrix Polynomial Approach to Frequency Domain Modal Parameter Estimation", PhD Dissertation, University of Cincinnati, 146 pp., 2001. 17. Brown D. L., “Review of Spatial Domain Parameter Estimation procedures and Testing Methods”, Proceedings, International Modal Analysis Conference, 23pp., 2009 18. ”A First Order, Extended State Vector Expansion Approach to Experimental Modal Parameter Estimation", Brown, D.L., Phillips, A.W., Allemang, R.J.,Proceedings, International Modal Analysis Conference,11 pp., 2005. 19. Pickrel, C. R., Foss, G.C, Phillips, S; Allemang,R.J.,Brown, D. L.,”New Concepts in Aircraft Ground Vibration Testing”, Proceedings, International Modal Analysis Conference, 6 pp., 2004
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Fiber Optic Strain Gage Verification and Polyethylene Hip Liner Testing Lucas Chavez1, Michael Martin2, Stephen O. Neidigk3, Phillip Cornwell4, R. Michael Meneghini5, Joe Racanelli6 1
New Mexico Institute of Mining and Technology, Socorro, NM, 87801 Texas A&M University, College Station, TX, 77840 3 University of New Mexico, Albuquerque, NM, 87106 4 Rose-Hulman Institute of Technology, Terre Haute, IN 47803 5 New England Musculoskeletal Institute, University of Connecticut Health Center, Farmington, CT 06030 6 Stryker Orthopaedics, Mahwah, NJ 07430 2
Abstract To optimize stability in total hip arthroplasty, the use of larger femoral heads necessitates a polyethylene liner of reduced thickness. An understanding of the mechanical properties, particularly resistance to fatigue failure, of highly-crosslinked polyethylene is critical to determine the optimal parameters for clinical use. The primary purposes of this study were to characterize the X3TM highly cross-linked polyethylene (HCLPE) liner peripheral face strain field in multiple orthopaedic acetabular shell constructs under physiological loading and to evaluate the usefulness of fiber optic strain gages in this type of biomedical application. The first phase of this study involved measuring X3 HCLPE material properties in tension and compression using uniaxial fiber optic strain gages and resistance based uniaxial and multi-axial (rosette) strain gages to gain greater insight into the complexities and limitations of the use of fiber optic strain gages with X3 HCLPE. In the second phase, physical testing was used to evaluate the effect of HCLPE thickness on the hoop strain field of liner samples of three different thicknesses at three inclination angles and three head offsets that simulate potential in vivo clinical scenarios occurring in hip replacement. The results from these studies will be presented in this paper. 1
Introduction
1.1
Background
Statistics show that between 200,000 and 300,000 total hip replacements occur every year in the United States [1]. With such a large number of people receiving hip replacements, ensuring that these devices function properly is extremely important. One study of hip replacement reliability found that 3.9% of total hip replacement patients dislocated within the first 6 months after surgery and 30% dislocated after 5 years [2]. Several factors contributed to these problems including femoral component head size, acetabular component orientation, and excessive wear of the acetabular polyethylene liner which has been linked to osteolysis [3]. These components must be designed appropriately to withstand such problems in a hostile and dynamic environment. It has been shown that during normal-level walking the hip joint can encounter resultant forces from five to eight times the body weight of the individual [4]. During the swing phase of normal level walking, the fermoral head and acetabular component have been shown to separate up to 2.8 mm [3]. This dislocation can result in less than optimal head/liner contact leading to edge loading the rim of the acetabular liner. This loading condition may lead to increase strain in the liner rim and increased wear. 1.2
Motivation
Studies have been conducted to investigate the dislocation issue, and one potential solution is to increase the size of the femoral head [2]. Results indicated that increasing the femoral head diameter has a twofold advantage. First, an increased femoral head size increases the vertical femoral head displacement T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_11, © The Society for Experimental Mechanics, Inc. 2011
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(VHD) that the head must experience before dislocation can occur. As shown in Crowninshield et al., the VHD increases from 3.2 to 5.8 mm when the head size is increased from 22 to 40 mm with an acetabular component oriented at 45°. Also, when the head size changes from 22 to 44 mm, the prosthetic impingement free range of motion (PIF-ROM) is increased by approximately 30°. However, coupled with these two advantages is an obvious constraint problem, that is, with increased head size and fixed acetabular component size, the liner cup thickness must decrease. Therefore, the threat of liner wear becomes an even more prevalent problem in hip replacement [2]. Hip liners are traditionally made from ultra high molecular weight polyethylene (UHMWPE). This particular polymer has outstanding mechanical properties and has been used in orthopedics as a load bearing material in artificial joints for the last 40 years. The leading factor limiting the longevity of implants made from UHMWPE is wear [5]. The wear properties of the material can be increased by cross-linking through exposure to gamma sterilization. Stryker uses a number of PE materials for hip liners including N2Vac which are machined from compression molded GUR 1020 bar stock, packaged in nitrogen and sterilized using 30 kGy gamma irradiation and is packaged in a nitrogen environment to prevent oxidation. [6] A potential solution to the problem of creating thinner liners with sufficient wear performance is the use of TM highly cross-linked polyethylene materials. One such material is Polyethylene X3 . This material is manufactured by Stryker from a compression molded GUR 1020 sheet which is sliced into rectangular bars. These bars are further processed to received 30 kGy gamma irradiation (Co60 source) followed by an annealing step at 130 °C for 8 hours. This process is performed three times to accumulate a total dose of 90 kGy in these bars. Experiments have shown that this material has a 97% lower wear rate than conventional polyethylene. In addition, this material maintains a high resistance to oxidation — a critical property for material survival in the human body — as well as good mechanical performance over time [7]. Liners with minimum radial thicknesses of four mm have been manufactured using this new material. It is difficult to measure the strains in thin liners or on the thin liner face with conventional methods such as resistance based strain gauges. These transducers are far too large to measure the peripheral strain field on the line face.. Fiber optic strain gauges, however, are extremely narrow. Some are only 0.23 mm in diameter, making them prime candidates for monitoring the strain field of a liner under load [8-12]. Due to their geometry, fiber optic strain gauges can be used in situations where bulkier, larger resistance strain gages are not feasible. Furthermore, they are inert to any type of electromagnetic interference. Fiber optic strain gages can also be integrated into structures such as composite materials, allowing for the measurement of internal strains. The fiber optic strain gages investigated in this study use Bragg Gratings. A Bragg Grating is a “periodic perturbation of the refractive index which is laterally exposed in the core of an optical fiber,” [13] where the refractive index is defined as the ratio of the velocity of light in a vacuum to that in a medium. The grating determines what wavelengths of the incoming light wave are reflected. When this grating is stretched due to a mechanical strain, the grating spacing changes and therefore a different wavelength is reflected. This change can be directly related to the strain of the fiber. 1.3
Purpose TM
The purposes of this study were to characterize the X3 highly cross-linked polyethylene (HCLPE) liner peripheral face strain field in multiple orthopaedic acetabular shell constructs under physiological loading and to evaluate the usefulness of fiber optic strain gages in this type of biomedical application. The first phase of this study was to determine X3 HCLPE material properties, such as elastic modulus and Poisson’s ratio, in tension and compression using uniaxial fiber optic strain gages and resistance based uniaxial and multi-axial (rosette) strain gages. Multiple tension and compression specimens were tested using ASTM standards to determine these properties. The strain in the specimens was monitored using standard resistance based strain gages, fiber optic strain gages, and an extensometer. By comparing the three sets of strain data, the validity of the fiber optic strain gage results was determined. Another goal of these tests was to provide insights into the complexities and limitations of the use of fiber optic strain gages with X3 HCLPE.
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In the second phase, physical testing was used to evaluate the effect of HCLPE linear radial thickness on the hoop strain field of liner samples of three different thicknesses at three inclination angles and three head offsets that simulate potential in vivo clinical scenarios occurring in hip replacement. Liners made from N2Vac and X3 were tested. The peripheral face strain field for each test configuration was monitored using one or two fiber optic strain gages.
2
Experimental Procedure
2.1
Strain Gage and Polyethylene Considerations
2.1.1 Adhesion of Gages Polyethylene is a difficult material to adhere to primarily due to its low surface tension. In order to properly adhere to polyethylene, the adhesive needs to have a lower surface tension than the polyethylene [14]. When this occurs, the adhesive has proper wetting. Figure 1 shows how proper wetting creates more surface area for adhesion. In an attempt to find a suitable adhesive for X3 HCLPE, different types of adhesive and surface preparations were investigated.
Figure 1: Diagram of proper wetting [8]
The first adhesive that was tried was M-Bond 200. This adhesive is commonly used for attaching strain gages, but when used in this application, resulted in very poor adhesion between the strain gages and the X3 HCLPE. This was evident in the stress-strain plot obtained in preliminary testing. The strain measured by the resistance and fiber optic strain gages deviated greatly from the strain measured by the extensometer. The second adhesive tried was Master Bond X17. Although this resulted in what appeared to be a good bond between the gages and the X3 HCLPE, the first tensile test also resulted in very poor correlation between the strain gages and the extensometer results. It was later determined that the shear strength of the epoxy was much too small for adequate strain transfer. Various surface preparations were also tried including Master Bond X17 as a primer with an epoxy as the adhesive, but with no improvement in the adhesion. The next adhesive tried was Barco Bond epoxy, because it has a high shear strength. With Barco Bond epoxy, it was difficult to smooth the epoxy out from underneath the gage, so the gage was measuring the strain in the epoxy and not in the polyethylene [15]. Finally, an adequate epoxy was found called Bondit B-481 TH. Bondit B-481 TH is a two part epoxy with a high modulus of elasticity compared to the polyethylene. This relatively high modulus transfers the strain from the specimen to the gage. In addition, this adhesive has a relatively low surface tension compared to the M-bond 200 and other epoxies. It also is viscous enough to smooth a thin layer of the adhesive under the gage. 2.1.2 Strain Gage Application To apply the resistance strain gages, the surface of the sample was sanded in two directions 90˚ apart to form cross hatching. After the surface was sanded, it was cleaned and degreased using methanol and
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Kimwipes. Gage tape was used in the conventional manner to stick the gage to the tape and orient it on the specimen. The tape was then peeled back to expose the underside of the gage for cleaning. Then, Bondit was applied to the gage. The tape was smoothed back onto the sample by pressing on it with a finger, keeping the gage from moving. To attach the fiber optic strain gage the surface was prepared in the same fashion as the resistance gages. The gage length was located by looking at the wavelength and applying a pressure across the manufacture specified length of the fiber optic. Once located, it was marked and laid across the desired section of the specimen and the ends were taped to the specimen. The gage length was then adhered to the sample using the Bondit. 2.2
Specimen Testing
2.2.1 Tensile Test The first test to be performed was a standard tensile test that was performed in accordance with ASTM D638-08 using a Type I sample with dimensions 6.35 mm (0.25 in.) thick and 165 mm (6.5 in.) long [16]. A specimen in the Instron grips is shown in Figure 2. The tests were run at 2.0 mm/min (0.079 in./min) and at room temperature. This load rate is slower than the ASTM standard in order to obtain more data points. The samples were instrumented with a fiber optic gage mounted longitudinally on one face of each sample and one resistance based rosette strain gage mounted on the opposite side of the same sample. The gages were mounted as close as possible to the middle of each specimen. An extensometer was then attached to the sample. A 5 kN load cell was used to measure the force. The samples were loaded to approximately 4% strain according to the extensometer. Good data from three samples were obtained. More samples were tested, but problems associated with adhering the gages resulted in bad data. 2.2.2 Compression Test
Figure 2: Tension test sample in Instron test machine
The second test to be performed was a compression test that was done in accordance with ASTM D695-08. A 25.4 mm (1.0 in.) diameter, 63.5 mm (2.5 in.) long cylindrical sample was used to minimize the barreling effect [17]. A photo of an instrumented sample is shown in Figure 3. In this figure the strain gage rosette and extensometer are pictured. The test was run at 2.0 mm per minute (0.079 in./min) and at room temperature. One longitudinal fiber optic gage and a resistance strain gage rosette were mounted on each sample 180° apart about the circumference of the cylinder. The gages were mounted as centrally as possible along the longitudinal axis of the cylinder. The compression fixture as shown in Figure 3 was mounted into the Instron testing machine. The longitudinal strain was measured from the fiber optic gage and the longitudinal and transverse strains were measured using the rosette. Good data from three samples were obtained. 2.3
Clinical Testing
Figure 3: Instrumented compression sample
The second aspect of this study was to investigate the hoop strain of various X3 HCLPE liners and to compare these results to finite element results. The liners were inserted into titanium acetabular shells
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that were in potted aluminum fixtures. A test matrix of the tests to be performed is shown in Table 1. A total of nine measurements were taken for each liner size, corresponding to three offsets for each inclination angle. In Table 1 N2Vac refers to Stryker’s nitrogen treated polyethylene. Table 1: Test matrix for clinical testing
Liner Material
Liner Diameter (mm)
Inclination Angle (Degrees)
Separation Offset (mm)
X3 HCLPE
36
45,55,65
0,1,2
X3 HCLPE
40
45,55,65
0,1,2
X3 HCLPE
44
45,55,65
0,1,2
N2Vac
44
45,55,65
0,1,2
Once the liner was in the shell, fiber optic strain gages were mounted on the top surface lip of the liner. The fiber optic strain gage was placed in the middle of the thickness of the lip of the liner as shown in Figure 4. The area was sanded and the gage was placed using similar mounting techniques as was used on the tension and compression samples. On the 44 mm and 40 mm liners only one gage was used. On the 36 mm liners two gages were used. The thickness of the lip was separated into thirds and the gages were attached. Bondit was used in the application of all of the gages.
Figure 4: Gage locations for the 36 mm liner (bottom) and the 40 mm and 44 mm liners (top)
A Cobal Chrome plated stainless steel femoral head was mounted into a 5 kN load cell. A 2450 N force was applied to the liner and maintained for 30 seconds. The liner in the acetabular fixture was held by a movable vice as shown in Figure 5. This vice was used to alter the inclination angle of the mold as well as the offset distance from the load line. The offset was created by placing one or two 1.06 mm thick aluminum pieces between the vice and the aluminum holders. The hoop strain results from testing the X3 HCLPE liners were compared to the results from shells made of Stryker’s conventional polyethylene liner material, N2Vac.
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Figure 5: Test fixture used to hold the acetabular liners showing the offset and angle of inclination
During hip liner testing the compressive load was applied using a displacement of 5 mm/min. According to ASTM Standard D638-08, the speed is to be chosen so that rupture of the specimen occurs within 0.5 to 5 minutes. This standard lists several speeds for particular geometries, and based upon this test’s geometry an estimation was made for the appropriate loading speed for the liner. Shown in Figure 6 is a typical loading curve for the liner, where the peak value is approximately 2450 N. Ϭ ͲϱϬϬ
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dŝŵĞ;ƐͿ Figure 6: Typical loading curve for clinical liner testing
2.4
Data Acquisition
Two data acquisition systems were used in this study. One system used a LabVIEW VI to record wavelength measurements from the fiber optic gages as well as the time stamp of each sample. The front panel is shown in Figure 7. This system tracked the peak reflection wavelength transmitted by the fiber.
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Figure 7: LabVIEW VI used to collect fiber optic gage readings
The second system acquired voltage readings from the resistive strain gages bridge circuit, load readings from the tensile machine, and strain measurements from the extensometer. Time stamps were recorded for each sample. For the subsequent analysis, the time signatures between the two systems needed to be aligned so that meaningful comparisons could be made. The LabVIEW program used to monitor the fiber optic gages was triggered at a specific load reading, either 25 N for the tensile tests or 100 N for the compression tests. A datum was established at this trigger point, and the time signature recorded by the other system at the trigger load was added to this datum. By doing this, all data was converted to the tensile machine time reckoning. 2.5
Calculations
To obtain useful data, the wavelength and voltage readings of the two types of strain gages needed to be converted to actual strain. To convert the wavelength readings into strain, the equation is ௱ఒ
ߝி ൌ ሺఒሻሺǤ଼ሻ
(1)
where ߣ is the wavelength recorded at the trigger point, ߂ߣ is the deviation from this datum wavelength, and 0.78 is the photo-elastic constant. The datum wavelength varied slightly from test to test, depending on numerous factors including the manufacture and possibly pre-stress on the fibers. The conversion equation for the resistance gages is ߝோ ൌ ܸோ
Ǥଶଽହ ଼
(2)
where ܸோ is the voltage output from the resistive strain gages, and the calibration constant is 2.95% strain for every 8 volts produced. Finally, the strain equation for the extensometer is ߝா ൌ
௱ ீಽ
(3)
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where ߂ ܮis the deviation from the datum length of the extensometer, ܩ . Once these strain values were determined, the corresponding stress values needed to be matched. This was not difficult for the resistance gages or extensometer, as the same data acquisition system was used to sample these and the load on the specimen. For the fiber optic readings, the converted time signatures as described above were used to match stress data as taken by the tensile machine with the strain data taken at the same time by the fibers. Stress values were calculated using the engineering, or nominal, stress definition ߪ ൌ
ி
(4)
where ܨand ܣare the load reading and nominal cross-sectional area of the specimen whose normal is parallel to the load line, respectively. Engineering stress and strain plots were generated from the data for each test. To determine a tangent modulus a second order polynomial fit was applied to the extensometer data from 0 to 0.2% strain. Using the equation for this polynomial, the tangent modulus was calculated at 0.02% strain. Poisson’s Ratio was also calculated using the data acquired from the axial and transverse resistance gages using ఌ
ߥ ൌ െ ఌ
(5)
where ߝ௭ and ߝ௬ are the nominal strain values in the transverse and axial directions, respectively. This ratio was calculated for every data point recorded. When the values seemed to reach a constant value, an overall average was calculated. 3
Finite Element Models
Finite element models of the acetabular component were constructed to obtain insight into the anticipated strain values to be obtained from the experiment. Models using linear and non-linear material models were constructed in ANSYS. A cross section of the femoral head, X3 HCLPE liner, and acetabular cup is shown in Figure 8. The model used a single plane of symmetry to reduce the size of the model and the femoral head was modeled as being hollow. The femoral head is shown in red and different loading conditions were considered by offsetting the head normal to the face of the liner. A 2 mm offset is shown in Figure 8. The hip liner is held by a titanium shell that, in a patient, would be attached to the pelvis. A Femoral head 2450 N vertical downward force was applied to the femoral head to model the loading conditions used in the experiment. Previous physical testing of X3 HCLPE provided the necessary properties used in the models. Figure 9 shows an example of typical strain field contours obtained from a finite element analysis. These results are for a 36 mm diameter cup with a 2 mm offset. Figure 9 showns the maximum principle strains. This figure shows that there is a larger strain around the point of the loading. The maximum principle strain for this run was 0.5%. Since the fiber optic gages were to be placed in the hoop direction directly below the point of loading, contour plots of the strain in this direction were also created. Figure 10 showns the strain in this direction. Thus, these values are what were expected to be seen in the experiment.
liner Acetabular cup
Figure 8: Cross section of FEA model
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Figure 9: Maximum principle strain contours
Figure 10: Strain in the z-direction (horizontal)
Finite element runs were made for 36 mm and 44 mm femoral heads with inclination angles of 45° and 65° and an offset of 2 mm. The resulting strain values corresponding to the direction associated with the fiber optic strain gages are summarized in Table 2. As seen in Table 2 the finite element models predict a larger strain in the thin liner than in the thick liner. One unusual result observed in the finite element models, and illustrated in Table 2, is that the thin liner was found to experience a tensile hoop strain, whereas the thick liner had a compressive hoop strain. In Table 2 the strain did not change significantly as a function of inclination angle. This is not to say that the total strain would not change, but rather that the strain that can be measured with the fiber optic strain gages did not change significantly. Table 2: Finite element results for the strain in the direction and location of the fiber optic strain gages. The femoral head was offset 2 mm.
Head Size (mm) Inclination (Degrees) Inner Strain (%) Middle Strain (%) Outer Strain (%) 45
-0.211
36 65
-0.197
45 44
Results
4.1
Specimen Testing
-0.197 0.433
N/A 65
4
-0.168 N/A
N/A 0.433
4.1.1 Tensile Test The first set of tests performed after determining an appropriate adhesive was the tensile tests. A typical stress-strain curve obtained from the tensile test data is shown in Figure 11. The loading and unloading are included in this figure to show the hysteresis associated with this material. As seen in Figure 11, the results from the fiber optic and resistance strain gages correlated very well with the results from the extensometer. The stress-strain curve shown is clearly non-linear with no clear linear elastic region. This result was not unexpected, because X3 HCLPE is a viscoelastic material.
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ŶŐŝŶŶĞĞƌŝŶŐ^ƚƌĞƐƐ;'WĂͿ
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ZĞƐŝƐƚĂŶĐĞ'ĂŐĞ
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ŶŐŝŶĞĞƌŝŶŐ^ƚƌĂŝŶ Figure 11: Typical tensile test result for X3 HCLPE
The long horizontal line associated with the fiber optic strain gage in Figure 11 corresponds to the point where the gage debonded from the sample. The fiber optic gages typically debonded at strains lower than the resistance gages. This is most likely due to the small contact area of the fiber for the epoxy in comparison to the resistance gage. Another peculiar feature in Figure 11 is the vertical line of strain associated with the resistance strain gage at about 3.75% strain. This occurred because the resistance strain measuring system became saturated at 10 V. In Figure 12 is shown the extensometer results from all three tension tests. Clearly the extensometer results from the three tests were very consistent. The loads for tensile tests T1 and T2 were applied up to a strain of 4%, but for test T3 the load was inadvertently only applied up to a strain of 1%.
ŶŐŝŶĞĞƌŝŶŐ^ƚƌĞƐƐ;'WĂͿ
Ϭ͘Ϭϭϰ Ϭ͘ϬϭϮ Ϭ͘Ϭϭ Ϭ͘ϬϬϴ Ϭ͘ϬϬϲ Ϭ͘ϬϬϰ dϭ dϮ dϯ
Ϭ͘ϬϬϮ Ϭ Ϭ
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ŶŐŝŶĞĞƌŝŶŐ^ƚƌĂŝŶ Figure 12: Tensile test results for X3 HCLPE found using the extensometer
Ϭ͘Ϭϰϱ
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The stress-strain curves found for the three tension tests using the resistance strain gages are shown in Figure 13. The resistance strain gage results were very consistent from one test to another. The results from test T1 and T2 only go up to about 3.5%, indicating that the system saturated before the 4% strain recorded by the extensometer.
ŶŐŝŶĞĞƌŝŶŐ^ƚƌĞƐƐ;'WĂͿ
Ϭ͘Ϭϭϰ Ϭ͘ϬϭϮ Ϭ͘Ϭϭ Ϭ͘ϬϬϴ Ϭ͘ϬϬϲ Ϭ͘ϬϬϰ dϭ dϮ dϯ
Ϭ͘ϬϬϮ Ϭ Ϭ
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ŶŐŝŶĞĞƌŝŶŐ^ƚƌĂŝŶ Figure 13: Tensile test results for X3 HCLPE found using resistance gages
The stress-strain curves found for the three tension tests using the fiber optic strain gages are shown in Figure 14. Once again the fiber optic gages gave consistent results from one test to another. The main difference between the fiber optic gage results and those from the other transducers is the maximum strain. For the fiber optic gages, the maximum strain is less than 2.5% because the bonds failed.
ŶŐŝŶĞĞƌŝŶŐ^ƚƌĞƐƐ;'WĂͿ
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Ϭ͘ϬϬϮ Ϭ Ϭ
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ŶŐŝŶĞĞƌŝŶŐ^ƚƌĂŝŶ Figure 14: Tensile test results found for X3 HCLPE using fiber optic strain gages
Ϭ͘ϬϮϱ
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Poisson’s ratio was calculated at every value of strain using the lateral and the transverse resistance strain gages. The calculated value was then plotted against axial strain. Figure 15 shows a typical curve derived using this method. This plot shows the effects of dividing two very small numbers with experimental error during initial loading by the vertical asymptote near the origin. The curve then quickly levels out at ν = 0.408. ϭ͘ϲ ϭ͘ϰ
WŽŝƐƐŽŶΖƐZĂƚŝŽ
ϭ͘Ϯ ϭ Ϭ͘ϴ Ϭ͘ϲ Ϭ͘ϰ Ϭ͘Ϯ Ϭ Ϭ
Ϭ͘ϬϬϮ
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ŶŐŝŶĞĞƌŝŶŐdžŝĂů^ƚƌĂŝŶ Figure 15: Poisson’s ratio for X3 HCLPE calculated from test T3
The material properties determined from the tensile tests are shown in Table 3. The tangent modulus was calculated using the extensometer readings, and Poisson’s ratio was calculated using the resistance gages for all three tests. The average tangent modulus calculated using the extensometer was 1.01 GPa. This value is slightly higher than non-cross-linked polyethylene, which has a tangent modulus of 0.830 GPa [5]. Table 3: Results from tension tests
Property
T1
T2
T3
Average
Elastic Modulus (GPa)
0.990
1.053
0.976
1.007
Poisson's Ratio
0.419
0.491
0.408
0.439
4.1.2 Compression Tests After the axial tension tests were completed, compression tests of X3 HCLPE were preformed. A typical stress-strain curve resulting from the compressive testing is shown in Figure 16. The compressive tests were only performed to a maximum strain of about 2%. In Figure 16 the vertical axis refers to the compressive stress. It is clear from Figure 16 that the results from the various transducers, that is, the fiber optic strain gage, resistance strain gage and extensometer did not correlate as well as they did for the tensile tests. The extensometer results and fiber optic strain gage results were fairly consistent, but the resistance strain gage reported a larger strain for a particular stress. This is most likely due to bonding issues associated with the epoxy. The results of various preliminary tests indicated that the results for the resistance strain gage and the fiber optic strain gage could vary significantly depending on the quality of the gage fixation, including the type of epoxy and the thickness of the epoxy layer. In the test shown in Figure 16, the gages differed by about 2 MPa at the peak strain.
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ŶŐŝŶĞĞƌŝŶŐ^ƚƌĞƐƐ;'WĂͿ
Ϭ͘Ϭϭϰ Ϭ͘ϬϭϮ Ϭ͘Ϭϭ Ϭ͘ϬϬϴ Ϭ͘ϬϬϲ džƚĞŶƐŽŵĞƚĞƌ
Ϭ͘ϬϬϰ
ZĞƐŝƐƚĂŶĐĞ'ĂŐĞ Ϭ͘ϬϬϮ
&ŝďĞƌKƉƚŝĐ'ĂŐĞ
Ϭ Ϭ
Ϭ͘ϬϬϱ
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Ϭ͘Ϭϭϱ
Ϭ͘ϬϮ
Ϭ͘ϬϮϱ
ŶŐŝŶĞĞƌŝŶŐ^ƚƌĂŝŶ Figure 16: Typical compression curves for X3 HCLPE
Three samples were tested in compression and the stress-strain curves resulting from the extensometer data are shown in Figure 17. From this figure it is clear that the extensometer resulted in very consistent results from one test to another. The maximum percent difference between the extensometer results occurred at the maximum strain and was equal to only 0.20%.
ŶŐŝŶĞĞƌŝŶŐ^ƚƌĞƐƐ;'WĂͿ
Ϭ͘Ϭϭϰ Ϭ͘ϬϭϮ Ϭ͘Ϭϭ Ϭ͘ϬϬϴ Ϭ͘ϬϬϲ Ϭ͘ϬϬϰ ϭ Ϯ ϯ
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ŶŐŝŶĞĞƌŝŶŐ^ƚƌĂŝŶ Figure 17: Compressive test results for X3 HCLPE found using the extensometer
Figure 18 shows the compression test results using the resistance strain gages. The resistance strain gage result from test C2 was found to deviate from the other tests quite drastically and in a non-linear fashion. After all the tests the samples were compared, and it was discovered that C2 had a thicker, nonuniform layer of epoxy which may explain the difference between this test and the others. The results from test C1 and C3 were consistent, but with a slight offset.
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Ϭ͘Ϭϭϰ
ŶŐŝŶĞĞƌŝŶŐ^ƚƌĞƐƐ;'WĂͿ
Ϭ͘ϬϭϮ Ϭ͘Ϭϭ Ϭ͘ϬϬϴ Ϭ͘ϬϬϲ Ϭ͘ϬϬϰ ϭ Ϯ ϯ
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ŶŐŝŶĞĞƌŝŶŐ^ƚƌĂŝŶ Figure 18: Compressive test results for X3 HCLPE found using resistive gages
Figure 19 shows the compression test results using the fiber optic strain gages. As seen in Figure 14 the results using the fiber optic strain gages were very consistent, although not as consistent as the extensometer result. Around 0.5% strain Test C3 showed an unusual jump in stress. The reason for this jump is not clear, although it may be due to the adhesive. The fiber optic gages for all three compression tests failed before the maximum strain recorded by the extensometer. These failures were likely due to the poor compressive strength of the epoxy and the small surface area for adhesion between the gage and the samples. The fiber optic gage that failed at the lowest strain was the one used for test C3, which debonded at about 1.1%. Ϭ͘Ϭϭϰ
ŶŐŝŶĞĞƌŝŶŐ^ƚƌĞƐƐ;'WĂͿ
Ϭ͘ϬϭϮ Ϭ͘Ϭϭ Ϭ͘ϬϬϴ Ϭ͘ϬϬϲ Ϭ͘ϬϬϰ ϭ Ϯ ϯ
Ϭ͘ϬϬϮ Ϭ Ϭ
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ŶŐŝŶĞĞƌŝŶŐ^ƚƌĂŝŶ Figure 19: Compressive test results for X3 HCLPE found using fiber optic gages
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Poisson’s ratio for compression was plotted at all values of axial strain as shown for C2 in Figure 20. These results were determined using data from the transverse and longitudinal resistance strain gages. From this test a Poisson’s ratio of about 0.4 was obtained. This compares favorably to the Poisson’s ratio obtained from the tensile tests. The large deviation for small levels of strain is due to dividing two small numbers with experimental error. At about 1.4% strain the Poisson’s ratio appears to rise slightly. This is different than the tensile test results, which diverged very little from the final value of Poisson’s ratio. This trend may be due to a characteristic of the plastic when loaded in compression, but more data is necessary to determine if this trend continues. Ϭ͘ϰϱ
WŽŝƐƐŽŽŶΖƐZĂƚŝŽ
Ϭ͘ϰ Ϭ͘ϯϱ Ϭ͘ϯ Ϭ͘Ϯϱ Ϭ͘Ϯ Ϭ͘ϭϱ Ϭ͘ϭ Ϭ͘Ϭϱ Ϭ Ϭ
Ϭ͘ϬϬϮ
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Ϭ͘ϬϬϲ
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ŶŐŝŶĞĞƌŝŶŐdžŝĂů^ƚƌĂŝŶ Figure 20: Poisson’s ratio for X3 HCLPE calculated from test C2
The tangent modulus and Poisson’s ratio obtained from the compression tests are summarized in Table 4. The tangent modulus was calculated using the extensometer data. In comparison with the average tangent modulus of 1.007 GPa for tension, the average tangent modulus for compression was 1.493 GPa.. The Poisson’s ratios obtained for tension and compression were very similar with only a 5% difference between the two average values obtained. Table 4: Results from the compression tests
4.2
Property
C1
C2
C3
Average
Tangent Modulus (Gpa)
1.428
1.518
1.534
1.493
Poisson's Ratio
0.399
0.406
0.453
0.419
Clinical Testing
After the tension and compression tests were completed, the tests simulating clinical loading were performed. In Figure 21 are shown the hoop strains obtained from the fiber optic strain gages closest to the load for the tests on the 36 mm diameter X3 HCLPE liners. As can be seen in this figure, the strain was found to generally increase as the offset increased for all inclination angles. As expected, the tests at 45° showed lower strain measurements than the tests at 65°. However, the tests at 55° showed lower strain than both of the other tests, at least at the strain gage location. Recall that only the hoop strain is being measured and not the total strain.
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Ϭ͘ϲ ϰϱĞŐƌĞĞƐ
Ϭ͘ϱ
WĞƌĐĞŶƚ^ƚƌĂŝŶ
ϱϱĞŐƌĞĞƐ ϲϱĞŐƌĞĞƐ
Ϭ͘ϰ Ϭ͘ϯ Ϭ͘Ϯ Ϭ͘ϭ Ϭ͘Ϭ Ϭ
Ϭ͘ϱ
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ϭ͘ϱ
KĨĨƐĞƚ;ŵŵͿ
Ϯ
Figure 21: Hoop strain obtained from the strain gages closest to the load for the tests on the 36 mm diameter X3 HCLPE liners.
Table 5 shows the numerical values for the 36 mm diameter liner results as shown in Figure 21. This table shows the percent strain as measured by the fiber optic gages for all parameter variations. The inner strain is the strain measured by the fiber optic gage that was closest to the load application. The outer strain is the strain measured by the fiber optic farthest from the load application. From this table it is clear that the hoop strain was smaller for the strain gages located farther from where the load was applied. The outer strain measurements on the 36 mm liner were quite small and showed no clear trend. This is most likely due to the fact that the strains at that point were too small to measure accurately. In Table 5 are also shown the results from the finite element analysis. Clearly, there was a significant difference between the analysis results and the testing results which requires further investigation.
dĂďůĞϱ͗ZĞƐƵůƚƐĨƌŽŵϯϲ ŵŵĚŝĂŵĞƚĞƌyϯůŝŶĞƌ Experimental Liner Diameter (mm)
Material
Load Angle (deg)
45
Offset (mm)
Inner Strain (%)
Outer Strain (%)
Inner Strain (%)
Outer Strain (%)
0
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FEA Analysis
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Figure 22 displays the clinical testing hoop strain results for the 40 mm diameter liner. As seen with the 36 mm diameter liner, the trends for the inner strain values are relatively consistent while the outer measurements are not. It can be seen that the strain transitions from compressive to tensile after the offset is increased to 2 mm for every test except for the data acquired from the 65° 1 mm offset. Also, except for the strain recorded at the 65° angle with a 2 mm offset, the strains increase with increasing angle. Strain gage closest to load
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Figure 22: Hoop strain obtained from the strain gages for the tests on the 40 mm diameter X3 HCLPE liners.
Figure 23 shows the clinical testing results for the 44 mm diameter liners made from N2Vac and X3 HCLPE. This size liner was the thinnest liner tested and had one fiber optic gage in the middle of the lip of the liner. Increasing the offset of the load did not create a trend in the strain values. The values of strain for the liner made from X3 HCLPE fluctuated between -0.02 and -0.07 percent strain for all loading situations except 65° with 2 mm offset where there was a significant increase in the strain. The results for the 44 mm diameter head results using the N2Vac material showed an increasing strain with increasing angle. Additionally, as with the 44 mm head made of X3 HCLPE, there is a general compressive to tensile change as offset is increased for each angle group. For a 65° angle the strain in the X3 HCLPE liner was higher than the strain in the N2Vac liner for every offset. For the 45° and 55° angles offset this was not the case – sometimes the strain in the X3 liner was higher and sometimes it was lower. Finite element results were available for only two of the test configurations. These results are shown in Table 6. The FEA results for the 45° loading angle with 2 mm offset were tensile, however the experimentally found values were compressive. However, the FEA results and experimental data for the 65° loading angle with 2 mm offset were both tensile, although the numerical values were quite different. The strain found through the numeric model was much larger than the strain determined experimentally.
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KĨĨƐĞƚ;ŵŵͿ Figure 23: Results from 44 mm diameter N2Vac and X3 HCLPE liners
Figure 6: Comparison of finite element results and experimental results for the 44 mm liner made from X3
Experimental
FEA Analysis
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From the above tables and analysis, it is clear that the finite element results did not agree with the experimental results. Because the fiber optic gages were shown to produce good data in tension and compression during the material testing, it is believed that the material models used for the finite element analysis should be improved. 5
Conclusions and Recommendations
In this study, standard tension and compression samples of X3 HCLPE were tested and hip liners subjected to realistic loadings were tested for various inclination angles and offsets. Bonding the strain gages to X3 HCLPE proved to be more difficult than anticipated. This was primarily due to the material’s low surface tension and the adhesive’s inadequate wetting. Bondit was found to provide a relatively good bond between each type of gage and the polyethylene, although in every failure of a fiber optic gage the adhesive failed before the fiber itself. Another problem experienced with the fiber optic strain gages was that they were very delicate and broke easily while being installed.
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The tension tests showed that the stress-strain curves obtained using data from the fiber optic strain gages and the resistance strain gages agreed well with results found using the data from the extensometer. For tension, the average tangent modulus calculated using the extensometer was about 1 GPa which was slightly higher than non-cross-linked polyethylene. In the compression tests there was more variability between the fiber optic strain gage, resistance strain gage, and extensometer results. The extensometer had the best correlation from one test to another. The variability in the other tests is most likely due to bonding issues with the epoxy. For compression, the average tangent modulus calculated using the extensometer was about 1.5 GPa and a Poisson’s ratio of about 0.4 was obtained in both the tension and compression tests. The results obtained from hip liners of various thickness indicated relatively small hoop strains in all cases. The shell for the 35 mm head had the largest measured hoop strains, which could be due to the choice of locations for the gages. Unfortunately, it was not possible to measure the transverse shear, so no observations can be made concerning the total strain as a function of thickness. The results from the finite element model did not correlate well with the experimental results. References [1] Frey, R. Encyclopedia of Surgery: A Guide for Patients and Caregivers. sugeryencyclopedia.com. [Online] Advameg, Inc., 2007. [Cited: 07 23, 2009.] [2] Crownshield, R., W. Maloney, D. Wentz, S. Humphrey, C. Blanchard. Bio Mechanics of Large Femoral Heads, What They Do and Don't Do. Clinical Orthopaedics and Related Research. 2004; 429:102-107. [3] Lombardi, A., T. Mallory, D. Dennis, R. Komistek, R. Fada, E. Northcut. An in vivo determination of total hip arthroplasty pistoning during activity. The Journal of Arthroplasty. 2000;15(6):702-709 [4] Paul, J.P. Approaches to design: Force actions transmitted by joints in the human body. Proc. R. Soc. Lond. B. 1976;192:163-172 [5] Kurtz, S., Ultra High Molecular Weight Polyethylene in Total Joint Replacement, The UHMWPE Handbook, 2002;263 [6]Yau, Shi-Shen.
[email protected]. Request for information on X3 and N2Vac. 19th August 2009. [7] Stryker. X3 Sequentially Annealed Irradiated Polyethylene. X3 The power of Technology. (Technical data sheet) 2006. [8] Kersey, A. D., M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C.B. Askins, M. A. Putnam, and E. J. Friebele, Fiber grating sensors, J. Lightwave Technology. 1997;15:1442-1463 [9] Todd, M., D. Inman, Optical-Based Sensing, in Damage Prognosis, John Wiley and Sons Inc. (Chichester, UK) 2004. [10] Lopez-Higuera J.M., Handbook of Optical Fiber Sensing Technology, John Wiley and Sons, Inc. (Chichester, UK), 2002. [11] Udd E, Fiber Optic Sensors: An Introduction for Scientists and Engineers, Wiley Interscience, 2006. [12] Limited, Roctest. Rocktest USA. [Online] 2004. [Cited: July 27, 2009.] http://www.roctest.com/modules/AxialRealisation/img_repository/files/documents/FOS.pdf.
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[13] Rao M, Bhalt M.R., Murthy C.R.L., Madhav K, Asokan S. Structural Health Monitoring Using Strain Gages, PVDF Film and Fiber Bragg Grating Sensors: A Comparative Study. Proc. National Seminar on Non-Destructive Evaluation, 2006:333-337 [14] Fourche G, An Overview of the Basic Aspects of Polymer Adhesion. Part 1: Fundamentals. Polymer Engineering and Science. 1995;35(12):957-967 [15] Briassoulis D, Schettini E. Measuring strains of LDPE films: the strain gage problems, Polymer Testing 21. 2002;507-512 [16] D695-08, Standard Test Method for Compressive Properties of Rigid Plastics [17] D638-08 Standard Test Method for Tensile Properties of Plastics
BookID 214574_ChapID 12_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII
February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc. 6XJJHVWLRQIRU ( YDOXDWLRQ0HWKRGVRI5LGH&RPIRUWDW /RZ )U HTXHQFLHV
7DND\XNL.2,=80,1REXWDND768-,8&+,6RJR2.8085$ Department of Engineering, Doshisha University, 1-3, Tataramiyakodani, Kyotanabe-city, Kyoto, 610-0321, Japan 6DFKLNR<$0$'$-LUR1,120,<$7DNDPLWVX7$-,0$ HONDA R&D Co., Ltd. 4630, Shimotakanezawa, Haga-cho, Haga-gun, Tochigi, 321-3393, Japan
$%675$&7 In the development stage of manufacturing automobiles, ride comfort performance is as important as driving performance. Evaluation methods treat many single direction vibrations. But when a car is running, the driver is exposed to a compound vibration. Therefore, a method that evaluates the riding comfort of compound vibration becomes important. In this paper, a method was constructed that evaluated the riding comfort of the coupled vibration. The vibration exciter used in this study consists of two hydraulic actuators. The subjects were exposed to the following: acceleration vector summation, vibration with a different frequency, phase difference, and the amplitude ratio in the low frequency band. The vibrational input to the subjects was vertical-anteroposterior and vertical-horizontal at an intended frequency from 1 to 3.5 Hz. When the frequency and the amplitude ratio increase, the discomfort increases. Additionally, when the vector summation increases, discomfort increase. ,1752'8&7,21 Riding comfort performance has become a trend in the car market in recent years as well as driving and control stability performances1-3). This trend is associated with the change from using automobiles as means of transportation to their use for driving enjoyment. Moreover, the performance difference between manufacturers has been reduced by exploiting CAE technology(4). Additionally, consumer needs are various. Thus, riding comfort performance improvement technology is focused on. An objective evaluation of riding comfort is crucial for the future development of cars. To derive riding comfort, such factors as habitat, the vehicle, the vibration, and human must be considered when evaluating a car. However, realistically understanding riding comfort is difficult. It is also difficult to obtain evaluation results that can be fed back to the design stage. The present study targeted the driver who received the vibrations. The sense of the person who causes by this vibration is narrowly treated as riding (6)
comfort . Riding comfort is a subjective psychology response, although an objective evaluation approach is necessary as mentioned above. Because experts have done traditional evaluations, no objective evaluation indexes and universal development indicators have been done7). Quantifying the human sensibility effectively judges the ride quality from measured physical values8-9). Riding comfort was evaluated by the behavior and the eyesight in past research10-11). However, a fundamental study of the input that influences the sense characteristics has not been performed. Additionally, many single shaft vibrations were treated in these evaluation approaches whose results don't necessarily agree with the coupling vibration input felt by the driver when a vehicle is running. Therefore, an evaluation approach must pay attention to the vibration feeling that receives the coupling vibration characteristics.
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_12, © The Society for Experimental Mechanics, Inc. 2011
135
BookID 214574_ChapID 12_Proof# 1 - 23/04/2011
136 The present study grasped the vibration feeling that received the vibration in the two axes. The input condition included the following: acceleration vector summation, the phase difference, the amplitude ratio, and the frequency to the low frequency band. The experiment presumed the driver's sense level by exciting the received vibration with a large-scale excitation machine.
'(6,*1,1*(;3(5,0(17$/6(783 The vibration exciter used in this study consisted of two hydraulic actuators, a 1.2×1.2 m shaking table, and hydraulic units. This exciter operates in the horizontal and vertical directions. The frequency and acceleration of the input vibration were controlled by a PC connected to this two-dimensional shaker. The seat, the steering wheel, the seat belt, and the pedals were attached to this table. Except for the pedals, all of these auto parts were mass-produced. The height from the table to the seat as well as the positional relationships between the seat and the steering wheel, seat belt, or pedals resembled the values of a sedan. Fig. 1 shows the experimental setup. The excitation experiment in this study was conducted using vibration inputs in three orthogonal directions: longitudinal (X), lateral (Y), and vertical (Z).
Fig.1 Experimental setup 0($685,1*+80$19,%5$7,216(16,7,9,7,(6 ,QSXWWLQJVXEMHFW¶VFRUUXJDWLRQ The behavior of the upper part of the body and the head in the low frequency band is closely related to the 12)
unpleasantness detected by past research } Vehicle A assumed that the riding comfort was bad, so vehicle B was selected because it was good. The driver's head acceleration was measured while we drove the vehicle on the road. Fig. 2 shows one example of the result. For the vehicle whose riding comfort was good, the absolute value of the Lissajous of the head’s acceleration was small. Moreover, the acceleration amplitude ratio was small, and the phase lag was large. The size of the acceleration vector summation of the translation 2-axis vibration, the frequency, the phase lag, and the amplitude ratio were assumed to be input parameters in the present study and became the driving posture on the excitation machine for the riding comfort evaluation experiments. The object’s frequency was set to 1.0-3.5 Hz. The input vibration was a sine wave. The values of the size of the acceleration vector harmony, the phase lag, and the amplitude ratio in three directions were set based on the floor face vibration during actual running. Each input parameter is explained in the following section.
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Fig.2 Head response when car running
6HWWLQJRIDFFHOHUDWLRQYHFWRUVXPPDWLRQ The acceleration vibration was calculated to 3 Hz of a vehicle’s floor face vibration. The vector summation of the acceleration in three directions was calculated based on it. As a result, the vector summation of the coupled 2
2
vibration was decided to be 1.62 m/s and 1.00 m/s . 3KDVHGLIIHUHQFHE\WZRD[LVYLEUDWLRQ The acceleration amplitude was set so that the vector summation when the phase difference of the X-Z vibration was 0° might become 1.62 m/s2. The acceleration amplitude in the X and Z directions is 0.35 m/s2 and 1.58 m/s2. When the vector summation is 1.00 m/s2, the acceleration amplitude in the X and Z directions is 0.22 m/s2 and 0.98 m/s2. The phase difference to input it to the subject is -90, -45, 0, 45, and 90°. The standard vibration is 0°. The phase difference was similarly set in the Z direction in the Y-Z coupled vibration. The acceleration in the Y direction was the same as the X direction. $PSOLWXGHUDWLRE\WZRD[LVYLEUDWLRQ The vector summation remained constant, and the acceleration amplitude in the X-Z direction was changed. The acceleration amplitude in the Y-Z direction was also changed. Both accelerations are shown in Tables 1 and 2. When the amplitude ratio shown in Tables 1 and 2 is shown in the input angle, it is -25, and -12.5, 0, 12.5, and 25°. The standard vibration was -12.5°. Amplitude ratio-12.5° and phase difference 0° are identical. Table 1 Condition of input vibration for amplitude ratio when vibration magnitude is 1.62[m/s2] Amplitude ratio[deg]
-25.0 -12.5 0.0 12.5 25.0
Z[m/s^2] X,Y[m/s^2]
1.47 1.58 1.62 1.58 1.47
-0.69 -0.35 0.00 0.35 0.69
Vector summation[m/s^2]
1.62 1.62 1.62 1.62 1.62
Table 2 Condition of input vibration for amplitude ratio when vibration magnitude is 1.00[m/s 2] Amplitude ratio[deg]
-25.0 -12.5 0.0 12.5 25.0
Z[m/s^2] X,Y[m/s^2]
0.91 0.98 1.00 0.98 0.91
-0.43 -0.22 0.00 0.22 0.43
Vector summation[m/s^2]
1.00 1.00 1.00 1.00 1.00
BookID 214574_ChapID 12_Proof# 1 - 23/04/2011
138 ,QSXWIUHTXHQF\ The input vibration frequency was set to six ways: 1, 1.5, 2, 2.5, 3, and 3.5 Hz. 5LGLQJFRPIRUWHYDOXDWLRQLQSKDVHODJDQGDPSOLWXGHUDWLR The experimental methodology employed to evalute the sense characteristic when the frequency changes is shown in Fig. 1. The subjects were 5-10 people. The subjects extracted by the public advertisement, received the purpose of the measurement and the explanation of the method enough from the principal investigator beforehand when participating about the research by oral and the document, and approved it by the document. A pair comparison method was employed for sensory evaluation in which subjects evaluated each comparison vibration for the standard vibration. A standard vibration score was defined as ten points. Subjects then evaluated the relative vibrations in the range of 0 to 20 points. Feeling more uncomfortable increased the score. The standard vibration was 1.0 Hz, and the comparison vibration was 1.5-3.5 Hz. After exposing the subjects to the standard vibration, they were randomly exposed to the chosen comparison vibration, which they evaluated by questionnaire.The vibration, which was linked to the standard vibration, became the comparison vibration that was input so the subject would not forget the normal vibration. The interval time of the standard and comparison vibrations was 0.5 s. Next we describe the experimental methodology to understand the sense characteristic when the vector summation changes. The standard vibration of each frequency of vector summation, 1.00 m/s 2 and 1.62 m/s2, was 2
input to the subjects. The standard vibration was vector summation 1.00 m/s . This summation vibration was 2
assumed to be ten points, and it was a comparison vibration for vector harmony 1.62 m/s . The comparison vibration was evaluated by 0-20 points. Next the experimental methodology is described to understand the sense characteristic when the phase difference changes. The subjects were 7-10 people. The acceleration input a sine vibration of 0.35 m/s 2 in the X or Y directions, and a sine vibration of 1.58 m/s2 was input in the Z direction. The acceleration in the X or the Y direction input a sine vibration of 0.22 m/s2 to 9-10 subjects. A sine vibration of 0.98 m/s2 was input in the Z direction. The input frequency was evaluated in each frequency by six kinds of paired comparison tests: 1, 1.5, 2, 2.5, 3, and 3.5 Hz. There were five points in the standard vibration evaluation. The comparison vibrations ranged from 0-10 points. The normal vibration is phase lag 0°. There were four kinds of comparison vibrations: -90 ,-45, 45, and 90°. The subjects put a slash on the questionnaires and replaced the slash position with a point. Fig. 3 shows one example. The experimental methodology is described to understand the sense characteristic when the amplitude ratio 2
changes. The subjects were 9-10 people. The vector summation input sine vibrations of 1.62 and 1.00 m/s . The standard and comparison vibrations were input and evaluated by the subjects. The evaluation method resembled the phase difference experiment.
Fig.3 Evaluation Sheet &RUUHFWLRQRIHYDOXDWLRQUHVXOWV The outliers were removed from the average scores of all subjects, based on the points in the evaluation average and the standard deviation. Next frequency weight was done to the points in the evaluation averages. The points in the evaluation of the 1.0 Hz of the phase difference and the amplitude ratio was set to 1.0 times. The weight of the points in the evaluation was done by 1.5-3.5 Hz.
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139 Because the standard vibration of the phase lag and amplitude ratio experiments is identical, the points in the evaluations of both should also be equal. Therefore, for the points to become equal in the normal vibration evaluations of the phase lag experiment, the evaluation points of the amplitude ratio experiment’s normal vibration were corrected. In addition, the ratio during correction was used as the evaluating value of the comparison vibration. Finally, the points in the evaluation ratio of 1.62 m/s2 to vector summation 1.00 m/s2 were calculated. The calculated ratio was multiplied by all the points in the evaluations of vector summation 1.62 m/s 2. 5HVXOWDQGGLVFXVVLRQE\H[FLWDWLRQH[SHULPHQW 9LEUDWLRQIHOWE\WKHUHFHLYHGFKDUDFWHULVWLFWRIUHTXHQF\ First, the X-Z coupled vibration is described. Fig. 4(a) shows the result of the vibration felt by the received 2
characteristic in vector summation 1.62 m/s to the frequency veering shown by the mean value and the standard deviation of the evaluation result. Vector summation 1.00 m/s2 is similarly shown in Fig. 4(b). The frequency and the evaluation points are shown in the horizontal and vertical axes. The evaluation points were a maximum of 1.5 Hz from Fig. 4(a) at vector summation 1.62 m/s2 in 2 Hz or less. The evaluation points increase by a frequency increase of 2 Hz or more. On the other hand, the evaluation points increased as the frequency increases from 1 to 3.5 Hz when Fig. 4(b) is seen. When the vector harmony changes from this in 2 Hz or less, the tendency to experience discomfort changes. When the frequency increases in 2 Hz or more regardless of the vector harmony, the discomfort feeling increases, too. Next, the Y-Z coupled vibration is described. Figs. 5(a) and (b) show the sense characteristics of the frequency of 2
2
vector harmony 1.62 m/s and 1.00 m/s . The evaluation points increased as the frequency increases. Therefore, the discomfort increases as the frequency increases, regardless of the vector summation.
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(a) Vector summation 1.62 [m/s ]
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Fig.4 The characteristics of vibratory sensibility for frequency at X-Z
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Fig.5 The characteristics of vibratory sensibility for frequency at Y-Z
BookID 214574_ChapID 12_Proof# 1 - 23/04/2011
140 9LEUDWLRQIHHOLQJUHFHLYLQJFKDUDFWHULVWLFE\HDFKLQSXW The discomfort of each input parameter was reproduced in a color map based on the corrected evaluation points. The color map changes from blue to red to show unpleasantness. The X-Z coupled vibration is described. A color map of the phase difference and the amplitude ratio in vector 2
summation 1.62 m/s is shown in Figs. 6(a) and (b), respectively. A color map of the phase difference and the 2
amplitude ratio in vector summation 1.00 m/s is similarly shown in Figs. 7(a) and (b). Figs. 6(a) and 7(a) show the frequency in their horizontal axes, and the phase difference is shown in their vertical axes. The horizontal axes of Fig. 6(b) and 7(b) show the frequency. Their vertical axes show the amplitude ratio. Discomfort increased from 3 Hz in Fig. 6(a). Moreover, a large shift of the discomfort by the phase difference change is not seen in the same frequency. Fig. 6(b) showed remarkable discomfort in an amplitude ratio of –25° and 25°. The discomfort change is small for the change in an amplitude ratio in 1 Hz. The discomfort becomes small at 0 phase lag° in each frequency (Fig. 7(a)). Fig. 7(b) shows remarkable discomfort in an amplitude ratio of –25° and 25°. The discomfort change of the amplitude ratio change is remarkable for the vector summation of the two conditions. Moreover, the vibration of an amplitude ratio of 25° influences the discomfort regardless of the vector summation. The Y-Z coupled vibration is described. A color map of the phase lag and the amplitude ratio in vector summation 1.62 m/s2 is shown in Figs. 8(a) and (b), respectively. A color map of the phase lag and the amplitude 2
ratio in vector summation 1.00 m/s is similarly shown in Figs.9(a) and (b). Discomfort increases when it increases in the frequency shown in Figs. 8(a) and 9(a). However, a large shift of 2
2
discomfort was not confirmed by the phase difference changes. Vector summations 1.00 m/s and 1.62 m/s were compared. The change in the color map was remarkably large in vector summation 1.62 m/s2. When the vector summation increases, the change of the unpleasantness increased. When the absolute value of the amplitude ratio increases, the discomfort also increases, as shown in Figs. 8(b) and 9(b). This tendency appears remarkably in 2
vector summation 1.62 m/s . The discomfort increases when the frequency and the amplitude ratio are large.
(a)Phase difference - Freqency
(b) Amplitude ratio - Freqency 2 Fig.6 The characteristics of vibratory sensibility at X-Z_1.62[m/s ]
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(a) Phase difference - Freqency
(b) Amplitude ratio - Freqency 2
Fig.7 The characteristics of vibratory sensibility at X-Z_1.00[m/s ]
(a) Phase difference - Freqency (b) Amplitude ratio - Freqency Fig.8 The characteristics of vibratory sensibility at Y-Z_1.62[m/s2]
(a) Phase difference - Freqency
(b) Amplitude ratio - Freqency 2
Fig.9 The characteristics of vibratory sensibility at Y-Z_1.00[m/s ]
&21&/86,216 The present study focused on the sense characteristics of vibration when input by two axes to grasp the contribution characteristics of the input parameters. The acceleration vector harmony, the phase lag, the amplitude ratio, and the frequency were assumed to be input parameters based on a vehicle’s floor vibration. The vibration feeling that absorbed the characteristic was determined, and the input parameter was specified that influenced the discomfort. The following conclusions were obtained in the present study: 1䠊 Discomfort depends on the increase and the decrease of the acceleration vector summation. 2䠊 The dependency of phase difference to discomfort is low. 3䠊 The amplitude ratio pattern was specified that influenced discomfort. 4䠊 In the targeted bandwidth, discomfort also showed an increasing tendency as the frequency increased. 5()(5(1&(6
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142 [1]
K.Amasaka and S.Nagasawa, Base and Application for evaluation of sensitivity, Japanese Standards Association, (2003) [2] S. Doi et al., Vehicle Dynamics Performance Evaluations Through Human Dynamic Properties, Toyota Central R&D Labs., Inc., Vol.30 No.3 pp3-15 (1995) [3] Y.Akatsu , An Evaluation Method of Improving Ride Comfort, JSAE, Vol.52 No.3 pp47-52 (1998) [4] S.Kaneko et al., Vehicle Dynamic-Performance Evaluation Based on "KANSEI", JSAE, Vol.54 No.11 pp70-74 (2000) [5] Y.Murata et al., Development of High Performance Seat Cushion Pad, JSAE, Vol.60 No.7 pp62-67 (2006) [6] K. Yamazaki, Evaluation of ride quality, ASJ , Vol.46 No.2 pp157-161 (1990) [7] H.Inagaki, T. Taguchi, E. Yasuda, and S. Doi, Evaluation of Seat Kansei Quality, Toyota Central R&D Labs., Inc. Review, Vol. 35, No.4, (2000) [8] H. Chikaoka, Invitation to Design engineering of "KANSEI" , NIKKEI MECHANICAL No.519 pp36-59 (1997) [9] H.Iguchi, S. Doi, Comfort, Pleasantness and Usability on Human Engineering, JSAE, Vol.57 No.10 pp4-9 (2003) [10] K.Kato, T.Sonoda, S.Kitazaki, Effect’sof driver’s head motion and visual information on perception of ride comfort, JSAE, No.113-08 (2008) [11] E.Yasuda, S.Doi, M.Ishiguro, H.Tanaka, Analysis of Ride Comfort due to Human Vibration Characteristics Report2. Evaluation methods of Human sensitivity by Vibration Components, JSME Symposium Paper, pp216-220, (1996) [12] H.Abe, Evaluating Ride Comfort of Automobiles Considering Human Body Behaviors, (2006)
BookID 214574_ChapID 13_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
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• $%675$&7 While driving off-road vehicles, operators are exposed to whole body vibration that influences the fore-and-aft and vertical vibration. A dummy weight with no dynamic characteristics was installed on the seat as a load condition instead of a human body to evaluate the seat structure’s durability. A miniature model of a sitting human body was developed for seat durability evaluations. In a laboratory study, accelerations and forces in the vertical direction were measured at the seat face during whole body vibration in the fore-and-aft and vertical vibration. Four male subjects with body masses around 60 kg were chosen for the tests. They sat on a rigid seat with no backrest. A linear two-degree-freedom model, which was used to describe the dynamic behavior of a sitting driver, was assumed to be supported by two points: the floor and the seat face. The model parameters were identified by fitting the dynamic mass to the measured values. Good agreement was obtained between the experimental and simulation results. Based on the identified parameters, a miniature model of a seated human body was designed.
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IB m1 m2 g kF kB cF cB bF bB
L T Fs Ff
: Inertia of weight : Mass of weight : Mass of base plate : Accelaration of gravity : Stiffness of front spring : Stiffness of rear spring : Damping coefficient of front damper : Damping coefficient of rear damper : Distance between front spring and center of weight base : Distance between rear spring and center of weight base : Length of thigh : Rotation angle of weight : Reaction force on seat surface
h1 h2 T
xO x1 x2 zO z1 z2 FH F
: Height of weight gravity : Distance between weight and base plate : Moment around hip : Displacement of center of weight base in horizontal direction : Displacement of center of gravity of weight in horizontal direction : Displacement of seat in horizontal direction : Displacement of center of weight base in vertical direction : Displacement of center of gravity of weight in vertical direction : Displacement of seat in vertical direction : Force to support weight in horizontal direction : Integrated force of whole human body
: Reaction force on floor
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_13, © The Society for Experimental Mechanics, Inc. 2011
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The structure of the operator’s seat for a hydraulic excavator, which has no suspension mechanism on its undercarriage, is exposed to severe vibrations by excavation work or off-road travelling. The product’s reliability improvement is required to precisely measure the fatigue strength. The seat is an important component of the vehicle driver’s security. The durability of the seat structure is often evaluated through laboratory measurements. A dummy weight is installed on the seat instead of a human subject as a load condition; however previous works[1] [2] suggest that the human body behaves like a vibration system, so the results of the two cases are different. The human body’s biodynamic response to vibration is complex. The drivers of heavy road and off-road vehicles like hydraulic excavators are exposed to vertical and fore-and-aft direction whole body vibration. Much work on this issue has been done for the vertical direction [3]. Mathematical models of the vertical apparent mass of an upright sitting human body have been developed[4]. Human body response measurements in the fore-and-aft and lateral directions have been reported[5]. A numerical approach[6] has also designed fore-and-aft suspensions with higher performance. The main objective of this paper is WR construct a vibration response model of a seated human body in a median sagittal plane and to produce a miniature model based on it for seat durability evaluations. A linear two-degree-freedom model is used to describe the sitting driver’s dynamic behavior. The model assumed that the reaction force on the floor supported driver to the torque of the body’s upper part around the hip during horizontal vibration. The human body is modeled in two dimensions to analyze the vibration of the horizontal and vertical directions. The model parameters are identified by fitting the apparent mass and frequency response functions to the measured values. Accordingly, first, the dynamic characteristics of the human subjects were acquired in the vertical and fore-and-aft directions. Second, we present a model based on the dynamic characteristics and use a frequency domain identification method to identify the system. Finally, we present a miniature model composed of springs, dampers, and an iron part and verified the miniature model’s validity.
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0HDVXULQJDSSDUDWXV The dynamic characteristics of the seated human subjects were measured by a vibration exciter (Fig.1), which is a vibration platform supported on two hydraulic actuators that operates in the horizontal and vertical directions. The frequency and acceleration of the input vibrations are controlled by a PC connected to this two-dimensional shaker. A seat on which the subjects sat with no backrest and no cushion was bolted to the vibration exciter’s platform. Four male subjects with body masses between 60.5 and 64.0 kg and height between 169 and 176 cm were selected for the tests. They sat in a relaxed upright posture with their feet on the floor. The acceleration and the reaction force were measured on the floor and the bearing surface. Force plates were attached to the flat seat surface on which the subjects sat and the floor on which the subject’s feet rested. The platform’s acceleration was monitored in the X and Z directions. Measurements were carried out by sweeping frequency under the constant amplitude of the vibration in each detection. The sweeping rate was 1-40 Hz / 90 2 bsec, and the amplitude was 1m/s .
Fig. 1 Vibration exciter seat
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Referencing the contact force on the seat’s surface to the exposed vertical acceleration, we obtain dynamic mass M , which forms a relationship between contact force Fs and exposed acceleration z at the same point in the frequency domain:
M( f )
Fs ( f ) z( f )
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Referencing the moment of the upper body around the hip to the exposed fore-and-aft acceleration, we obtain the frequency transfer function, which forms a relationship between contact force F on the floor and exposed acceleration x in the frequency domain:
O( f )
T( f ) x( f )
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where each contact point is assumed to be freely supported and the moment caused around the hip is supported by the reaction force on the leg floor side. 02'(/ 7KHRU\ To model the system, the following were the main simplifications. We assumed that the seated human body’s motion only occurs in the median sagittal plane. The weight’s rotary center has translatory motion in the vertical direction. The system was modeled with the subject’s feet on the floor and with no backrest. This model is comprised of springs, dampers, and mass. The center of mass was assumed to be exactly over the center of rotation, which is point O. The springs and dampers are located at distance bi from O point. The movement of the model’s base plate is identical to the floor face input. The rotation center’s movement in the horizontal direction is identical to the base plate.
z1
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x: z:
yym1 x1 f H 0 yym1 z1 f F f B
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0
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T:
yyI BT1
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T1 =0 was obtained in the following form:
f H (h2 z1 z 2 ) cos T 1 f B bB cos T 1 f F bF cos T 1 m1 gh1 sin T 1
(5)
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The positions of each point can be determined from the following equations:
x1
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z1 z O h1 cos T 1 xO x 2
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zF
z O bF sin T1
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zB
z O bB sin T 1
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The force of the springs and the dampers is calculated by the rotation and the relative displacement of the weight and the base plate. Forces f F , f B , f H are shown below:
fB
k F z F z 2 c F ( z F z 2 ) k B z B z 2 c B ( z B z 2 )
fH
m1 x1
fF
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This model is supported at two points (floor and seat) and assumes that the reaction force on the floor supports the torque around the hip. Then the reaction force on seat surface Fs is obtained from an integrated force in consideration of moment T around the hip:
F
Fs
T L
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where integrated reaction force F on both support points of the model is derived from
Fs F f
F
f F f B m2 z2
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T
F f L1
f F bF f B bB m2 z2 b2
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To identify the parameters of the proposed mode, the sum of the complex error squares is defined as an error function using both the measured and the simulated results:
error _ M
1 N M meas n M sim n , error _ O ¦ M N n1 meas n
1 N Omeas n Osim n ¦ O Nn1 meas n
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Variable N specifies the number of evaluated frequency points. Error is minimized by a conjugated gradient routine to identify the searched set of parameters.
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Figure 3 illustrates the dynamic mass of the four subjects measured in the Z direction with a vertical acceleration signal. The estimated dynamic masses for all subjects exhibited the same trends and had a dominant resonance frequency between 5-7 Hz. The magnitudes of the dynamic masses at the dominant resonance frequencies for all subjects were found to be between 89-109 kg. The measured torque transmissibilities are shown in Fig. 4 and have a dominant resonance frequency at 2 Hz or less.
BookID 214574_ChapID 13_Proof# 1 - 23/04/2011
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Fig. 3 Magnitude of dynamic mass of four seated subjects
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Fig. 4 Magnitude of torque transmissibility of four seated subjects
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The values of the unknown parameters corresponding to each of the subjects were identified (Table 1). Fig. 5 shows the dynamic mass of the identified model and one seated subject, and Fig. 6 shows the torque transmissibility of the identified model and one subject. The simulated results of the model’s dynamic mass and torque transmissibility have good agreement with the experimental results for one subject, as in the system’s dominant resonance frequencies. 䢳䢶䢲
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Fig. 5 Magnitude of dynamic mass of experimental and simulated results for one subject
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Fig. 6 Magnitude of torque transmissibility of experimental and simulated results for one subject
Table 1 Identified model parameters of a subject bF [mm] h2 [mm] kF [N/m] kB [N/m] bB [mm] 10 10 10 24.5 24.5 cF [Ns/m] cB [Ns/m] m1[kg] m2 [kg] IB [kg mm2] 0.47 0.47 45.8 10.1 8.43
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&RQVWLWXWLRQ The miniature model was based on the identified model (Fig. 7) and consists of an iron seat base, weight, springs, and hydraulic dampers. Its weight is supported on a set of four springs and four hydraulic dampers on the base plate and the weight is restrained to the guide of a linear slider by a base middle point. The four dampers and coil springs were set up in symmetry mainly on weight in parallel. For the weight, the springs, and the dampers, simple detachment is possible. The miniature model’s characteristics can be adjusted by exchanging them, depending on the dynamic characteristics of the subjects.
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Weight
Table 2 Major dimensions of miniature model Damper
Guide unit
Height Width Depth Weight
Spring
400-600[mm] 470[mm] 370[mm] 30-70[kg]
Base plate
Fig. 7 Pictorial view of miniature model The coil springs are fixed to the coping cylindrical ditch on the lower part of the weight and the top surface of the miniature model’s base. The dampers are fixed on the lower part of the weight and the base with a knuckle joint with a rotary degree of freedom. The linear guide with a degree of freedom in the vertical direction consists of a slide unit and a shaft that is perpendicularly fixed to the base. A pin supports the weight of the slide unit. The weight has two degrees of freedom for the base; one is the rotary degree of freedom and the other is the vertical translation degree of freedom. The boundary dimensions of the miniature model are shown in Table 2. $GMXVWDELOLW\RIG\QDPLFFKDUDFWHULVWLFV Figure 8 illustrates the dynamic mass of the model in the Z direction vibrating for each k . Fig. 9 shows the torque transmissibility of the model exposed to the X direction vibrating for each k. Both the dominant resonance frequencies in the X and directions increase with k . Fig. 10 illustrates the dynamic mass of the model in the Z direction vibrating for each h1 . Fig. 11 shows the torque transmissibility of the model in the X direction exposed to the vibration for each h. The dominant resonance frequencies in the X direction don’t depend on h1 , and the dominant resonance frequencies in the direction decrease with h1 . 䢳䢶䢲
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Fig. 8 Magnitude of dynamic mass with different k values
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Fig. 9 Magnitude of torque transmissibility response with different k values
The miniature model’s mass is determined from the static load of the human subject, and the dominant resonance frequency of the dynamic mass can be adjusted by changing the miniature model’s springs. However, the influence reaches the dominant resonance frequency of the torque transmissibility. On the other hand, the dynamic mass doesn't depend on h1 because only the dominant resonance frequency of the torque transmissibility is influenced by h1 . The dominant resonance frequency of the dynamic mass and the torque transmissibility of the miniature model can be adjusted based on the human subject’s characteristics.
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Fig. 10 Magnitude of dynamic mass with different h1 values
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Fig. 11 Magnitude of torque transmissibility with different h1 values
&21&/86,216 Dynamic masses in the Z direction and the torque transmissibility in the direction exposed to the X axial vibration were acquired for human subjects. The model of the seated human body, which was identified by the dynamic characteristics of four human subjects, can analyze the response of two-axis vibration in the X and Z directions. The miniature model was based on our proposed model. The dominant resonance frequencies of the miniature model can be adjusted by changing the position of the center of gravity and the spring constant. Changing the position of the center of gravity can only adjust the dominant resonance frequencies of the vibration response characteristics in the direction without influencing the dynamic mass. 5()(5(1&(6 [1] RAKHEJA S, Seated occupant apparent mass characteristics under automotive postures and vertical vibration. Journal of Sound and Vibration, Vol.253 No.1, pp57-75 (2002) [2] RAKHEJA S, Effects of sitting postures on biodynamic response of seated occupants under vertical vibration, International Journal of Industrial Ergonomics, Vol.34 No.4, pp289-306 (2004) [3] H. Ne´ lissea, S. Patra, S. Rakheja, J. Boutin, P.-E´ . Boileau, Assessments of two dynamic manikins for laboratory testing of seats under whole-body vibration, International Journal of Industrial Ergonomics 38 (2008) 457–470 [4] LIANG Cho-chung, CHIANG Chi-feng, A study on biodynamic models of seated human subjects exposed to vertical vibration, International Journal of Industrial Ergonomics 36 pp869 –890 (2006) [5] STEIN George Juraj, MUCKA Peter, CHMURNY Rudolf, Measurement and modelling of x-direction apparent mass of the seated human body-cushioned seat system, Journal of Biomechanics 40 pp1493–1503 (2007) [6] FLEURY Gerard, MISTROT Pierre, Numerical assessment of fore-and-aft suspension performance to reduce whole-body vibration of wheel loader drivers, Journal of Sound and Vibration 298 pp672–687 (2006)
BookID 214574_ChapID FM_Proof# 1 - 23/04/2011
BookID 214574_ChapID 14_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Tutorial Guideline VDI 3830: Damping of Materials and Members L. Gaul Institute of Applied and Experimental Mechanics, University of Stuttgart Pfaffenwaldring 9, 70550 Stuttgart, Germany
[email protected]
Committee Background It was Nov 10, 1982 when Prof. Federn, Prof. Gaul, Prof. Mahrenholtz, and Dr. Pieper VDI decided to work out a guideline on damping in the VDI/FANAK C13 Committee “Material Damping”. They were joined by Prof. Ottl, Prof. Kraemer, Prof. Pfeiffer, Prof. Markert, Prof. Wallaschek, and Mr. Hilpert VDI lateron in their names order. The idea was to comprise distributed theoretical and experimental knowledge and to homogenize the nomenclature of this subject. At the very beginning, important knowledge was provided by the books J.D. Ferry: Viscoelastic Properties of Polymers John Wiley & Sons, New York, 1960 B.J. Lazan: Damping of Matrials and Members in Structural Mechanics Pergamon Press, Oxford, 1968 Important contributions to the subject were made at conferences in the USA, such as – Damping Lynn Rogers – The Role of Damping in Vibration and Noise Control, ASME Boston Lynn Rogers, Lothar Gaul – Damping Sessions at IMAC Lothar Gaul et al and in Germany by the colloquium – Daempfungsverhalten von Werkstoffen und Bauteilen Kolloquium, TU Berlin, 1975 VDI-GKE H. Fuhrke, K. Federn, R. Gasch Results of five guidelines worked out by the named committee VDI-Richtlinie3830, Blatt 1-5 have been presented at the conference Schwingungsdaempfung (Vibration Damping) October 16 and 17, 2007, Wiesloch near Heidelberg providing information about
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_14, © The Society for Experimental Mechanics, Inc. 2011
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– modelling – numerical methods (Finite Elements, Boundary Elements, Modal Analysis) – experimental techniques for determining material damping properties from measured components or system characteristics along with – passive and adaptive practical applications.
The guideline VDI 3830 “Damping of Materials and Members” The guideline VDI 3830 consists of the following parts: Part 1 Classification and survey Part 2 Damping of solids Preliminary note 1 Physical phenomena 2 Linear models 3 Nonlinear models Part 3 Damping of assemblies Preliminary note 1 From the material to the homogeneous member 2 Laminated members 3 Damping in joints 4 Damping due to fluids 5 Damping by squeezing 6 Assemblies Part 4 Models for damped structures Preliminary note 1 Basic model 2 Structures with finite number of degrees of freedom 3 Calculation of viscoelastic components using the boundary element method Part 5 Experimental techniques for the determination of damping characteristics Preliminary note 1 Remarks on experimental techniques 2 Experimental techniques and possible instrumentation 3 Special experimental techniques for determining damping characteristics under aggravated conditions 4 Experimental Modal Analysis (EMA) 5 Experimental techniques for the damping measurement of subsoil
Introduction All dynamic processes in mechanical systems are more or less damped. Consequently, damping is highly relevant in those fields of technology and applied physics which deal with dynamics and vibrations. These include • machine-, building-, and structural dynamics, • system dynamics, • control engineering, and • technical acoustics, because damping in these cases often has a considerable effect on the time history, intensity, or even the existence of vibrations. Important applications are:
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• transient vibrations (transient effects associated with the onset or decay of vibrations, shock-induced vibrations, reverberation effects) • resonance vibrations (unavoidable with random excitation) • wave propagation • dynamic-stability problems Accordingly, a multitude of scientific publications dealing with damping, or taking it into account at least, are found in technical literature. Due to different theory approaches, objects, and task definitions in the applications listed above, the designations, the characterisation of damping, the experimental techniques, and the analytical and numerical methods are not harmonised. The dynamic behaviour of damped structures can, in special cases, be calculated using generally valid material laws for inelastic materials based on continuum mechanics taking into account boundary effects (e.g. joints). In general, this approach is too elaborate or expensive, or not at all practicable. In most cases, therefore, phenomenological equivalent systems or mathematical models tailored to the task definition are used which are only valid assuming a special state of stresses and/or a special time history. Harmonic (sinusoidal) time histories are a preferred special case where complex quantities describe the elastic and damping properties. These depend on a number of parameters: material data, rate of deformation, frequency, temperature, number of load cycles, etc. In the case of nonlinear behaviour there is also a dependence on the amplitude. For certain problems, it is sufficient to state, for one deformation cycle, the energy dissipated in a unit volume or within the system, or the energy released into the environment at the system boundaries, often related to a conveniently chosen elastic energy in a unit volume or in the system as a whole. In structural dynamics, the use of modal damping ratios has proven useful, which do no longer contain detailed information about the damping. This guideline is not a textbook; it cannot be a compilation of generally mandatory rules. It is intended • to contribute to a better understanding of the physical causes of damping, • to facilitate interdisciplinary cooperation by defining harmonised terms and pointing out the relations between different approaches to the modelling of damping, and • to allow an overview of the state of knowledge and experience gathered in various fields of application and research, in order to promote the application of existing knowledge. This guideline is structured in accordance with its objective. It starts off with the notion of damping and the causes of damping before dealing with different modelling approaches for the linear and nonlinear behaviour of solids, and establishing cross-references between these approaches. Linear viscoelastic materials being the best investigated. Their behaviour is discussed in great detail. They are followed by the damping of assemblies, relevant to the user, by its mathematical characterisation and its relation to material damping. Models for damped structures are discussed next, and the application of the boundary element method (BEM) is explained. Finally, as statements on damping rely on experiments, Part 5 describes established experimental techniques, possible instrumentation for the determination of damping characteristics, and analytical methods.
The notion of damping Damping in mechanical systems is understood to be the irreversible transition of mechanical energy into other forms of energy as found in time-dependent processes. Damping is mostly associated with the change of mechanical energy into thermal energy. Damping can also be caused by releasing energy into a surrounding medium. Electromagnetic and piezoelectric energy conversion can also give rise to damping if the energy converted is not returned to the mechanical system.
Classification of damping phenomena The physical causes of damping are multifarious. In addition to friction, wave propagation or flow effects, other possible causes are phase transitions in materials or energy conversion by piezoelectric, magnetostrictive, or electromechanical processes.
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Forces associated with damping are non-conservative. They can be internal or external forces. If both action and reaction forces in a free body diagram the damping force, are effective within the system boundaries, the effect is said to be an internal damping effect. Where the reaction force is effective outside the system boundaries, the effect is an external damping effect. Examples of internal damping are: • material damping due to nonelastic material behaviour • friction between components, e.g. in slide ways, gears, etc. • conversion of mechanical vibration energy into electrical energy by means of the piezoelectric effect and dissipation due to dielectric losses Examples of external damping are: • friction against the surrounding medium • air-borne-sound radiation into the environment • structure-borne-sound radiation into the ground Phenomenologically, the damping in a mechanical system can be composed of the following contributions: • Material damping The energy dissipation within a material, due to deformation and/or displacement, is called material damping. Its physical causes are, in essence: – in solids • heat flows induced by deformation (thermomechanical coupling) • slip effects • microplastic deformations • diffusion processes – in fluids • viscous flow losses • Contact-surface damping Relative motion, friction Contact-surface damping is caused by relative motions in the contact surfaces of joined components such as screwed, riveted, and clamped joints. The physical causes are: – friction due to relative motions in the contact surface – pumping losses in the enclosed medium due to relative motion in a direction normal to the contact surface (e.g. gas pumping) The term “structural damping” includes: • Damping in guides This includes energy dissipation in longitudinal guides (e.g. slides) and circular guides (e.g. journal bearings). • Electromechanical damping Electromechanical damping can be caused by piezoelectric, magnetostrictive, or electromagnetic effects. • Energy release to the surrounding medium This includes: – air damping – fluid damping – bedding damping
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Notes on modern, computer-based analytical and measurement programs Whereas the mass and stiffness matrices of relatively complex structures can be readily determined nowadays using threedimensional CAD drawings, automatic grid generation, and subsequent FEM analysis, an appropriate calculation model cannot usually be established which sufficiently precise information on damping. More precise damping parameters contains can be determined experimentally. “Experimental Modal Analysis” (EMA) has become established as the suitable tool worldwide. It uses measured frequencyresponse curves between appropriately chosen excitation points and measuring points, and modern curve-fitting techniques for identifying the modal parameters: natural frequencies, eigenmodes, and modal damping ratios. In the case of simple structures, the system can be excited by means of a hammer inpact. In the case of complex components and considerable damping, excitation using one or several exciters has proven convenient, allowing to control exciter amplitudes and energy distribution for selected frequency ranges. The system response is often measured by means of piezoelectric accelerometers or laser-optical sensors. Modern measurement and analytical systems offer the possibility to identify discrete damping couplings provided that the substructures have been separately investigated beforehand. Link modules allow to establish the connection between the results of experimental modal analysis and the calculated FEM analysis (e.g. matching of nodal points and coordinate axes through interpolation). Quality criteria such as MAC (Modal Assurance Criterion) compare the relations (such as orthogonality) between the eigenmodes found in terms of the scalar product of the eigenvectors. Additional normalisation using the mass or stiffness matrix allows a quantitative assessment. After model updating on the modal level, including damping ratios determined by experiment, operation vibrations can be calculated for any load function. The simulation model which was developed step by step can thus be verified under practical conditions.
Content of tutorial The content of the guideline VDI 3830 is explained in the tutorial along with physics, theory, numerical approaches, and practical applications taken from review articles and archival publications of the tutor and his coworkers focussed on damping topics. 1. L. Gaul: The Influence of Damping on Waves and Vibrations Mechanical Systems and Signal processing (1999) 13(1),1-30 Wave propagations and vibrations are associated with the removal of energy by dissipation or radiation. In mechanical systems damping forces causing dissipation are often small compared to restoring and inertia forces, However, their influence can be great and is discussed in the present survey paper together with the transmission of energy away from the system by radiation. Viscoelastic constitutive equations with integer and fractional time derivatives for the description of stress relaxation and creep of strain as well as for the description of stress-strain damping hysteresis under cyclic oscillations are compared. Semi-analytical solutions of wave propagation and transient vibration problems are obtained by integral transformation and elastic-viscoelastic correspondence principle. The numerical solution of boundary value problems requires discretization methods. Generalized damping descriptions are incorporated in frequency and time domain formulations for the boundary element method and the finite element method. 2. L. Gaul and R. Nitsche: The Role of Friction in Mechanical Joints Appl Mech Rev vol 54, no 2, March 2001, 93-109 Vibration properties of most assembled mechanical systems depend on frictional damping in joints. The nonlinear transfer behavior of the frictional interfaces often provides the dominant damping mechanism in a built-up structure and plays an important role in the vibratory response of the structure. For improving the performance of systems, many studies have been carried out to predict, measure, and/or enhance the energy dissipation of friction. This article reviews approaches for describing the nonlinear transfer behavior of bolted joint connections. It gives an overview of modeling issues. The models include classical and practical engineering models. Constitutive and phenomenological friction models describing the nonlinear transfer behavior of joints are discussed. The models deal with the inherent nonlinearity of contact forces (e. g. Hertzian contact), and the nonlinear relationship between friction and relative velocity in the friction interface. The research activities in this area are a combination of theoretical, numerical, and experimental investigations. Various solution techniques, commonly applied to friction- damped systems, are presented and discussed. Recent applications are outlined with regard to the use of joints as semi-active damping devices for vibration control. Several application areas for friction damped systems due to mechanical joints and connections like shells and beams with friction boundaries are presented. This review article includes 134 references.
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3. A. Schmidt and L. Gaul: Finite Element Formulation of Viscoelastic Constitutive Equations Using Fractional Time Derivatives Nonlinear Dynamics 29, 37-55, 2002 Fractional time derivatives are used to deduce a generalization of viscoelastic constitutive equations of differential operator type. These so-called fractional constitutive equations result in improved curve-fitting properties, especially when experimental data from long time intervals or spanning several frequency decades need to be fitted. Compared to integer-order time derivative concepts less parameters are required. In addition, fractional constitutive equations lead to causal behavior and the concept of fractional derivatives can be physically justified providing a foundation of fractional constitutive equations. First, three-dimensional fractional constitutive equations based on the Gr¨ unwaldian formulation are derived and their implementation into an elastic FE code is demonstrated. Then, parameter identifications for the fractional 3-parameter model in the time domain as well as in the frequency domain are carried out and compared to integerorder derivative constitutive equations. As a result the improved performance of fractional constitutive equations becomes obvious. Finally, the identified material model is used to perform an FE time stepping analysis of a viscoelastic structure. 4. L. Gaul and M. Schanz: A comparative study of three boundary element approaches to calculate the transient response of viscoelastic solids with unbounded domains Comput. Methods Appl. Mech. Engrg. 179 (1999) 111-123 As an alternative to domain discretization methods, the boundary element method (BEM) provides a powerful tool for the calculation of dynamic structural response in frequency and time domain. Field equations of motion and boundary conditions are cast into boundary integral equations (BIE), which are discretized only on the boundary. Fundamental solutions are used as weighting functions in the BIE which fulfil the Sommerfeld radiation condition, i.e., the energy radiation into a surrounding medium is modelled correctly. Therefore, infinite and semi-infinite domains can be effectively treated by the method. The soil represents such a semi-infinite domain in soil-structureinteraction problems. The response to vibratory loads superimposed to static pre-loads can often be calculated by linear viscoelastic constitutive equations. Conventional viscoelastic constitutive equations can be generalized by taking fractional order time derivatives into account. In the present paper two time domain BEM approaches including generalized viscoelastic behaviour are compared with the Laplace domain BEM approach and subsequent numerical inverse transformation. One of the presented time domain approaches uses an analytical integration of the elastodynamic BIE in a time step. Viscoelastic constitutive properties are introduced after Laplace transformation by means of an elasticviscoelastic correspondence principle. The transient response is obtained by inverse transformation in each time step. The other time domain approach is based on the so-called ‘convolution quadrature method’. In this formulation, the convolution integral in the BIE is numerically approximated by a quadrature formula whose weights are determined by the same Laplace transformed fundamental solutions used in the first method and a linear multistep method. A numerical study of wave propagation problems in 3-d viscoelastic continuum is performed for comparing the three BEM formulations. 5. L. Gaul, H. Albrecht and J. Wirnitzer: Semi-active friction damping of large space truss structures Shock and Vibration 11 (2004) 173-186 The authors dedicate this paper to the memory of Professor Bruno Piombo. We commemorate him as a vital contributor to our science. From the experience of sharing conferences and workshops with Bruno since many years, learning from his expertise and appreciating his advice, the first author mourns the loss of a good friend whose works and words will be kept in our minds and hearts. The present approach for vibration suppression of flexible structures is based on friction damping in semi-active joints. At optimal locations conventional rigid connections of a large truss structure are replaced by semi-active friction joints. Two different concepts for the control of the normal forces in the friction interfaces are implemented. In the first approach each semi-active joint has its own local feedback controller, whereas the second concept uses a global, clipped-optimal controller. Simulation results of a 10-bay truss structure show the potential of the proposed semi-active concept.
BookID 214574_ChapID 15_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII
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T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_15, © The Society for Experimental Mechanics, Inc. 2011
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BookID 214574_ChapID 16_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Second Order Blind Source Separation techniques (SO-BSS) and their relation to Stochastic Subspace Identification (SSI) algorithm J. Antoni+, S. Chauhan* +
Department of Mechanics, University of Technology of Compiegne, Centre de Recherche de Royallieu, BP 20529 – 60205, Compiegne, France *
Bruel & Kjaer Sound and Vibration Measurement A/S Skodsborgvej 307, DK 2850, Naerum, Denmark Email:
[email protected],
[email protected]
Abstract Blind Source Separation (BSS) and Independent Component Analysis (ICA) techniques are emerging signal processing techniques that aim at identifying statistically independent sources from a linear mixture of such sources without requiring (or with little) a priori information about the input source signals. Recently, it has been shown that these techniques can also be utilized for Operational Modal Analysis (OMA), or Output-only Modal Analysis, where system characteristics are identified only on the basis of information available from the measured outputs. Second Order Blind Source Separation (SO-BSS) techniques are BSS algorithms that employ diagonalization of output correlation matrices in order to recover information about the original sources and the mixing matrix. Work presented in this paper aims at establishing a link between SO-BSS techniques (such as AMUSE and SOBI) and Stochastic Subspace Iteration (SSI) algorithm, which is a well known OMA algorithm. The paper presents the mathematical theory behind these algorithms and shows how these algorithms are related. In this manner, this work helps in enhancing the overall understanding of BSS techniques and their subsequent use for modal analysis purposes.
1. Introduction Blind Source Separation (BSS) techniques have been recently, a subject of research amongst structural dynamics community for the purpose of modal parameter extraction [1-4]. These works have shown, how second order BSS techniques (SO-BSS) can be used for this purpose and have also stated various advantages and limitations of these techniques with regards to modal parameter extraction. Stochastic Subspace Iteration (SSI) technique [5, 6], on the other hand, is a well known Operational Modal Analysis (OMA) algorithm which is based on parametric stochastic state space model. Because they originate from different disciplines, SO-BSS and SSI appear to be two different and competing approaches to solve the OMA problem. Further, the mathematical principles behind the two algorithms are quite rigourous and it’s difficult to see the common thread amongst the two approaches. The objective of this paper is to show that they are actually more similar than expected, in the sense that they both achieve nearly the same factorization of the covariance matrix of observations at several time lags, with slightly different algorithms. As a matter of fact, it is shown that AMUSE is a particular case of SSI. Another result of this paper is the assumptions under which SO-BSS is valid to solve the OMA problem, which, as per the current state of art, appear to be more restrictive as compared to those of SSI: in theory, SO-BSS requires the system to be undamped; however, it happens to be very robust against this assumption for most practical purposes. T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_16, © The Society for Experimental Mechanics, Inc. 2011
177
BookID 214574_ChapID 16_Proof# 1 - 23/04/2011
178 Paper is arranged in the following manner; Section 2 describes briefly the basic principles of blind source separation, along with mathematical details of two SO-BSS techniques, Algorithm for Multiple Unknown Signals Extraction (AMUSE) and Second Order Blind Separation (SOBI). Section 3, lists the problem statement and assumptions with regards to estimating modal parameters of n degree of freedom dynamic system. Along with the basic formulation of the problem in state space form, this section also illustrates the main objective of OMA algorithms in mathematical terms. Next section uses this theoretical background to understand application of SO-BSS techniques for OMA purposes. Finally, Section 5 establishes the connection between the two approaches, showing relation between AMUSE and SSI, and algorithmic level similarities between SOBI and SSI.
2. Second Order based Blind Source Separation (SO-BSS) Blind Source Separation (BSS) can be seen as an extension to Principal Component Analysis (PCA) which aims at recovering the source signals from the set observed instantaneous linear mixtures without any a priori knowledge of the mixing system. Yet another term which is often used in conjunction with BSS is Independent Component Analysis (ICA) which refers to the subset of BSS techniques that typically use higher order statistics (HOS) based methods for recovering source signals. Mathematically, BSS problem can be formulated as
x(t ) = As (t )
(01)
where x(t) is a column vector of m output observations representing an instantaneous linear mixture of source signals s(t) which is a column vector of n sources at time instant t. A is an m X n matrix referred to as “mixing system” or more commonly as “mixing matrix”. There are number of good references giving a detailed discussion of the subject of BSS and ICA and interested readers can refer to these works for more details [7-13]. In addition to these resources readers can also refer to the special issue on ICA and BSS published by Mechanical Systems and Signal Processing in 2005 [14], which focuses mainly on the use of these techniques for vibration and structural dynamics related applications. Unlike HOS based methods which exploit statistical independence of non Gaussian sources, Second order statistics (SOS) methods exploit weaker conditions for separating the sources assuming that they have a temporal structure with different autocorrelation functions (or power spectra) and are mutually uncorrelated (having zero cross-correlation). Generally speaking, SO-BSS refers to the class of BSS algorithms which seeks to separate a mixture of sources by forcing their mutual decorrelation. Under the assumption of stationarity of the sources, the problem is identifiable provided that the sources also have different spectra. More precisely, decorrelation at two different time-lags is a necessary condition for the problem to find a unique solution. This is the basic idea behind AMUSE, which is one of the simplest SO-BSS algorithm. However, AMUSE is not very robust, since it uses only two time lags. A more stable solution is provided by SOBI which alleviates the difficulty of selecting the good time-lags by forcing decorrelation at several of them jointly. The two SO-BSS algorithms, that are the focus of this paper (AMUSE and SOBI), differ in the manner in which they solve the blind identification problem, using second order statistics (or more precisely correlation functions at different time lags) and temporal structure of the sources (sources having different power spectra and being mutually uncorrelated). Discussion to follow details AMUSE and SOBI algorithms. The fundamental difference in the two approaches will be clearer in lieu of these explanations of the two algorithms. This will also lay down the theoretical background that will help in establishing the link of these algorithms with Stochastic Subspace Iterative (SSI) algorithm.
2.1 Algorithm for Multiple Unknown Signals Extraction (AMUSE) [7, 15] AMUSE belongs to the class of SO-BSS techniques that utilizes the concept of simultaneous diagonalization for the purpose of solving the instantaneous mixing model shown in Eq. (1). For the case, where number of sources are equal to number of sensors (or observations) i.e. m = n, AMUSE carries out simultaneous diagonalization of two symmetric matrices according to the steps listed below, thus obtaining the mixing matrix and the sources.
BookID 214574_ChapID 16_Proof# 1 - 23/04/2011
179 1. Estimate the covariance (mean removed correlation) matrix of the output observations
1 Rˆ x (0) = N where
N
¦ x(k ) k =1
xT (k )
(02)
Rˆ x (0 ) is the covariance matrix at zero time lag and N is the total number of time samples
taken. 2. Compute EVD (or SVD) of
Rˆ x (0 )
Rˆ x (0 ) = U x ¦ x V xT = V x Λ xV xT = Vs Λ sV sT + V N Λ N V NT
(03)
where Vs is m X n matrix of eigenvectors associated with n principal eigenvalues of ȁs = diag{Ȝ1, Ȝ2, ….., Ȝn} in descending order. Vn is m X (m-n) matrix containing the (m-n) noise eigenvectors associated with noise eigenvalues ȁn = diag{Ȝn+1, Ȝn+2, ….., Ȝm}. The number of sources n is thus estimated based on the n most significant eigenvalues (or singular values in case of SVD). 3. Perform pre-whitening transformation
x (k ) = Λ s 2VsT x(k ) = Qx(k ) −1
(04)
4. Estimate the covariance matrix of the vector x (k ) for specific time lag other than Perform SVD on the estimated covariance matrix.
1 Rˆ x (τ ) = N
N
¦ x (k ) k =1
x T (k − τ ) = U x ¦ x VxT
τ
=0 (say
τ
=1).
(05)
5. The mixing matrix and source signals can be estimated as 1 Aˆ = Q +U x = Vs Λ s2U x
(06)
y (k ) = sˆ(k ) = U xT x (k )
(07)
In general, it has been observed that time lag τ = 1 is often a good choice. AMUSE performs well for colored sources with different power spectra shapes which means that the eigenvalues of the time-delayed covariance matrix are distinct. The accuracy of AMUSE however deteriorates in presence of additive noise.
2.2 Second Order Blind Identification (SOBI) [7, 16] AMUSE suffers from the limitation that since it’s based on EVD/SVD based simultaneous diagonalization of only two matrices, good choice of τ (the time lag other than zeroth time lag), corresponding to which the correlation matrix
Rˆ x (τ ) is to be diagonalized along with Rˆ x (0 ) , becomes very important. Choice of τ is
very critical and though τ = 1 is often the criterion, performance of AMUSE deteriorates significantly in presence of noise and it’s not easy to find a suitable τ . This deficiency of AMUSE can be dealt with by exploring average eigen-structure of more matrices than just two as is the case with AMUSE.
Yet another way of limiting disadvantages of AMUSE is by adopting joint diagonalization process instead of simultaneous diagonalization using EVD/SVD techniques. SOBI algorithm is a SO-BSS technique that utilizes joint diagonalization procedure to find an orthogonal matrix that diagonalizes a set of covariance (zero mean correlation) matrices [7, 17, 18]. SOBI works well for simple colored sources with distinct power spectra (or distinct autocorrelation functions) and like AMUSE, it operates on time delayed covariance matrices. SOBI utilizes the pre-whitening transformation similar to that described in case of AMUSE. This is followed by estimation of set of covariance matrices for a preselected set of time lags (τ 1 ,τ 2 ,......,τ L )
1 Rˆ x (τ i ) = N
N
¦ x (k )x (k − τ ) = QRˆ (τ )Q T
i
k =1
x
i
T
(10)
BookID 214574_ChapID 16_Proof# 1 - 23/04/2011
180 Joint approximate diagonalization (JAD) is performed on the above matrices;
Rx (τ i ) = U D(τ i ) U T , to
estimate the orthogonal matrix U that diagonalizes a set of covariance matrices. Several efficient algorithms are available for this purpose including Jacobi techniques, Alternating Least Squares, Parallel Factor Analysis etc. [17, 18]. Finally the sources and signals can be estimated using the same equations as explained earlier with AMUSE. It should be noted that D (τ i ) is a diagonal matrix that has distinct diagonal
entries. However, it is difficult to determine a priori a single time lag τ at which the above criterion is satisfied. Joint diagonalizaton procedure avoids this difficulty by providing an optimum solution considering a number of time lags.
Typically, most SO-BSS algorithms seek the separating matrix W as the joint diagonalizer of the set of covariance matrices R yy [τ ] at several time-lags τ in some set A based on the minimization of the general cost function
~ C (W ) = ¦ wτ . off (W H R yy [τ ]W )
2
(11)
τ ∈A
where
2
stands for any valid matrix norm, off( ) for the operator that extracts the off-diagonal elements of a
matrix, and where
{wτ ;τ ∈ A} is collection of positive weights. As clear from above discussion, AMUSE and
SOBI turn out to be particular cases performing the above minimization.
3. Problem statement, notations and assumptions 3.1 Equations of motion and assumptions Eq. (12) describes the matrix differential equation of an n-degree-of-freedom system with mass, stiffness, and damping matrices
M ∈ ℜn×n , K ∈ ℜ n×n , C ∈ ℜ n×n Mx(t ) + Cx (t ) + Kx(t ) = F (t )
(12)
n×1
n×1
Where F (t ) ∈ ℜ is the forcing vector which entails the displacement vector x (t ) ∈ ℜ . Note that number of forces acting on the system is considered to be same as the number of measured responses. The vector of measured signals (the observations) is given by,
y (t ) = x i (t ) + ν (t ), ∈ ℜ n×1
(13)
where i = 0,1,2 depending on whether displacements, velocities or accelerations are measured, and a vector of measurement noise. It is further assumed that, as in OMA, F (t ) and random stationary, broadband, and mutually uncorrelated vector processes.
ν (t )
ν (t )
is
are two zero-mean
The equation of motion, as presented above, can also be described in the state space form. State-space formulation of the equation of motion is a key point to highlight the connection between SO-BSS and SSI. As will be shown in this section, it results in a expression consisting covariance matrices of the observations, which clearly points out that SO-BSS and SSI are two different routes to tackle the same factorization problem. From linear system theory, the discrete-time state-space formulation of the equation of motion is
X [ k + 1] = Ad X [ k ] + Bd F [ k ] y[ k ] = Cdi X [ k ] + Ddi F [ k ] + ν [ k ] where
x[k ] = [ x (kTs )t , x(kTs )t ]t ∈ ℜ 2 n×1 with Ts the sampling period,
y[k ] = y ( kTs ),ν [k ] = ν (kTs ), Ad = exp{ AcTs }∈ ℜ 2 n×2 n with
(14)
BookID 214574_ChapID 16_Proof# 1 - 23/04/2011
181
ª− M −1C Ac = « ¬ I n×n
− M −1 K º 2 n× 2 n » ∈ℜ 0 n× n ¼
(15)
Δt
Bd = ³ Bc exp{ Ac t}dt ∈ ℜ 2 n×n with Bc = [( M −1 ) t 0 n×n ]t ∈ ℜ 2 n×n , and Cdi ∈ ℜ m×2 n and Ddi ∈ ℜ m×2 n are two 0
matrices whose exact expressions depend on whether displacements (i=0), velocities (i=1) or accelerations (i=2) are measured in Eq. (13).
3.2 OMA Objectives The objective of OMA, whether it is approached by a SO-BSS or a SSI strategy, is to recover the modal parameters of the system, i.e. the poles Ȝi and the eigenvectors iji, from the observations y(t) only. Specifically, in both cases the unknowns of the problem are the diagonal matrix
Λ = diag (λ1 , λ2 , λ3 ,....., λn ) ∈ ℜ n×n of poles (for simplicity it will be assumed that all poles are distinct, i.e.
λi ≠ λ j , i ≠ j )
and
(λ M + Ȝ C + K )ϕ 2 i
i
i
the
modal
matrix
Φ = [ϕ1 , ϕ 2 , ϕ 3 ,....., ϕ n ]∈ ℜ n×n of eigenvectors such that
= 0 for any i.
One slight difference in philosophy is that SSI attempts to estimate all the modal parameters by once, whereas SO-BSS techniques utilizes the modal expansion theorem [20] and the concept of modal filters [19] to estimate the modal parameters [3]. SO-BSS first estimates the modal matrix Ɏ, and then separate modal coordinates η (t ) ∈ ℜ by uncoupling the "mixture"
x(t ) = Φη (t ) η (t ) = Φ −1 x(t )
(16)
from which the poles are finally estimated from standard single-degree-of-freedom techniques. Just as for Eq. (12), the set of difference equations (14) can be decoupled by performing a modal decomposition
X [k ] = Ψq[k ] where
(17)
q[ k ] ∈[ ]2 n×1 is the vector of state-space modal coordinates and Ψ ∈[ ]2 n×2 n the state-space modal
matrix such that
Ad = ΨΣ d Ψ −1 with Ȉd a diagonal matrix. The state-space modal parameters are related to
the "physical" modal parameters ȁ and Ɏ as
ªΦη[k ]º q[k ] = Ψ −1 « » ¬Φη[k ]¼ 0 n× n º ªΦΛ Φ *Λ* º ªexp{ΛTs } Σd = « and Ψ = « » exp{Λ*Ts }»¼ Φ* ¼ ¬ 0 n×n ¬Φ
(18)
(19)
where * stands for the conjugate operator. These equivalences will turn out useful in the rest of the paper to establish the connections between SO-BSS -- which is explicitly expressed in terms of ȁ and Ɏ -- and SSI -which is more concisely expressed in terms of Ȉd and ȥ.
3.3 Covariance equation The starting point for both SO-BSS and SSI is the covariance matrix of the observations y(t) which, under the assumption of stationarity, is defined as
BookID 214574_ChapID 16_Proof# 1 - 23/04/2011
182 K 1 y[k + τ ] y[k ]t = y[k + τ ] y[k ]t ¦ K →∞ 2 K + 1 k =− K
R yy [τ ] = lim
with
t
the transpose operator and
(20)
the time-averaging operator. It is now shown how the covariance
matrix factorizes in terms of the unknown modal parameters, a central result to understand the mechanisms of SO-BSS and SSI. From the theory of linear system applied to Eq. (14), it is an easy matter to establish that
R yy [τ ] = Cd Adr −1[ Ad RXX [0] Adt + Bd RFF [τ ]Ddt + Rνν [τ ]], τ > 0
(21)
or, in terms of the state-space modal parameters,
R yy [τ ] = LΣ rd−1 R + LΣ rd−1 Bd RFF [τ ]Ddt + Rνν [τ ], τ > 0 with
(22)
L = Cd Ψ and R = Σ d Rqq [0]Ψ t Cdt .
Several important inferences can be drawn at this stage. First of all, it is seen that the covariance matrix depends on three terms which account respectively for the free response of the system, the forced response, and the measurement noise. Obviously only the first term is to be used in OMA, since it contains nothing else than the dynamics of the system. The second term, which involves the dynamics of the unknown forces, is zero only when displacements or velocity are measured (Dd = 0), but not accelerations (Dd 0). In the later case, however, that term can be made small as possible with respect to the first term by considering large enough values of τ , say τ ≥ τ 0 > 0 , when the forces are broadband enough as compared to the resonances of the system (i.e. provided that
RFF [τ ] decreases faster than Σ rd−1 ), which is a fundamental
assumption for OMA. Following a similar reasoning, the third term can be neglected provided the measurement noise is broadband enough. Therefore Eq. (22) simplifies to
R yy [τ ] = LΣ rd−1 R, τ ≥ τ 0 for some
(23)
τ0 > 0 .
Secondly, from OMA point of view, it is clear that main objective of OMA algorithms comes down to identifying the first two factors (or parts of) of the covariance matrix expression in Eq. (23), where the left one (the state-space modal participation factors L = Cd Ψ ) directly gives access to the eigenvectors of the system, and the middle one, Ȉd, to the poles of the system -- see Eq. (19). Rest of the paper, demonstrates how SO-BSS and SSI perform the same factorization to achieve OMA objectives, in seemingly different yet related manner.
4. Application of SO-BSS Techniques for OMA SO-BSS techniques seek identification of parts of the left factor L appearing in Eq. (23). Rearranging Eq. (23) as
R yy [τ ] = LRqq [τ ]LH , τ ≥ τ 0 where
H
stands for the transpose conjugate operator and where the property
(24)
Rqq [τ ] = Σ rd Rqq [0] has been
used. This is the usual symmetrical form that enters SO-BSS algorithms. It is immediately seen that for SOBSS to correctly consider L as the joint diagonalizer of the covariance matrix R yy [τ ] at time lags τ ≥ τ 0 ,
Rqq [τ ] must indeed be a diagonal matrix at these time-lags. For this to happen, the modal coordinates should have nearly disjoint spectra in the frequency domain (uncoupled resonances) or, in other words, are approximately uncorrelated in the time domain [1, 2]. Physically for an arbitrary loading, this is the case when, and only when, the damping of the system is light and the system does not have closely spaced
BookID 214574_ChapID 16_Proof# 1 - 23/04/2011
183 modes. It should however be noted that SO-BSS seems very robust against this assumption, and that in many occasions it seems to return reasonable results even with significantly damped modes. Under this assumption, it can be shown that
Rqq [τ ] = q[ k + τ ] q[ k ]H ≈
0 n× n º 1 ªΔ exp(ΛτTs ) « * 0 n× n Δ exp(Λ τTs )»¼ 2¬
(25)
n× n
with Δ ∈[ ] an arbitrary diagonal matrix. Inserting this result into Eq. (24), and performing some elementary manipulations, it can be shown that
{
}
LRqq [τ ]LH ≈ Φ R ℜ (ΛΛ* ) t Δ exp{ΛτTs } Φ tR
(26)
with i = 0,1,2 depending on whether displacements, velocities or accelerations are measured. In all three cases, the real part of the modal matrix, Φ R , is found as the joint diagonalizer of the response observation’s covariance matrix for time-lags
τ ≥τ0.
Note that the term joint diagonalization applies to SO-BSS techniques like SOBI that use joint diagonalization approach to solving the instantaneous mixing model problem. Above discussion hold true for simultaneous diagonalization based approaches, like AMUSE, as well. Important aspect of SO-BSS techniques is, as shown above, that they seek a matrix that diagonalizes the covariance matrix (or matrices). It has been further observed that joint diagonalization should be performed for time-lags greater than some strictly positive value, τ 0 > 0 , in order to circumvent the effects of the unmeasured forces and of the measurement noise in the covariance matrix; this is in contrast with the usual habit to diagonalize the covariance matrix at τ = 0 in a first step, as explicitly done in AMUSE or in SOBI. Some of these techniques are explained in [7]. Due to the mathematical formulation of the SO-BSS problem, these techniques find modal matrix which is real. This is also one of the reasons, why these methods will serve well while identifying lightly damped systems. The resulting diagonalized matrices are then seen to contain, in their diagonal entries, the covariance functions of the modal coordinates, i.e. Rη iηi [τ ] ∝ exp{−γ iτTs } cos{ jωiτTs } from which the modal parameters Ȧi (eigen-frequencies) and
ζ i = tan −1 (γ i / ωi )
(damping ratios) can be directly and easily
estimated from standard single-degree-of-freedom techniques.
5. SO-BSS and SSI Stochastic Subspace iteration algorithm (SSI) [5, 6] is one of the most commonly used OMA algorithms. It is a general algorithm that purposely seeks the left, middle and right factors of the covariance matrix in Eq. (23). It starts from building the (mK1 X mK2) block Hankel covariance matrix
R yy [τ ] β1.R yy [τ + 1] ª « β1.Ryy [τ + 1] β12 .Ryy [τ + 2] « [ H K1 , K 2 ] = « « «¬β K1 −1.Ryy [τ + K1 − 1] β K1 −1β1.R yy [τ + K1 ]
β K 2 −1.Ryy [τ + K 2 − 1] º » β1β K 2 −1.R yy [τ + K 2 ] » » » β K1 −1β K 2 −1.R yy [τ + K1 + K 2 − 2]»¼
where {ȕi > 0; i = 1,2,…..max(K1, K2) - 1} is a sequence of user-defined weights. defined Hankel covariance matrix in Eq. (27) factorizes as
(27)
From Eq. (22), the so-
BookID 214574_ChapID 16_Proof# 1 - 23/04/2011
184
[H
K1 , K 2
]
L ª º « β .LΣ » 1 d » Στ −1 R β .RΣ =« 1 d « » d « K1 −1 » «¬ β K1 −1.LΣ d »¼
[
]
β K 2 −1.RΣ dK 2 −1 , τ ≥ τ 0
(28)
Following this, estimation of L and Σ d is a two-step procedure: i) Compute the singular value decomposition
[ H K1 , K 2 ] = UDV H truncated to 2n terms (i.e. such that
D = diag {d1 , d 2 ,....., d 2 n }∈ [ ] with d1 d2 … d2n), and & & −1 / 2 ii) Compute the singular value decomposition D U H [ H K1 , K 2 ]VD −1 / 2 = QΣQ H (where [ H K1 ,K 2 ] is U ∈[
]nK ×2n , V ∈ [ ]2 n×nK 1
2
2 n× 2 n
, and
block-shifted Hankel matrix) and form the estimates
Lˆ = I n×2 nUD −1/ 2Q Σˆ d = Σ
(29)
where InX2n is made up of the first n rows of the identity matrix.
5.1 The Connection between AMUSE and SSI Since they both involve a singular/eigen-value decomposition of the observation covariance matrices, SSI and AMUSE bear a striking similarity. It is not difficult to observe this connection and realize that AMUSE is a particular case of SSI. Indeed, by taking τ = 0 and K1 = K 2 = 1 in Eq. (21, 22), and shrinking the order of the system to n instead of 2n in step (ii) above, the two algorithms are strictly equivalent. As a matter of fact, if the system has no (or light) damping, all the poles λi are aligned on the imaginary axis with multiplicity two, so that only n unknowns are sought (their imaginary parts). Under the same assumption, the eigenvectors are real, thus dividing by two the number of unknowns in L. This explains why SSI will work in this particular case with only half the usual system order. However, considering time-lag τ = 0 is probably not recommended, as explained in section 4.
5.2. The connection between SOBI and SSI Unlike AMUSE, the connection between SOBI and SSI is not as obvious. This section explains this connection by means of the minimizing cost functions that the two algorithms use. To understand this Eq. (27, 28) are revisited with the generalization that K1 > 1, K2 > 1. Eqs. (27, 28) also show that SSI is a suboptimal way of solving the least square problem
[
C ( L, Σ d ) = H K1 ,K 2
with respect to L and Ȉd, where
]
L ª º « β .L Σ » 1 d » Στ −1 R β .RΣ −« 1 d « » d « K1 −1 » ¬« β K1 −1.LΣ d ¼»
[
2
β K 2 −1.RΣ dK 2 −1
]
(30)
A = ¦ Aij2 stands for the Frobenius norm of a matrix A. The above cost 2
ij
function expands as
C ( L, Σ d ) =
K1 −1 K 2 −1
¦ ¦β β . R k
k =0
l =0
l
yy
[τ + k + l ] − LΣτd−1+ k +l R
2
(31)
BookID 214574_ChapID 16_Proof# 1 - 23/04/2011
185 or, after a convenient change of notations, simply as
C ( L, Σ d ) = ¦ wτ . R yy [τ ] − LΣτd−1 R
2
(32)
τ ∈A
A = {τ ,.....,τ + K1 + K 2 − 2} and wτ a function of the ȕk's. In the special case where damping is
with
negligible (or light), this further simplifies as
{
}
C ( L, Σ d ) = ¦ wτ . R yy [τ ] − Φ R ℜ (ΛΛ* ) i Δ exp{ΛτTs } Φ tR τ ∈A
2
(33)
as per the results in section 4 (Eq. 26). Minimizers of the cost function shown in Eq. (33) will be the same as the following one,
~ C ( L, Σ d ) =
¦ wτ . τ ∈ A1
{
}
Φ −Rt R yy [τ ]Φ −R1 − ℜ (ΛΛ* ) i Δ exp{ΛτTs }
2
(34)
which trivially expands as
~ C (Φ R , Λ ) = ¦ wτ . off (Φ −Rt R yy [τ ]Φ −R1 ) τ ∈A
2
{
}
+ ¦ wτ . diag (Φ −Rt R yy [τ ]Φ −R1 ) − ℜ (ΛΛ* )i Δ exp{ΛτTs } τ ∈A
2
(35)
= Coff (Φ R ) + Cdiag (Φ R , Λ ) This is a combination of two cost functions: •
a cost function
Coff (Φ R ) which is minimized by a real modal matrix Φ R that diagonalizes the
covariance matrix •
R yy [τ ] at several time lags, and
a cost function
{
Cdiag (Φ R , Λ ) which is minimized by a diagonal matrix Λ such that
}
ℜ (ΛΛ ) Δ exp{ΛτTs } * i
matrix Φ
−t R
R yy [τ ]Φ
matches
the
diagonal
entries
of
the
diagonalized
covariance
−1 R .
At this point, several important conclusions can be drawn. First of all it is seen that under the assumption of negligible damping, SSI intrinsically contains the same cost function as SO-BSS techniques -- compare Eqs. (11) and (34, 35) -- that forces the joint diagonalization of the observation covariance matrix at several timelags. Additionally, SSI contains another cost function that forces the diagonal entries of diagonalized covariance matrix to be proportional to ℜ{exp{ΛτTs }} with Ȝi the poles of the system. In contrast, SO-BSS does not place such a constraint; the returned diagonalized covariance matrix is theoretically arbitrary, with no imposed structure. By means of above discussion, a relationship between SSI and SO-BSS techniques is identified on the basis of which it is shown that SSI generalizes SO-BSS algorithms in the sense that it intrinsically contains the same cost function. Thus on one hand, SSI will, in principle, perform as well as SO-BSS to recover the modal matrix as a joint diagonalizer. On the other hand, since SSI contains another cost function that forces an imposed structure to the diagonalized covariance matrix in terms of the poles of the system, which SOBSS does not, SSI is expected to perform better than SO-BSS in general. Yet another aspect to consider is the ability of SSI to find even the complex modal vectors and handle the cases with heavy damping. So far most of the studies, that have been conducted for investigating utilization of SO-BSS techniques, suggest that they are more applicable (in their basic formulation) to systems having negligible or light damping (thus ensuring modal vectors to be primarily real) as theoretically SO-BSS techniques do not accommodate high damping scenarios.
BookID 214574_ChapID 16_Proof# 1 - 23/04/2011
186 Further, in its current form, utility of SO-BSS techniques is limited to cases where one has more number of output responses in comparison to modes of interest. SO-BSS techniques are not able to deal with situations where there are fewer sensors than the number of modes. SSI, on the other hand, is capable of returning the estimates of the poles Ȝi and the modal participation matrix L even when number of sensors are limited; provided a sufficiently high model order (K1, K2 in Eq. 27) is chosen such that mK1 2n and mK2 2n, where m is number of sensors and n is number of modes of interest.
6. Conclusions and Scope for Future Work As interest in blind source separation techniques increases in the structural dynamics community, it is important to understand their advantages and limitations in comparison to existing modal parameter estimation techniques. Though these algorithms have emerged as a result of research carried out in medical imaging and wireless communication disciplines, they bear striking similarities to mathematical concepts involved in Operational Modal Analysis. Understanding the underlying commonalities and mathematical fundamentals between existing OMA techniques and BSS methods will greatly help in extending the use of BSS techniques for structural dynamics and other related applications. This paper highlights how SSI and SO-BSS techniques (such as AMUSE and SOBI) factorize the covariance matrices to obtain modal parameters. It is explained by means of this work that SSI and SO-BSS techniques share similar mathematical foundations. It is shown that AMUSE is a particular case of SSI and that SO-BSS techniques including SOBI minimize an objective function which is also a part of the objective function being minimized by SSI. This cost function can be explained in terms of modal expansion theorem and modal filters and minimization of this cost function reveals the modal filters which also happen to be unscaled modal vectors of the system. In addition to minimizing this objective function, SSI algorithm also minimizes a second cost function putting more constraints on the poles of system. On the basis of this understanding, along with other details, and the fact that SO-BSS have tighter assumptions (light damping and distinct modes, number of sensors to be equal to or more than modes of interest), it can be argued that SSI is a more powerful technique than SO-BSS for OMA related applications. However, having drawn above conclusions, it is also important to note that this study helps in bridging the gap between the two approaches and thus, lays foundation for further research in this subject. In future, results from this study can be used as the basis for further improvement of SO-BSS techniques for their application to OMA. This includes extension of these techniques to structures having heavily damped modes with complex mode shapes and structures with closely-spaced modes. Further, for applicability of SO-BSS techniques to practical OMA situations, it is important to investigate their robustness against noise and situations where number of sensors is smaller than number of modes of interest.
References [1] [2]
[3]
[4] [5] [6] [7] [8]
Kerschen, G., Poncelet, F., Golinval, J.C. (2007), “Physical Interpretation of Independent Component Analysis in Structural Dynamics”, Mechanical Systems and Signal Processing (21), pp. 1561-1575. Poncelet, F., Kerschen, G., Golinval, J.C. (2006), “Experimental Modal Analysis Using Blind Source Separation Techniques”; Proceedings of ISMA International Conference on Noise and Vibration Engineering, Katholieke Universiteit Leuven, Belgium. Chauhan, S., Martell, R., Allemang, R. J. and Brown, D. L. (2007), “Application of Independent Component Analysis and Blind Source Separation Techniques to Operational Modal Analysis”, Proceedings of the 25th IMAC, Orlando (FL), USA. McNiell, S.I., Zimmerman, D.C. (2008), “A Framework for Blind Modal Identification Using Joint Approximate Diagonalization”, Mechanical Systems and Signal Processing (22), pp. 1526-1548. Van Overschee, P., De Moor, B. (1996), Subspace Identification for Linear Systems: TheoryImplementations-Applications, Kluwer Academic Publishers, Dordrecht, Netherlands. Brincker, R., Andersen, P. (2006), “Understanding Stochastic Subspace Identification”, Proceedings of th the 24 IMAC, St. Louis, Missouri. Cichocki, A., Amari, S.; “Adaptive blind signal and image processing”, John Wiley and Sons, New York, 2002. Hyvarinen, A., Karhunen, J., Oja, E.; “Independent Component Analysis”, John Wiley and Sons, New York, 2001.
BookID 214574_ChapID 16_Proof# 1 - 23/04/2011
187 [9] [10] [11] [12] [13] [14] [15] [16]
[17] [18] [19]
[20]
Hyvarinen, A., Oja, E.; “Independent component analysis: Algorithms and applications”, Neural Networks, Vol. 13, p. 411-430, 2000. Cardoso, J.F.; “Blind signal separation: statistical principles”, Proceedings of the IEEE, Vol. 86, Number 10, pp. 2009-2025, October, 1998. Lathauwer, L.D., Bart De Moor, Vandewalle, J.; “An introduction to independent component analysis”, Journal of Chemometrics, Vol. 14, pp. 123-149, 2000. ICA Central, http://www.tsi.enst.fr/icacentral/ Tony Bell’s ICA Webpage, http://www.cnl.salk.edu/~tony/ica.html “Special Issue: Blind source separation”, Mechanical Systems and Signal Processing, Vol. 19 (6), pp. 1163-1380, November, 2005. Tong, L., Soon, V.C., Huang, Y., Liu, R.; “AMUSE: a new blind identification algorithm”, Proceedings of IEEE ISCAS, pp. 1784-1787, Vol. 3, New Orleans, LA, 1990. Belouchrani, A., Abed-Meraim, K.K., Cardoso, J.F., Moulines, E.; “Second order blind separation of correlated sources”, Proceedings of International Conference on Digital Signal Processing, pp. 346-351, 1993. Cardoso, J.F., Souloumiac, A.; “Jacobi angles for simultaneous diagonalization”, SIAM Journal of Matrix Analysis and Applications, Vol. 17, Number 1, pp. 161-164, January, 1996. nd Hori, G.; “A new approach to joint diagonalization”, Proceedings of 2 International Workshop on ICA and BSS, ICA’ 2000, pp. 151-155, Helsinki, Finland, June 2000. Shelly, S.J.; “Investigation of discrete modal filters for structural dynamic applications”, PhD Dissertation, Department of Mechanical, Industrial and Nuclear Engineering, University of Cincinnati, 1991. Allemang, R.J.; “Vibrations: Experimental modal analysis”, Structural Dynamics Research Laboratory, Department of Mechanical, Industrial and Nuclear Engineering, University of Cincinnati, 1999, http://www.sdrl.uc.edu/sdrl_jscript_homepage.html
BookID 214574_ChapID FM_Proof# 1 - 23/04/2011
BookID 214574_ChapID 17_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
8WLOL]DWLRQRI%OLQG6RXUFH6HSDUDWLRQ7HFKQLTXHVIRU0RGDO$QDO\VLV B. Swaminathan+, B. Sharma+, S. Chauhan* +
Structural Dynamics Research Lab, University of Cincinnati, Cincinnati, OH, USA *
Bruel & Kjaer Sound and Vibration Measurement A/S Skodsborgvej 307, DK 2850, Naerum, Denmark
Email:
[email protected][email protected],
[email protected]
$EVWUDFW In past few years, there have been attempts at utilizing Blind Source Separation (BSS) and Independent Component Analysis (ICA) techniques for modal analysis purposes. Most of the early work in this regard has been promising, though restricted to application of these techniques to analytical and laboratory based experimental structures. It is felt that in order to make these techniques applicable to more challenging scenarios, they need to be modified keeping in view the demands of modal parameter estimation procedure. This includes making them more robust and applicable to handle complex scenarios (for e.g. closely coupled modes, heavily damped modes, low signal-to-noise ratio, etc.). This forms the motivation for this paper which aims at tuning BSS / ICA methods for modal analysis purposes in an effective and efficient manner. Amongst other methods, it is shown how to incorporate signal processing techniques, modify BSS techniques to handle data in specified frequency ranges, extract modal parameters from limited output channels, etc. to derive most benefits out of these algorithms.
,QWURGXFWLRQ Blind Source Separation techniques emerged from the medical imaging and wireless communication fields as image or signal processing techniques. Antoni [1] demonstrated how these techniques can be used for blind separation of vibration components and laid down the associated principles. This was followed by works that attempted at showing the application of these techniques for modal analysis [2-5]. These early works aimed at establishing a connection between BSS techniques and Operational Modal Analysis (OMA). These studies showed encouraging results, by means of application of BSS techniques to analytical and simple laboratory structures and estimating modal parameters in the process. As fundamental principles of application of BSS techniques for OMA became clear and understood, their limitations in this regard also came into light. Mathematical formulation of BSS techniques shows that they are more suitable to handle systems having no or negligible damping [5]. Early simulations, however, suggest that this does not pose serious challenge as long as the modal coordinates [4] have distinct spectra and are mutually uncorrelated. However, the verdict is still not clear and more rigourous study needs to be conducted. Yet another limitation, on account of mathematical foundations of BSS techniques, is that they find real modal vectors, which might not be the case for most real-life structures. A framework to extend BSS techniques to take this limitation into account was recently suggested in [6]. Further, the effect of noise on the performance of these techniques is not yet evaluated (for modal analysis applications). Typically, these techniques work directly on the raw data without much signal pre-processing and thus their performance is more sensitive to the quality of the data acquired. Current formulation of most BSS algorithms also poses a limitation on account of the fact that one can estimate only as many modes as the number of output responses being measured. T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_17, © The Society for Experimental Mechanics, Inc. 2011
189
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The work presented in this paper attempts at addressing some of these issues by means of utilizing Second Order Blind Identification (SOBI) [7, 8] algorithm. In light of above discussion, a modified formulation of SOBI algorithm is provided in the paper that aims at circumventing the limitation that only as many modes can be found as the number of sensors measuring the output response. This modification also includes use of signal processing techniques such as averaging and windowing to reduce the effect of noise. Suggested modifications, thus improve the overall performance of SOBI algorithm and contributes towards making it more suitable for OMA purposes and widely applicable to more realistic structures. Section 2 presents SOBI in its original form and its modified form developed in this work, along with revisiting the link between BSS and OMA. Section 3 demonstrates the positive effect these modifications have on overall results, by means of studies conducted on a 15 degree-of-freedom analytical system and a lightly damped circular plate. Finally, based on these results, conclusions are drawn in Section 4.
6HFRQG2UGHU%OLQG,GHQWLILFDWLRQ 62%, >@ 7KHRUHWLFDO%DFNJURXQG Mathematically, an instantaneous BSS problem, in time domain, can be formulated as
x t
As t
(01)
where x(t) is a column vector of m output observations representing an instantaneous linear mixture of source signals s(t), which is a column vector of n sources at time instant t. A is an m X n matrix referred to as “mixing system” or more commonly as “mixing matrix”. SOBI is a BSS algorithm that separates the sources assuming that they have a temporal structure with different autocorrelation functions (or power spectra) and are mutually uncorrelated (having zero crosscorrelation). It utilizes the concept of joint diagonalization for achieving this goal. Following are the steps involved in this algorithm. Note that, number of sources are considered equal to number of sensors (or observations) i.e. m = n. 1. First the covariance (mean removed correlation) matrix of the output observations is estimated
Rˆ x 0
1 N
N
¦ x k x k T
(02)
k 1
where Rˆ x 0 is the covariance matrix at zero time lag and N is the total number of time samples taken. 2. Compute EVD (or SVD) of Rˆ x 0
Rˆ x 0 U x ¦ x VxT
Vx / xVxT
Vs / sVsT VN / NVNT
(03)
where Vs is m X n matrix of eigenvectors associated with n principal eigenvalues of ȁs = diag{Ȝ1, Ȝ2, ….., Ȝn} in descending order. Vn is m X (m-n) matrix containing the (m-n) noise eigenvectors associated with noise eigenvalues ȁn = diag{Ȝn+1, Ȝn+2, ….., Ȝm}. The number of sources n is thus estimated based on the n most significant eigenvalues (or singular values in case of SVD). 3. Perform pre-whitening transformation
x k / s 2VsT x k Qx k 1
4. Estimate the covariance matrix of the vector other than W =0.
Rˆ x W i
1 N
N
(04)
x (k ) for a preselected set of time lags (W 1 ,W 2 ,......,W L )
¦ x k x k W T
i
(05)
k 1
5. Perform Joint Approximate Diagonalization on the above set of covariance matrices
Rx W i U D W i U T
(06)
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to estimate the orthogonal matrix U that diagonalizes a set of covariance matrices. Several efficient algorithms are available for this purpose including Jacobi techniques, Alternating Least Squares, Parallel Factor Analysis etc. [9, 10]. 6. The mixing matrix and source signals can be estimated as
Aˆ Q U x y k It should be noted that
1
V s / s 2U x
(07)
sˆ k U x k
(08)
T x
D(W i ) is a diagonal matrix that has distinct diagonal entries. However, it is difficult to
determine a priori a single time lag W at which the above criterion is satisfied. Joint diagonalizaton procedure avoids this difficulty by providing an optimum solution considering a number of time lags.
%66DQG20$ The basic fundamental behind application of ICA / BSS techniques to modal analysis goes back to the concept of expansion theorem [12] and modal filters [14]. According to the expansion theorem [15], the response vector of a distributed parameter structure can be expressed as
x (t )
f
¦I K r
r
(t )
(09)
r 1
where ĭr are the modal vectors weighted by the modal coordinates Șr. For real systems, however, the response of the system can be represented as a finite sum of modal vectors weighted by the modal coordinates. In this manner, mathematically, expansion theorem yields similar formulation as expressed in Eq. (01), with source vector s(t) and mixing matrix A being analogous to modal coordinate response vector Ș(t) and modal vector matrix [ĭ], and hence BSS techniques like SOBI can be applied to obtain modal vectors and modal coordinates (which define the modal frequency and damping). To obtain a particular modal coordinate Și from response vector x, a modal filter vector ȥi is required such that
\ iT Ii
0,
for i z j
(10)
for i
and
\ iT I i z 0,
j
(11)
so that
\ iT x (t ) \ iT ¦ I rK r (t )
(12)
N
r 1
or
\ iT IiK i
(13)
Thus modal filter performs a coordinate transformation from physical to modal coordinates. Multiplying the system response x with modal filter matrix ȌT results in uncoupling of the system response into single degree of freedom (SDOF) modal coordinate responses (Ș).
6XJJHVWHG0RGLILFDWLRQV Mixing model shown in Eq. (1) does not take into consideration the additive noise, which is often present while dealing with real life structures. From operational modal analysis application point of view, one of the drawbacks of SOBI is in dealing with significantly noisy output response signals. In typical OMA scenario, such as response measurements taken over a bridge or a building, SNR (Signal-to-Noise Ratio) might not be good and this can deteriorate the performance of the algorithm. Apart from whitening (Eq. 4), SOBI doesn’t involve any other signal pre-processing step and thus errors can creep in due to additive measurement and process noise that is generally random in nature. One of the ways, to overcome this issue and improve performance of SOBI, is by minimizing the errors in measurement, i.e. effect of noise and bias, by means of techniques like averaging and use of windowing
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functions. Thus, instead of working directly on raw data, SOBI algorithm can be modified to work with correlation functions that are obtained after inverse Fourier transforming the averaged power spectra. This means that in step 5, instead of applying joint diagonalization procedure to covariance matrices calculated using complete data, it is performed on covariance matrices obtained after averaging procedure is carried out for noise minimization. Welch Periodogram method [11] is one such method that can be used for obtaining averaged output response power spectra. Along with averaging the power spectra, windowing and overlapping can also be used to reduce the leakage (bias) errors. This process of noise minimization, using power spectra averaging based technique, also provides a mechanism to extend the effectiveness of SOBI algorithm by making it possible to apply this technique within frequency band of interest. This is a significant improvement over SOBI in its original form, as it overcomes the limitation of identifying only as many sources as number of measured responses. Thus step 5, in the previous section, can be preceded by selecting averaged power spectra in the frequency band of interest, say between Ȧ1 and Ȧ2,
Gˆ x Z Z1 ,Z2 . This is followed by inverse Fourier transforming the power spectra in
selected frequency range to obtain corresponding Covariance matrix.
~ Rˆ x W 1 Gˆ x Z Z1 ,Z2
(14)
Step 5 then involves performing joint diagonalization of these covariance matrices, thus restricting SOBI to estimate modes in the specified frequency range. Advantages of these modifications are shown in the next section where modified version of SOBI is applied to a 15 Degree-of-Freedom analytical system and a lightly damped circular plate.
&DVH6WXGLHV 'HJUHHRI)UHHGRP$QDO\WLFDO6\VWHP A 15 DOF analytical M-C-K system is considered for analysis using the modified version of SOBI algorithm (Figure 1). This system has some moderately damped (1-4 %) modes; with a pair of closely spaced modes around 53.3 Hz and also some locally excited modes in 100-200 Hz frequency range.
)LJXUH $QDO\WLFDO'HJUHHRI)UHHGRP6\VWHP Theoretical modal frequency (in Hz) and damping (in % Critical) values for the system are shown in Table 1. Response data is obtained at each DOF by exciting the system at these DOFs by uncorrelated random forces. Simulated response data is sampled at a frequency of 1024 Hz and a total of 163840 response samples are collected. &RPSOHWH)UHTXHQF\5DQJH%DVHG$QDO\VLV In this analysis, data is analyzed in complete frequency range from 0-512 Hz. Table 1 lists the estimates of frequency and damping obtained using SOBI in its original form and its modified form. The estimates of modal frequency and damping are obtained by applying SDOF frequency domain methods on the power spectra of the obtained sources [12]. These are compared against the theoretical values of frequency and damping. It should be noted that while using modified version of SOBI, frequency band of interest is chosen
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to cover the complete frequency range. Thus, effectively this is similar to SOBI original, except that modified version of SOBI works with covariance functions obtained after inverse Fourier transforming the averaged complete power spectra, where as original form of SOBI works directly on the data without performing any prior signal processing. Figure 2 shows the plots of power spectra of SDOF modal coordinates obtained using original SOBI. All 15 modes are easily obtained using original SOBI. Similar plots are obtained for modified version of SOBI as well.
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Table 1 shows that the comparison on the basis of modal frequency and damping estimates is quite satisfactory though damping values obtained using SOBI (both algorithms) are slightly overestimated. Both SOBI approaches, original and modified, show fairly similar results. 7DEOH &RPSDULVRQRI0RGDO3DUDPHWHU(VWLPDWHVIRUWKH'2)6\VWHP 7KHRUHWLFDO
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The mode shapes of this system are complex. Except for the first mode at 15.98 Hz, which has fairly real mode shape, other modes have complex mode shapes. Since, BSS techniques (such as SOBI in its original form) are typically suited to give real mode shapes (although modified SOBI does produce complex mode shapes), it is expected that results might not be as accurate as expected. However, as shown by Table 2, when mode shapes obtained using the two algorithms are compared with theoretical mode shapes, the comparison between theoretical mode shapes and SOBI Original mode shapes is pretty good. MAC (Modal Assurance Criterion) [12] comparison for the mode shapes corresponding to the two closely spaced modes (around 53.3 Hz) is not as good as other modes, which highlights that BSS techniques such as SOBI find it difficult to handle effectively the cases involving closely spaced modes. Modified SOBI’s performance, on the other hand, is inferior in comparison (Highlighted in Table 2). One of the possible reasons can be the fact that the output response data obtained for this analytical system is free from noise and use of signal processing techniques such as averaging, windowing and overlapping invariably introduces some errors into the processed power spectra. Further, the modal vectors obtained using modified SOBI are complex (due to discrete Fourier transformation) and it is observed that if only the real part of the modal vectors is compared with the theoretical modes, MAC is significantly improved. 7DEOH 0$&&RPSDULVRQ 7KHRUHWLFDO 0RGHV
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1.00
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0.91
0.52
0.98
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0.48
0.96
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0.99
61.624
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0.99
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136.55
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$QDO\VLVLQD/LPLWHG)UHTXHQF\%DQG Table 3 highlights the real advantage of Modified SOBI which is not apparent when it is applied to complete frequency range like Original SOBI. In this case Modified SOBI is applied to the data in three different frequency ranges of 10-40 Hz, 35-85 Hz and 100-200 Hz. MAC values (highlighted in Table 3) for the two closely spaced modes at 53.3 Hz and also 59.4 Hz and 61.6 Hz mode show significant improvements in comparison to results obtained using Modified SOBI in complete range. MAC number for these four modes improves from 0.4-0.6 range to around 1 (around 0.95 for the closely spaced modes). In fact, the results are even superior to those obtained using Original SOBI in which case MAC is 0.91 and 0.88 (see Table 2). A typical plot of power spectra of SDOF modal coordinates (for a chosen frequency range of 10-40 Hz) is shown in Figure 3. There are only two proper estimates of modal coordinates in the selected frequency range (as indicated in Figure 3) and these estimates correspond to the modes at 15.99 Hz and 30.819 Hz. Rest of the 13 estimates can be neglected.
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53.427
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4.298
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2.609
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2.3839
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150.82
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Analysis conducted on 15 DOF analytical system verifies that, when applied in frequency bands, Modified SOBI is as effective (and in certain cases better) than original SOBI algorithm and compares well with the theoretical modal parameters.
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This study is now followed by analysis on an experimental structure to assess the utility of Modified SOBI further, especially in terms of assessing its performance in limited response sensors scenario (Number of sensors measuring output response is less than number of modes of interest).
/LJKWO\'DPSHG&LUFXODU3ODWH A lightly damped aluminum circular plate (Figure 4) is considered for experimental validation of Modified SOBI method. 30 responses are taken on the plate in a configuration shown in Figure 5. Plate is randomly excited by tapping it with fingers all over its surface. Data is acquired for a period of 5 mins. at a sampling rate of 1600 Hz, thus providing 4,80,000 samples.
)LJXUH ([SHULPHQWDO6HW8SIRU/LJKWO\'DPSHG&LUFXODU3ODWH
)LJXUH /LJKWO\'DPSHG&LUFXODU3ODWH 6HQVRU/RFDWLRQV Since there are 30 responses observed over the plate, due to algorithmic limitations one can at most identify 30 modes in the frequency range of interest (0-700 Hz) using original SOBI. For comparison purposes, an EMA test is also performed by exciting the plate by means of random excitation at three locations using
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electrodynamic shakers. Using EMA algorithms, such as Polyreference Time Domain (PTD) [13], a total of 21 modes are identified. This is in agreement with the corresponding Complex Mode Indicator Function (CMIF) plot [12] (Figure 6) which also indicates the presence of 21 modes. These modes are listed in Table 4 and are considered the reference against which performance of both SOBI algorithms (original and modified forms) will be evaluated.
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Table 4 lists the modal parameters obtained using the three approaches; EMA, original SOBI and modified SOBI. It should be noted that in case of modified SOBI, modal parameters are obtained by performing analysis in various frequency bands where as original SOBI estimates the modes by analyzing complete frequency range (0-800 Hz). Further, modal frequency and damping, in case of SOBI algorithms, is obtained by applying SDOF frequency domain methods to power spectra of obtained sources (as done in case of 15 DOF system). Advantage of modified SOBI is apparent in this analysis as it is able to identify all the expected modes in comparison to original SOBI which is able to identify only 19 modes, though overall estimates for these modes are in good agreement with the values obtained using EMA. Modified SOBI, on the other hand, provides satisfactory estimates for all the modes except for the discrepancy in frequency estimates which can be explained in terms of different frequency resolution used in the two approaches. MAC plot between modes obtained through EMA and those obtained using modified SOBI, shown in Figure 7, indicates that mode shapes corresponding to the two closely spaced around 133 Hz are not matching well. However, if real part of mode shapes obtained from modified SOBI is considered and compared with EMA, the MAC improves significantly. These MAC values are listed in Table 4. This behaviour is similar to that observed while analyzing closely spaced modes in 15 DOF system. Some higher modes (above 650 Hz) show similarity to some of the other lower modes. This is, perhaps, due to limited spatial resolution due to which these modes appear to have similar mode shape as the lower modes. 7DEOH &RPSDULVRQRI0RGDO3DUDPHWHU(VWLPDWHVIRUWKH&LUFXODU3ODWH (0$0RGHV
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56.938
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96.811
0.663
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96.425
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133.595
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132.74
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133.865
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219.70
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223.183
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222.92
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231.96
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0.217
350.99
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350.97
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358.23
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/LPLWHG6HQVRU%DVHG$QDO\VLV One of the main contributions of this paper is to demonstrate improved ability of modified SOBI algorithm to deal with situations where number of modes of interest exceeds the sensors measuring the response. This analysis brings to fore this advantage of modified version of SOBI. As mentioned before, one of the limitations of SOBI is that the number of modes that can be identified is at most equal to the number of sensors measuring the response. Following analysis shows that this limitation can be overcome by applying the proposed modified version of SOBI in a number of frequency bands, instead of complete frequency range as is the case with original SOBI. For this case, eighteen channels (out of thirty) are selected; other channels are not considered (See Figure 8). Goal is to identify all twenty one modes, given the response information corresponding to these eighteen channels only.
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)LJXUH 3RZHU6SHFWUDRI6'2)0RGDO&RRUGLQDWHV 2ULJLQDO62%,/LPLWHG&KDQQHO6WXG\ When original SOBI is applied on this limited dataset, it is known beforehand that at most eighteen modes could be identified due to algorithmic limitations. However, power spectra of estimated modal coordinates (Figure 9); indicate that only nine modes are properly identified. This underlines the drawback on part of original SOBI and severely affects its utilization for OMA purposes. Modified SOBI takes care of this limitation effectively since it can be applied in limited frequency ranges. Before discussing the results obtained using modified SOBI algorithm, effect of reducing the number of sensors from original thirty to eighteen on the estimation of modes is discussed. One of the potential dangers of reducing the number of sensors is that some of the modes might not be observable due to limited spatial resolution. This is also indicated by the auto MAC plot for the EMA modes shapes defined by the selected points, as shown in Figure 10. Indeed, some of the modes appear to be identical to some of the other modes, for e.g. closely spaced modes around 133 Hz are very identical, which is also the case for closely spaced modes around 571 Hz. On inspecting the mode shapes for these two pair of modes, the reasons for this observation becomes even clearer. It turns out that these modes are torsional modes and each closely spaced mode differs from the other only with respect to relative motion between the points, torsional motion being shifted by 45 deg. in the two cases. Removing immediately adjacent points to the ones selected has resulted in affecting the observability of these modes and thus it is expected that these modes might not be estimated. It is important to understand that, in this analysis, EMA mode shapes for comparison purposes are obtained by truncating the original EMA mode shapes (mode shapes obtained during analysis in previous section where complete data from all 30 sensors is used). It is observed that even EMA algorithms are not able to estimate these modes if data corresponding to the selected 18 channels is used. CMIF plot based on power spectra of the 18 selected channels also supports this (Figure 11). Some of the other modes also show similarity to other modes due to observability issues (Modes around 57 Hz and 233 Hz, 355 Hz and 670 Hz). The observability can be improved by altering the selection of channels on the physical structure, and results for a second set of 18 response points on the structure (discussed in Appendix A) indicate improved estimation (all the modes are identifiable) and better resolution between some of the closely lying modes.
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There are still, however, nineteen valid modes that need to be identified by means of information available from the eighteen channels. Estimates of these modes using modified SOBI algorithm along with corresponding MAC values (compared with EMA mode shapes defined by selected points) are shown in Table 5. All the estimates compare well with the EMA estimates, underlying the improved performance of SOBI algorithm with proposed modifications. Cross MAC plot between EMA and modified SOBI estimates, shown in Figure 12, is very similar to the auto MAC plot for EMA estimates (Figure 10), i.e. certain EMA modes that looked similar to each other due to limited spatial resolution, look similar to corresponding modes obtained using modified SOBI as well (for e.g. modes around 232 Hz and 57 Hz, and 355 Hz and 670 Hz). 7DEOH 0RGDO3DUDPHWHU&RPSDULVRQIRU/LPLWHG6HQVRU&DVH (0$0RGHV
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)LJXUH &URVV0$& (0$YV0RGLILHG62%, /LPLWHG&KDQQHOV&RQILJXUDWLRQ
&RQFOXVLRQV It has been shown, by means of work presented in this paper, how modifying the original Second Order Blind Identification algorithm can improve its effectiveness and utility for Operational Modal Analysis purposes. These modifications take into consideration more elaborate signal processing techniques, such as averaging, windowing etc., to reduce the effect of noise that is generally present while acquiring response data on real life structures, there by improving performance of SOBI. One of the main advantages of the suggested modification is that it enables SOBI to be applied even to situations where number of sensors measuring the response is lesser than the number of modes to be estimated. It is, however, important to note that this does not avoid observability related issues that might still be inherently present due to limited spatial resolution. In a more practical scenario (experimental analysis of the circular plate), it is shown that the modified form of SOBI outperforms its original version. This is attributed to incorporation of better noise reduction signal processing capabilities in the algorithm and extending its applicability to limited frequency ranges. Results presented in this paper are encouraging and form good foundation for future research in this area. One of the obvious research direction is to assess the performance of modified SOBI on real life structures that are typical OMA applications, like buildings, bridges etc. SOBI utilizes the concept of modal expansion theorem, according to which response of a system can be decomposed into several single degree of freedom systems (defined by resonant frequencies of the system) by means of modal vectors of the system which act as modal filters. Typically, modal expansion theorem is valid for systems having distinct modes and negligible or very light damping. Damping constraints also imply that expansion theorem is defined for systems with real modal vectors. Although one should not draw conclusions only on the basis of work presented in this paper, yet it is interesting to note that SOBI can identify heavily damped modes as well as modes with complex modal vectors (as shown in section 3.1, 15 DOF system). However, estimation of closely spaced modes does pose some challenges. In view of this discussion, future work in this research should concentrate on improving SOBI further. This requires making SOBI capable of handling closely spaced modes, and assessing its performance more rigourously for practical structures and for systems with heavily damped and complex modes.
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$FNQRZOHGJHPHQWV We would like to thank Dr. Randall J. Allemang, SDRL, University of Cincinnati, for his expert guidance and feedback which were critical for completion of the paper.
5HIHUHQFHV [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
Antoni, J.; “Blind separation of vibration components: principles and demonstrations”, Mechanical Systems and Signal Processing, Vol. 19 (6), pp. 1166-1180, November 2005. Kerschen, G., Poncelet, F., Golinval, J.C. (2007), “Physical Interpretation of Independent Component Analysis in Structural Dynamics”, Mechanical Systems and Signal Processing (21), pp. 1561-1575. Poncelet, F., Kerschen, G., Golinval, J.C. (2006), “Experimental Modal Analysis Using Blind Source Separation Techniques”; Proceedings of ISMA International Conference on Noise and Vibration Engineering, Katholieke Universiteit Leuven, Belgium. Chauhan, S., Martell, R., Allemang, R. J. and Brown, D. L. (2007), “Application of Independent Component Analysis and Blind Source Separation Techniques to Operational Modal Analysis”, Proceedings of the 25th IMAC, Orlando (FL), USA. Antoni, J., Chauhan, S. (2010); “Second Order Blind Source Separation (SO-BSS) and its relation to Stochastic Subspace Identification (SSI) algorithm”, To be presented at 28th IMAC, Jacksonville (FL), USA. McNiell, S.I., Zimmerman, D.C. (2008), “A Framework for Blind Modal Identification Using Joint Approximate Diagonalization”, Mechanical Systems and Signal Processing (22), pp. 1526-1548. Cichocki, A., Amari, S.; “Adaptive blind signal and image processing”, John Wiley and Sons, New York, 2002. Belouchrani, A., Abed-Meraim, K.K., Cardoso, J.F., Moulines, E.; “Second order blind separation of correlated sources”, Proceedings of International Conference on Digital Signal Processing, pp. 346-351, 1993. Cardoso, J.F., Souloumiac, A.; “Jacobi angles for simultaneous diagonalization”, SIAM Journal of Matrix Analysis and Applications, Vol. 17, Number 1, pp. 161-164, January, 1996. Hori, G.; “A new approach to joint diagonalization”, Proceedings of 2nd International Workshop on ICA and BSS, ICA’ 2000, pp. 151-155, Helsinki, Finland, June 2000. Stoica, P., Moses, R.L.; “Introduction to Spectral Analysis”, Prentice-Hall, 1997. Allemang, R.J.; “Vibrations: Experimental Modal Analysis”, Structural Dynamics Research Laboratory, Department of Mechanical, Industrial and Nuclear Engineering, University of Cincinnati, CN-20-263663/664, Revision - June 1999. Vold, H., Kundrat, J., Rocklin, T., Russell, R.; “A multi-input modal estimation algorithm for minicomputers”, SAE Transactions, Volume 91, Number 1, pp. 815-821, January, 1982. Shelly, S.J.; “Investigation of discrete modal filters for structural dynamic applications”, PhD Dissertation, Department of Mechanical, Industrial and Nuclear Engineering, University of Cincinnati, 1991. Meirovitch, L.; “Analytical Methods in Vibrations”, Macmillan, 1967.
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$SSHQGL[$ This section details results from a limited-sensors based study for an altered combination of response points on the lightly damped circular plate. This study is aimed at evaluating the effect of the selection points on the structure, on estimation of all the modes using the Modified SOBI methods. Eighteen points are chosen out of the available thirty responses, with the selection shown in Figure A.1.
)LJXUH$ 6HOHFWHG&KDQQHOV Modal estimates for this set of channels are listed in Table A.1. It is observed that this combination of channels is able to resolve the torsion modes around 133 Hz with better distinction. However, this selection of points too is unable to distinctly observe the closely lying modes around 571 Hz, although results are better in comparison to configuration shown in Figure 8, Section 3.2. This indicates the importance of selecting proper measurement locations in order to observe and estimate all modes of interest.
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BookID 214574_ChapID 18_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII
February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc. Minimally Intrusive Torsional Vibration Sensing on Rotating Shafts
0DUWLQ:7UHWKHZH\ Department of Mechanical and Nuclear Engineering Penn State University University Park, PA 16802 USA 0LWFKHOO6/HEROG 0DUN:7XUQHU Systems and Operations Automation Department Applied Research Laboratory Penn State University University Park, PA 16802 USA
$%675$&7 Time Interval Measurement System (TIMS) is a common torsional vibration sensing method used for rotating equipment. The technique uses high-speed counters to detect “zero crossings” in a carrier signal that is generated by a multiple pulse per revolution encoder on the shaft. The “zero crossings” are based on the passage timing of discrete intervals from an incremental geometric encoder (i.e., gear, optical encoder) on a rotating shaft. A variety of transducers have been used to sense the encoder interval passages, including Hall effect and reflective light intensity transducers. The encoder and sensor require physical attachment to the rotating shaft and surrounding hardware. Furthermore, the combinations must work in concert with each other. Usually the sensing transducer requires precise positioning with respect to the shaft encoder. The physical attachment and installation of the encoding and sensing devices can range from problematic to extremely difficult depending on the application. To make sensing of torsional vibration on a rotating shaft easier and minimally intrusive a combination of an adhesive backed “zebra” tape and a specially modified laser tachometer is used. The laser tachometer is considerably easier as permits a greater range of standoff, targeting and mounting options. Tests are performed a on mechanical diagnostics test bed with a 30 hp electrical drive connected to a 70 hp load motor by a shaft and couplers to demonstrate the issues. Torsional vibration measurements are acquired with the laser tachometer-“zebra” tape, Hall effect-gear and a precision optical encoder. Results are presented and discussed from the various torsional vibration sensing systems. The application illustrates the ease of set up for the laser tachometer-“zebra tape” combination and the high data quality that can be obtained. ,1752'8&7,21 The measurement of torsional vibration on rotating machinery is inherently more difficult than translational vibration. A significant amount of work has been devoted to developing torsional measurement techniques and has produced a variety of schemes. Previously reported sensing methods include the use of lasers [1,2], in line torque sensors [3], angular accelerometers [4] and time passage encoder based systems [5,6,7]. Encoder based systems have become increasingly popular. The method uses a fixed angular increment device (i.e., gear) installed on the rotating shaft. A transducer is used to sense the passage of the respective angular increments producing a time variant pulse train. The pulse train has a primary base carrier frequency equal to the multiplication of the rotation speed and the number of encoder angular increments. The shaft’s torsional vibration manifests itself as a modulation of this carrier frequency. Therefore the multiple pulses-per-revolution (ppr) shaft encoder signal must be demodulated to produce a time variant torsional vibration signal. There are two general
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_18, © The Society for Experimental Mechanics, Inc. 2011
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Transducer
Analog demodulator
A-D converter
Angular encoder
Transducer
Timer/Counter
Angular encoder
Conditioning amplifier )LJXUH. Analog torsional vibration measurement schematic.
Conditioning amplifier )LJXUH. Digital Time Interval torsional vibration measurement schematic.
categories of methods used for the demodulation; 1) analog [8], and; 2) digital [9]. Both of these methods have been used for rotating equipment and both can experience difficulty in extracting potentially low level vibration feature components in the torsional vibration spectra. A common feature of both measurement methods is the time passage sensing of the encoder angular increments. A number of angular encoder-passage sensing transducer combinations have been previously used. Each method has its inherent limits and advantages. The work in this paper will discuss previously used sensing methods and explores a new transducer based on a high speed laser tachometer. The new laser system will be compared to experiences with other methods to assess its capabilities. $1$/2*$1'',*,7$/7256,21$/9,%5$7,210($685(0(17 The similarities between the analog and digital torsional measurement schemes can be seen in Figures 1 and 2. The transduction and instrumentation are essentially the same. The primary difference between the measurement schemes is the demodulation method. The schematic of an analog torsional vibration measurement system is shown in Figure 1, and has five principle components. 1. 2. 3. 4. 5.
An incremental angular encoder attached to the shaft. A transducer to detect passage of the encoder segments. A transducer conditioning amplifier. An analog demodulator consisting primarily of a frequency to voltage circuit. An analog-to-digital converter/FFT analyzer to estimate the spectrum.
The transducer produces a pulse train reflecting the passage of angular encoder segments of the rotating shaft. A frequency to voltage to converter demodulator circuit accepts the pulse train and produces an analog output voltage signal proportional to the torsional vibration. The spectrum is estimated by discretizing the analog signal and applying a FFT algorithm. The digital method in Figure 2 has four components: 1. 2. 3. 4.
An incremental angular encoder attached to the shaft. A transducer to detect passage of the encoder segments. A transducer conditioning amplifier. A high speed timer/counter circuit.
The digital processing approach uses a timer/counter to sense the passage times of the pulse train from the rotating shaft. The difference between the passage times and a reference pulse train is computed. The respective time differences are then used to perform the demodulation. The output is a discrete time array with the element amplitudes proportional to the torsional vibration.
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)LOHG6HQVRU,QVWUXPHQWDWLRQDQG,QVWDOODWLRQ Most of the field applications to date have focused on rotating equipment already in service. Therefore, all the torsional vibration instrumentation must be retrofit to existing equipment designs and therefore, work within their respective installation constraints. The torsional vibration instrumentation consists of two primary components; 1) an incremental shaft encoder, and; 2) a sensor to detect the passage of the encoder segments. The installation of both components in a retrofit environment can affect the ability to acquire high quality torsional vibration. An incremental encoder (i.e., timing gear) must be attached to the shaft. For example, a toothed ring type encoder requires a bolted flange for its physical attachment to the shaft. Therefore, the choice of mounting locations is predetermined. Even for more flexible encoding schemes such as adhesive “zebra tape”, which does not require a flange, simple accessibility to open sections of the rotating shaft can be limited. Because of these location constraints the available torsional sensing location may not be ideal and can potentially affect the ability to sense the response of the desired signal features. Four examples of retrofit installations to measure torsional vibration are shown in Figure 3. Figure 3A shows a timing gear that used a Hall Effect transducer (not visible) to sense the passage of tooth segments. The timing gear was already available on the turbo-machine, being used for speed measurement. Hence, the adaptation to
)LJXUH$60 tooth gear encoder installed on high speed turbo machinery.
)LJXUH%Zebra tape encoder and fiber optic transducer on a large diameter shaft (22 inch).
Fiber Optic Probe Zebra tape
)LJXUH&. Zebra tape encoder and fiber optic transducer.
)LJXUH'. Zebra tape encoder and fiber optic transducer in limited rotating shaft access installation.
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measure torsional vibration was straightforward. Both installations in Figures 3B and 3C used adhesive backed tape with optical passage sensors. The relatively open accessibility to the rotating shaft in these examples affords flexibility in placing the encoding tape at optimal measurement locations to enable the measurement of the desired characteristics. By contrast, the installation shown in Figure 3D has very limited access to the rotating shaft. Hence, in this retrofit application the encoding device may be located at a less than ideal location and hinder the measurement of important response features. However, the hardware design presents no other options and the choice is to either use whatever is available or to not try at all.
)LJXUH Modal Shop LaserTach [10]
Sensors must be used to capture the passage of the encoder segments to determine the torsional vibration. The transducers often must be placed in close proximity (0.010 – 0.050 inches) to the encoding device to accurately sense the segment passages. The availability of sensor mounting locations is highly equipment dependent and may require long reaches to be used as seen in Figures 3C and 3D. These mounting demands can create some adverse conditions that may potentially affect the measured torsional vibration signature. /$6(57,0(3$66$*(6(1625 Through laboratory testing and field experience the torsional data sensing and instrumentation has evolved in an effort to make if more easily adaptable technology for industry. Most torsional measurement installations to date were custom designed utilizing cumbersome encoding gears and standoff sensitive sensors as discussed in the previous section. As a next step in the evolution is a non intrusive sensing system using a customized laser tachometer with a zebra surface on the rotating shaft. A prototype laser time interval passage sensing sensor was adapted from a Modal Shop LaserTach [10], shown in Figure 4. The commercial-of-the-shelf device is capable of supplying a once per revolution TTL key-phasor output for rotating equipment operating up to 30,000 rpm (500 Hz). Installation is considerably easier with standoff distances up to 51 cm (20 inches). The ICP device was modified by the ModalShop to accommodate the higher continuous interval sampling for torsional time passage sensing. /DERUDWRU\7HVWLQJ A torsional feature health monitoring test rig has been designed and constructed at Penn State’s Applied Research Laboratory as depicted in Figure 5. The rig consists of a 30 horsepower DC electric motor connected to a 75 horsepower DC electric motor with a 25 mm (1 inch) shaft. The 30 hp motor serves as the drive motor while the torsional loading is supplied via the 75 hp load motor. The prototype high speed laser was mounted near the drive motor with a standoff distance of approximately 50 mm (2 inches). The incremental encoder medium was a disk hub with zebra retro-reflective zebra style tape. The target tape had 70 pulses-per-revolution. In addition, a comprehensive instrumentation suite was deployed on the test rig including: 1. A 16,384 pulses-per-revolution glass encoder. 2. A 49 tooth gear type encoder and Hall Effect sensor were mounted near a spline coupling on the drive motor. 3. A 61 tooth gear type encoder and Hall Effect sensor were mounted near a spline coupling on the load motor. 4. Horizontal and vertical proximity probes were placed on the drive motor to measure dynamic shaft displacements. 5. Horizontal and vertical proximity probes were placed on the load motor to measure dynamic shaft displacements. 6. Horizontal and vertical accelerometers were placed on both bearing pillow blocks. 7. A key-phasor laser tachometer was mounted on between the load motor and lateral load assembly.
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The passage times of the incremental encoders were sensed and recorded with a National Instruments PCI-6602 Timer/Counter Board using an 80MHz clock reference. A typical test sequence entailed spinning the shaft and collecting approximately 30 minutes of data. Once recorded, the encoding tooth passage times were processed with a specialized torsional vibration algorithm based on the Time Interval Measurement System method. A high spectral resolution spectrum was ultimately produced. A representative torsional spectrum from the Modal Shop LaserTach is shown in Figure 6. The spectrum is very high quality, clearly showing the first system torsional resonance )LJXUH Rendering of torsional vibration test bed with 30 hp drive around 25 Hz. Also obvious is the shaft motor and 70 hp load motor connected by 25 mm diameter shaft. rotational related order content. The spectrum in Figure 6 is computed directly from the recorded time passage array without any compensatory processing applied to minimize the order content [11]. Such methods have been shown to be highly beneficial. The torsional spectra originating from Modal Shop LaserTach sensor was subsequently compared to the results from the more traditional Hall Effect-toothed gear encoder combination. The spectral results were essentially identical. The only differences were the in the spectral resolution resulting from the variation in effective sample rates from the number of encoder segments available in each. 6800$5< The measurement of torsional vibration from a rotating shaft is inherently more difficult than translational movement. This work has described some of the pragmatic sensing issues related to Time Interval Measurement System approach. Application and torsional testing explored the use of a prototype laser sensor developed from a Modal Shop LaserTach. Experience with the modified Modal Shop LaserTach showed several advantages for torsional time passage measurements.
Torsional Spectrum 40 20
1. Installation and setup was robust and easy. Fundamentally it was point and shoot, requiring little adjustment.
0
3. The LaserTach-retroreflective zebra target combination is suitable for either analog or digital demodulation. 4. The LaserTach is susceptible to torsional vibration data corruption from lateral mount motion in the same
dB
-20
2. The meticulous and iterative standoff adjustments required with Hall Effect transducers were nonexistent. Performance with respect to standoff has been observed to be robust, with changes not affecting data quality. Furthermore, the sensor response is not susceptible to target standoff variations due to dynamic shaft motion.
-40 -60 -80 -100 -120 0
50
100
150
Frequency (Hz)
)LJXUH Torsional vibration spectrum measured from test bed in Figure 5 with modified LaserTach sensor and retroreflective zebra tape target.
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manner as a Hall Effect sensor. Similar compensation methods may be used by employing three sensors around the circumference of the incremental encoder [12]. Furthermore, the greater allowable standoff distances allows the sensor installation to be entirely detached from the rotating equipment and potentially eliminating the problem all together. 5. The LaserTach-retroreflective zebra target combination is particularly well suited for retrofit applications of existing rotating equipment due to its ease of installation and setup. 6. The time passage data quality proved to be more consistent and robust than the Hall Effect sensor data. Current rationale points to a combination of items 1 and 2 as the reason. The modified LaserTach has become the sensor of choice for torsional testing on the test rig in Figure 5. 7. The modified LaserTach is susceptible to the same encoder segment irregularity corruption as with the Hall Effect-Toothed Gear combination. Similar compensatory methods, however can be applied which have been shown to be effective. The effort has proved successful in providing a torsional vibration Time Interval Measurement sensing system that is; 1) easily adaptable; 2) non-intrusive, and 3) robust and reliable.
$&.12:/('*(0(176 Portions of this work were supported by the Electric Power Research Institute (EPRI Contract EP-P9801/C4961). The content of the information does not necessarily reflect the position or policy of the EPRI, and no official endorsement should be inferred. 5()(5(1&(6 1. Halliwell, N.A., Pickering, C.J.D., and Eastwood, P.G., “The Laser Torsional Vibrometer: A New Instrument”, Journal of Sound and Vibration Vol. 93, pp. 588-592, 1984. 2. Li, X., Qu, L., Wen, G., and Li, C., “Application of Wavelet Packet Analysis for Fault Detection in ElectroMechanical Systems Based on Torsional Vibration Measurement”, Mechanical Systems and Signal Processing, Vol. 17, No. (6), pp. 1219-1235, 2003. 3. Honeywell Sensotec, 2080 Arlington Lane, Columbus, OH, USA, 2003. 4. Seidlitz, S., “Engine Torsional Transducer Comparison”, SAE Paper No. 920066, 1992. 5. Fu, H., Yan, P. “Digital Measurement Method on Rotating Shaft Torsional Vibration”, American Society of Mechanical Engineers, DE-Vol. 60, Vibration of Rotating Systems, pp. 271-275, 1993. 6. Wang, P., Davies, P., Starkey, J.M., and Rouston, R.L., “A Torsional Vibration Measurement System”, IEEE Transactions on Instrumentation and Measurement, Vol. 41, No. 6, pp. 803-807, 1992. 7. Hernandez, W., Paul, D., and Vosburgh, “On-Line Measurement and Tracking of Turbine Torsional Vibration Resonances Using a New Encoder Based Rotational Vibration Method (RVM)”, Society of Automotive Engineers, Paper No. 961306, 1996. 8. CoppTek, Barrington, IL, USA, 2003. 9. Vance, J. M., Rotordyamics of Turbomachinery, John Wiley & Sons, New York, 1988. 10. http://www.modalshop.com/electronics.asp?ID=19, 2009. 11. Resor, B.R., Groover, C.L., Trethewey, M.W., Maynard, K.P., “Natural Frequency Identification in Torsional Vibration with High Level Order Content,” 22nd IMAC, Dearborn, Michigan, USA, January 26-29, 2004. 12. Trethewey, M.W., and Lebold M.S., “Identification of Torsional Vibration Features in Electrical Powered Rotating Equipment”, 27th IMAC, Orlando FL, USA, February, 2009.
BookID 214574_ChapID 19_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Zebra Tape Butt Joint Algorithm for Torsional Vibrations Karl Janssens, Piet Van Vlierberghe, Philippe D’Hondt, Ton Martens, Bart Peeters, Wilfried Claes LMS International, Interleuvenlaan 68, B-3001 Leuven, Belgium,
[email protected]
ABSTRACT At the development stage of automobile constructions, many powertrain configurations may be considered. Due to complicated coupling mechanisms, the natural frequencies of the combined system are difficult to predict. It is therefore necessary to run tests on automotive subsystems under realistic operating conditions. To completely understand the dynamic behaviour, a study of both torsional and translational motions along the driveshaft is necessary. Torsional vibrations can conveniently be measured at any point along a rotating shaft by using a zebra tape that overcomes the drawbacks of existing systems which are either expensive or require time-consuming and difficult shaft modifications. Zebra tapes however require a dedicated DSP processing due to the butt joint of the two tape ends. This paper presents a new algorithm to correct the tacho moments obtained from pulses generated with a high quality zebra tape glued on a rotating shaft with torsional vibrations. Due to the misalignment of the stripe pattern at the joint of the two tape ends, the tacho moments are not evenly distributed over the shaft. The angle between subsequent tacho moments is exactly the same except at the butt joint. The algorithm first identifies the position and angular interval at the butt joint by using an angle estimator function and a dedicated spline interpolation and FIR bandpass filter. This information then allows to reconstruct the exact angle evolution and perform a proper torsional vibration analysis. First, the principles of the butt joint correction algorithm are discussed. Then, an error analysis is performed on a simulated dataset, evaluating the accuracy of the algorithm in the presence of torsional vibrations.
1
INTRODUCTION
Noise and vibration performance plays an important role in the development of rotating components, such as engines, drivelines, transmission systems, compressors and pumps. The presence of torsional vibrations and other specific phenomena require the dynamic behaviour of systems and components to be designed accurately in order to avoid comfort and durability related problems. However the complicated coupling mechanisms of mechanical systems make it difficult to accurately predict the dynamic behaviour of the combined system. Where both lateral and axial vibrations have a direct impact through spring-like elements (e.g. in bearings) on the vibration of the casing and other attached machinery components, torsional vibrations impact these observed vibrations in an indirect way, via cross-coupling with lateral vibrations [1]. It is therefore necessary to run tests under realistic operating conditions. Torsional vibrations can very conveniently be measured at any point along a rotating shaft by using zebra tape combined with an optical probe. The rotating velocity can be measured by timing the duration of the passage of the alternating light & dark stripes glued to the shaft [2-3]. Such systems overcome the drawbacks of existing systems which are either expensive or require time-consuming and difficult shaft modifications. Zebra tapes however suffer from the fact that the last stripe at the butt joint has in general not the same width as the other stripes, causing an erroneous torsional signal that disturbs the analysis of the data. Butt joint correction methods for this error are described in literature [e.g. 4-5], but these require relatively low rotation speeds and limited torsional vibrations of the shaft. T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_19, © The Society for Experimental Mechanics, Inc. 2011
213
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214 This paper presents a new butt joint correction algorithm which can deal with fast rotating shafts with strong torsional vibrations. The algorithm first identifies the position and angular interval at the butt joint by using an angle estimator function and a dedicated spline interpolation and FIR bandpass filter. This information then allows to reconstruct the exact angle evolution and perform a proper torsional vibration analysis. The principles of the algorithm are first presented in section 2. Then, an error analysis is conducted in section 3, investigating the accuracy of the algorithm in the presence of torsional vibrations.
2 BUTT JOINT CORRECTION ALGORITHM In order to explain the algorithm, let us consider a zebra tape as in figure 1 with N = 60 stripe pairs or tacho pulses per rotation and a butt joint error x of 0.4. The butt joint error x is defined as:
x=
ǻș N −1 ǻș
(1)
with ǻθ the constant angle between subsequent stripe pairs and ǻθN the angle at the butt joint which is larger than ǻθ in the current example. The angles ǻθ and ǻθN can be expressed as:
ǻș =
2ʌ 2ʌ = = 0.1040 rad N + x 60.4
ǻșN = ǻș * (1 + x ) =
(2)
2ʌ * 1.4 = 0.1456 rad 60.4
(3)
Let us assume that the zebra tape is measuring a rotating shaft with a base rotation speed fstationary of 20 Hz and which is subject to torsional vibrations. Suppose that the torsional vibrations have a frequency of two revolutions per shaft rotation (fm = 2* fstationary = 2*20 Hz = 40 Hz), a zero phase and an amplitude of 20 % of the base rotation speed (Am = 0.2* fstationary = 0.2*20 Hz = 4 Hz). Figure 2 shows the frequency evolution of the shaft over a period of 1.2 revolutions. The pulse moment train of the zebra tape and the corresponding angle evolution θtrue(t) in function of time are shown in figures 3 and 4. The time steps ǻT between the pulses are not equal because of the torsional vibrations. The last time step ǻTN at the butt joint is larger in this example because of the increased angle ǻθN.
2π ǻĬ1
T1
ǻT1
ǻĬ2
T2
ǻT2
ǻĬ3
T3
ǻT3
ǻĬi
Ti
ǻTi
ǻĬN=60
TN=60
ǻTN=60
Figure 1: Zebra tape with N = 60 stripe pairs and a butt joint error x of 0.4.
= butt joint
TN+1 = 61
BookID 214574_ChapID 19_Proof# 1 - 23/04/2011
215 Frequency evolution of shaft in function of time 28 26
Frequency [Hz]
24 22 20 18 16 14 12 0
0.01
0.02
0.03 Time [s]
0.04
0.05
0.06
Figure 2: Frequency evolution of rotating shaft (fstationary = 20 Hz, fm = 40 Hz, Am = 4 Hz) during 1.2 shaft cycles.
Train of pulse moments 2
1.5
1
0.5
0 0
0.01
0.02
0.03 Time [s]
0.04
0.05
0.06
Figure 3: Pulse moment train of the zebra tape with indication of the butt joint.
Angle evolution in function of time 8 7
Angle [rad]
6 5 4 3 2 1 0 0
0.01
0.02
0.03 Time [s]
0.04
0.05
0.06
Figure 4: Angle evolution in function of time with indication of the butt joint.
Now the problem with zebra tape measurements is that the angle evolution θtrue(t) is unknown. The only thing we know is: 1) the train of measured pulse moments T1, T2,…, TN,… etc., 2) the number of zebra stripe pairs N, 3) the angle evolution θtrue(t) increases with a constant ǻθ = 2π/(N+x), except at the butt joint where ǻθN = ǻθ*(1+x) and 4) the angle evolution θtrue(t) only contains torsional vibrations but no discontinuity at the butt joint. The unknown parameters are 1) the butt joint error x, which is value between -1 and 1 (0.4 in our example) and 2) the location of the butt joint (at time steps ǻTN, ǻT2N, etc. in our example).
BookID 214574_ChapID 19_Proof# 1 - 23/04/2011
216 2.1 Butt joint location is known Let us first assume that the location of the butt joint is known and that the butt joint error x is the only unknown parameter. In order to estimate this butt joint error, we introduce the following model description for the angle evolution θtrue(t):
ș true ( t ) = ș estimator ( t ) − ș error ( t )
(4)
where θestimator(t) is the angle estimator function and θerror(t) is the error term. The angle estimator function θestimator(t) would be the response of the zebra tape if it were perfect and had no butt joint. This function is perfectly known. It increases with ǻθestimator = 2π/N per pulse moment, even between the time steps N and N+1. The error function θerror(t) is an unknown function, but it has a known shape. It increases with ǻθerror = 2π/N 2π/(N+x) per pulse moment and falls back to zero each time after N pulses. The error term θerror(t) can also be expressed as:
ș error ( t ) = k * ș error ,x =1( t )
(5)
where θerror,x=1(t) is the error term when the butt joint error x is equal to 1 and k is an unknown parameter. The error term θerror,x=1(t) increases with ǻθerror,x=1 = 2π/N - 2π/(N+1) per pulse moment and is perfectly known. By substituting (5) in (4), we obtain equation (6) where k is the only unknown parameter in the right hand term:
ș true ( t ) = ș estimator ( t ) − k * ș error ,x =1( t )
(6)
Figure 5 shows the evolution of θtrue(t), θestimator(t) and θerror,x=1(t) over a period of 1.2 shaft cycles when the zebra tape is measuring a rotating shaft with frequency evolution as in figure 2. The detailed view on the θtrue(t) and θestimator(t) functions at the butt joint shows that the two functions coincide each time after N pulses.
True, estimator and error angle functions 8 6.8 6.6
7
6.4
6
Angle [rad]
6.2 6 5.8 5.6
Angle [rad]
5
5.4 5.2
4
5 0.04
0.042
0.044
0.046 Time [s]
0.048
0.05
0.052
3
2
1
0 0
0.01
0.02
0.03 Time [s]
0.04
0.05
0.06
Figure 5: Time evolution of θtrue (red), θestimator (blue) and θerror,x=1(black).
BookID 214574_ChapID 19_Proof# 1 - 23/04/2011
217 Figure 6 shows the same information as in figure 5 but this time with the trend removed from the functions. This figure nicely shows the discontinuity in the θestimator(t) and θerror,x=1(t) functions. This discontinuity is clearly not present in the θtrue(t) function which only contains torsional vibrations.
True, estimator and error angle functions after detrending and spline interpolation 0.25
Angle [rad]
0.2
0.15
0.1
0.05
0 0
0.01
0.02
0.03 Time [s]
0.04
0.05
0.06
Figure 6: Time evolution of θtrue (red), θestimator (blue) and θerror,x=1(black) after detrending.
The key step of the algorithm involves the application of a FIR bandpass filter which filters the torsional vibrations from the different terms in equation (6). Before doing so, the functions are first resampled by spline interpolation from a non-equidistant time axis to an equidistant one at high sampling frequency. The FIR filtering operation can be expressed as follows:
filtered(ș true ( t )) = filtered(ș estimator ( t )) − k * filtered(ș error,x =1( t ))
(7)
Since the θtrue(t) function only contains torsional vibrations, the left term of this equation is equal to zero and we obtain:
k * filtered(ș error,x =1( t )) = filtered(ș estimator ( t ))
(8)
where k now remains the only unknown parameter. This parameter can be easily estimated from the filtered θestimator(t) and θerror,x=1(t) functions by using a Least Squares (LS) method. Once this is achieved, the butt joint error x can be calculated as in equation (9). Finally, the evolution of θtrue(t) can be reconstructed and a torsional vibration analysis can be performed.
ș error (T1 ) = k * ș error,x =1(T1 )
2ʌ 2ʌ 2ʌ 2ʌ − = k *( − ) N N+ x N N +1
x=(
2ʌ )−N 2ʌ 2ʌ 2ʌ −k *( − ) N N N+1
(9)
Figures 7 and 8 show the original and filtered θestimator(t) and θerror,x=1(t) functions, each sampled at 12000 Hz. The left and right cut-off frequencies of the applied FIR bandpass filter were respectively 80 Hz (twice the torsional vibration frequency = 2*fm = 2*40 Hz = 80 Hz) and 600 Hz (torsional vibration nyquist frequency = fstationary*N/2 = 20*60/2 = 600 Hz) and the number of filter taps was 1000. The filtered functions clearly have similar shapes which basically only differ in magnitude by a factor k as expressed in equation (8). The value of k was estimated from this data and then used to calculate the butt joint error x. The obtained value of x was 0.400055 which is very close to the true value of 0.4.
BookID 214574_ChapID 19_Proof# 1 - 23/04/2011
218 The accurate estimation of the butt joint error is also reflected in figure 9, comparing the left and right hand terms of equation (8). The model fit is excellent with only very small residual errors between the two curves.
Original and filtered angle estimator function
Angle [rad]
0.2 0.15 0.1 0.05 0 -0.05 1
1.01
1.02
1.03 Time [s]
1.04
1.05
1.06
Figure 7: Original (full line) and filtered (dashed line) angle estimator function θestimator(t).
Original and filtered angle error function
Angle [rad]
0.2 0.15 0.1 0.05 0 -0.05 1
1.01
1.02
1.03 Time [s]
1.04
1.05
1.06
Figure 8: Original (full line) and filtered (dashed line) error function θerror,x=1(t).
Least squares model fit 0.02
0.015
0.01
Angle [rad]
0.005
0
-0.005
-0.01
-0.015
-0.02 1
1.01
1.02
1.03 Time [s]
1.04
1.05
1.06
Figure 9: Least squares model fit: i) left hand term of equation (8) in black, ii) right hand term in blue.
BookID 214574_ChapID 19_Proof# 1 - 23/04/2011
219 2.2 Butt joint location is unknown In the algorithm presented so far, the butt joint error x was the only unknown parameter. The location of the butt joint at time steps ǻTN, ǻT2N, etc. was supposed to be known. This assumption was necessary to make sure that the θerror,x=1(t) function has its discontinuity at exactly the same time step as the θestimator(t) function. In practice, however, the butt joint location is unknown and must be identified as well. This is done by running the above algorithm in section 2.1 for all possible time locations of the discontinuity in the θerror,x=1(t) function. As an example, figures 10 and 11 show the θestimator(t) and θerror,x=1(t) functions and the Least Squares (LS) fit of the filtered responses when the butt joint location is assumed at time steps ǻTN+12, ǻT2N+12, etc. Because of the wrong assumption of the butt joint location, the filtered responses are shifted in time, resulting in a poor model fit. As a quality indicator, the Mean Square Error (MSE) of the model fit was calculated for each of the runs with different butt joint location. The MSE results are presented in figure 12, showing a clear absolute minimum or best model fit for the true butt joint location at time steps ǻTN, ǻT2N, etc. The estimated butt joint error x according to this butt joint location was 0.400055 like discussed before.
Angle estimator and error function for butt joint at time step 12
Angle [rad]
0.2 0.15 0.1 0.05 0 -0.05 1
1.01
1.02
1.03 Time [s]
1.04
1.05
1.06
Figure 10: Angle estimator function θestimator(t) and error function θerror,x=1(t) when the butt joint location is assumed at time steps ǻTN+12, ǻT2N+12, etc.
Least squares model fit for butt joint at time step 12 0.02
0.015
0.01
Angle [rad]
0.005
0
-0.005
-0.01
-0.015
-0.02 1
1.01
1.02
1.03 Time [s]
1.04
1.05
1.06
Figure 11: LS model fit when the butt joint location is assumed at time steps ǻTN+12, ǻT2N+12, etc.: i) left hand term of equation (8) in black, ii) right hand term in blue.
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220
MSE for all possible locations of butt joint
-5
2
x 10
MSE
1.5
1
0.5
0 0
10
20
30 Location of butt joint
40
50
60
Figure 12: Mean Square Error (MSE) of model fits for all possible butt joint locations (i = 1,2,…, 60).
3 ERROR ANALYSIS An error analysis was performed by running the above presented algorithm for several torsional vibration signals, each time with a different frequency (fm) and amplitude (Am). The calculations were done for three test cases: 1) zebra tape with N = 60 stripe pairs, butt joint error x = 0.4, FIR filter of 1000 taps, 2) zebra tape with N = 60 stripe pairs, butt joint error x = 0.1, FIR filter of 1000 taps, 3) zebra tape with N = 180 stripe pairs, butt joint error x = 0.4, FIR filter of 4000 taps. The results for the three test cases are given in tables 1-3, showing the butt joint error estimations and their errors (in %) for torsional vibration signals with different frequencies f m = i*fstationary (i = 2, 4, 6, 8) and amplitudes Am = j*fstationary (j = 0, 0.01, 0.1, 0.2, 0.4, 0.6, 0.8). In general, the estimation results are very good, especially for torsional vibration signals with amplitude lower than 0.6*fstationary. For rotating shafts with stronger torsional vibrations, the estimation results are less accurate, certainly in the first two test cases when using a zebra tape of 60 stripe pairs and a FIR filter of 1000 taps. Here, the errors ran up to 2 % and even higher as shown in tables 1-2. For shafts with very strong torsional vibrations, we recommend to use a zebra tape with more stripe pairs and a sharper FIR filter with more filter taps. This is illustrated in table 3, showing the results for a zebra tape with 180 stripe pairs and a FIR filter of 4000 taps. In this case, we still obtain excellent results for the most difficult signals with errors below 0.2 %. Finally, one can also notice from the first two tables that the accuracy of the algorithm is not affected by the butt joint error itself. The estimation errors are of similar magnitude for both the zebra tapes with butt joint error of 0.4 and 0.1 respectively.
1) N = 60 stripe pairs 2) Butt joint error x = 0.4 3) Number of filter taps: 1000 fm = i*fstationary
Am = j*fstationary
Estimated butt joint error x
Percentual error (%)
i=2
j=0
0.4000000
0.00000 %
j = 0.01
0.4000026
0.00025 %
j = 0.1
0.4000268
0.00267 %
j = 0.2
0.4000548
0.00547 %
j = 0.4
0.4001157
0.01157 %
j = 0.6
0.4001938
0.01937 %
j = 0.8
0.4005163
0.05162 %
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221 i=4
i=6
i=8
j=0
0.4000000
0.00000 %
j = 0.01
0.4000047
0.00046 %
j = 0.1
0.4000508
0.00507 %
j = 0.2
0.4001099
0.01099 %
j = 0.4
0.4003155
0.03154 %
j = 0.6
0.4020687
0.20686 %
j = 0.8
0.3977163
0.22837 %
j=0
0.4000000
0.00000 %
j = 0.01
0.4000008
0.00007 %
j = 0.1
0.4000035
0.00034 %
j = 0.2
0.3999821
0.00179 %
j = 0.4
0.3992727
0.07273 %
j = 0.6
0.3909545
0.90455 %
j = 0.8
0.3782426
2.17574 %
j=0
0.4000000
0.00000 %
j = 0.01
0.4000014
0.00013 %
j = 0.1
0.3999611
0.00389 %
j = 0.2
0.3996019
0.03981 %
j = 0.4
0.3956493
0.43507 %
j = 0.6
0.3887520
1.12480 %
j = 0.8
0.4016221
0.16220 %
Table 1: Butt joint error estimations and their errors (in %) for test case 1: zebra tape with N = 60 stripe pairs, butt joint error x = 0.4, FIR filter of 1000 taps.
1) N = 60 stripe pairs 2) Butt joint error x = 0.1 3) Number of filter taps: 1000 fm = i*fstationary
Am = j*fstationary
Estimated butt joint error x
Percentual error (%)
i=2
j=0
0.1000000
0.00000
j = 0.01
0.1000023
0.00023
j = 0.1
0.1000243
0.00243
j = 0.2
0.1000501
0.00501
j = 0.4
0.1001065
0.01065
j = 0.6
0.1001724
0.01724
j = 0.8
0.1000895
0.00895
j=0
0.1000000
0.00000
j = 0.01
0.1000041
0.00041
j = 0.1
0.1000472
0.00472
j = 0.2
0.1001110
0.01110
j = 0.4
0.1003647
0.03647
j = 0.6
0.1021195
0.21195
j = 0.8
0.1104796
1.04796
j=0
0.1000000
0.00000
j = 0.01
0.1000007
0.00007
j = 0.1
0.1000261
0.00261
j = 0.2
0.1001207
0.01207
j = 0.4
0.1008784
0.08784
j = 0.6
0.0985559
0.14440
j = 0.8
0.0831674
1.68325
i=4
i=6
BookID 214574_ChapID 19_Proof# 1 - 23/04/2011
222 i=8
j=0
0.1000000
0.00000
j = 0.01
0.1000009
0.00009
j = 0.1
0.1000448
0.00448
j = 0.2
0.1001079
0.01079
j = 0.4
0.0979715
0.20284
j = 0.6
0.0846930
1.53069
j = 0.8
0.0952307
0.47692
Table 2: Butt joint error estimations and their errors (in %) for test case 2: zebra tape with N = 60 stripe pairs, butt joint error x = 0.1, FIR filter of 1000 taps.
1) N = 180 stripe pairs 2) Butt joint error x = 0.4 3) Number of filter taps: 4000 fm = i*fstationary
Am = j*fstationary
Estimated butt joint error x
Percentual error (%)
i=2
j=0
0.4000000
0.00000 %
j = 0.01
0.4000156
0.00156 %
j = 0.1
0.4001590
0.01589 %
j = 0.2
0.4003235
0.03234 %
j = 0.4
0.4006672
0.06671 %
j = 0.6
0.4010313
0.10312 %
j = 0.8
0.4014130
0.14130 %
j=0
0.4000000
0.00000 %
j = 0.01
0.4000017
0.00016 %
j = 0.1
0.4000181
0.00180 %
j = 0.2
0.4000368
0.00368 %
j = 0.4
0.4000774
0.00774 %
j = 0.6
0.4001342
0.01341 %
j = 0.8
0.4003549
0.03548 %
j=0
0.4000000
0.00000 %
j = 0.01
0.4000020
0.00020 %
j = 0.1
0.4000217
0.00216 %
j = 0.2
0.4000440
0.00439 %
j = 0.4
0.4001001
0.01000 %
j = 0.6
0.4002002
0.02001 %
j = 0.8
0.4003892
0.03891 %
j=0
0.4000000
0.00000 %
j = 0.01
0.4000013
0.00012 %
j = 0.1
0.4000145
0.00144 %
j = 0.2
0.4000321
0.00320 %
j = 0.4
0.4000838
0.00837 %
j = 0.6
0.4002083
0.02082 %
j = 0.8
0.4016780
0.16779 %
i=4
i=6
i=8
Table 3: Butt joint error estimations and their errors (in %) for test case 3: zebra tape with N = 180 stripe pairs, butt joint error x = 0.4, FIR filter of 4000 taps
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223 4
CONCLUSIONS
A new algorithm was developed that is able to accurately identify the location and angular interval at the butt joint of a zebra tape from the tacho moments measured on a rotating shaft with torsional vibrations. This information then allows to reconstruct the exact angle evolution and perform a torsional vibration analysis. The algorithm makes use of an angle estimator function and a dedicated spline interpolation and FIR bandpass filter. An error analysis was performed evaluating the accuracy of the algorithm for several torsional vibration signals, each time with a different frequency and amplitude. In general, the estimation results were found to be very good. The accuracy decreases with increasing torsional vibration amplitudes, but can be still kept high (less than 0.2% in our simulations) when using a zebra tape with more stripe pairs and and a sharper FIR bandpass filter.
REFERENCES [1] D.E. Bentley, Ch.T. Hatch, 2002. Fundamentals of rotating machinery diagnostics, p. 315 - 316. [2] J.M. Vance, 1988. Rotordynamics of turbomachinery, John Wiley & Sons, New York. [3] B.R. Resor, Ch.L. Groover, M.W. Trethewey, K.P. Maynard. Natural frequency identification in torsional vibration with high level order content. [4] P. Wang, P. Davies, J.M. Starkey, R.L. Routson. Torsional mode shape measurement on a rotating shaft. [5] B.R. Resor, M.W. Trethewey, K.P. Maynard, 2005. Compensation for encoder geometryand shaft speed variation in time interval torsional vibration measurement, Journal of Sound and Vibration 286, p. 897-920.
BookID 214574_ChapID FM_Proof# 1 - 23/04/2011
BookID 214574_ChapID 20_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
ACOUSTIC AND MECHANICAL MEASUREMENTS OF AN HYDRAULIC TURBINE’S GENERATOR IN RELATION TO POWER LEVELS AND EXCITATION FORCES F. Lafleur, S. Bélanger, L. Marcouiller and A. Merkouf, Institut de recherche d’Hydro-Québec, IREQ 1800 boul.Lionel-Boulet Varennes, Québec Canada J3X 1S1
ABSTRACT Recent measurements were performed on the existing generator of an hydraulic turbine to carry out a power increase diagnostic. These measurements were performed on the rotor and stator of the generator includes mechanical, thermal and air flow instrumentation. This paper will analyze the acoustic and mechanical measurements relative to the power level produced. Frequency analysis of noise, vibration and stress levels will be presented. The multi-channel analysis will enable us to make correlation analyses between signals and to link the different excitation frequencies to the electromagnetic or mechanical sources. INTRODUCTION A project aimed at increasing the power in existing generators is under way at Hydro-Québec’s research institute (IREQ). This project combines thermal, electromagnetic, mechanical and fluid numerical simulation and measurements in order to represent the generator behavior under different conditions. The simulation results are compared to measurements to ensure that the model of the generators represents reality. Once this exercise completed, an extrapolation is made to evaluate the real power this generator can provide without going beyond its mechanical and thermal limits. This approach will allow the utility to increase the nominal capacity of a number of generators compared to their nameplate rating by reducing the margin and without compromising their lifetime. The first phase of the project concentrated on a thermal and electromagnetic assessment of the stator of the generator. In order to extend the model to the full generator and to integrate a multiphysics approach, mechanical measurements are essential. Measurements were therefore performed recently on the rotor and stator of the generator using thermal, fluid, acoustic and mechanical instrumentation. This paper will describe the overall measurement setup with emphasis on details of the mechanical measurements. DESCRIPTION OF MEASUREMENTS A newly refurbished hydraulic power generator was instrumented and monitored. Figure 1 shows the rotor and stator of the generators during refurbishing. The generator’s specification is 65,000 and 74,750 kVA for summer and winter respectively. The power factor is 0.85; thus the output power is 55 MW and 64 MW.
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_20, © The Society for Experimental Mechanics, Inc. 2011
225
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226
Figure 1: Rotor and stator of the hydraulic generator Performance measurements Performance measurements of the generator were taken in several operating conditions, namely: Speed No Load (SNL), 0%, 70%, 85% and 100% of the nominal summer power specifications (55 MW). Temperature measurements Several temperature sensors (DTS fibers and thermocouples) were installed to measure the stator and the stator frame temperature as well as the air temperature in the generator enclosure. These temperature measurements were used to evaluate the temperature distribution for the numerical simulation. Flow measurements The speed flow measurements of the stator cooling ducts and enclosure were performed using specially designed static convergent cones with static pressure taps and flow meters [1]. These measurements were used to evaluate the mass and volume flow for future numerical simulation purposes. Acoustic and mechanical measurements Acoustic measurements were performed around the generator in all the above-mentioned operating conditions (see Figure 2 for the sound meter positions). These measurements were used to evaluate the overall noise level for monitoring the working environment and machinery by non-contact sensing. Frequency analyses were also performed to correlate the acoustic signals with the operating conditions. Mechanical instrumentation was installed on the rotor and stator. The rotor instrumentation includes 3 accelerometers installed in the axial, radial and tangential directions on one of the rotor’s 16 cross arms and 16 strain gages (one per cross arm, with the position optimized by finite-element analysis and previous measurements) (Figure 3). These signals were transferred to the acquisition system by RF transmission. The stator instrumentation allows acceleration measurements (radial and tangential) on the stator core and stator frame and relative displacement of the stator core to the stator frame to investigate their relative displacement vs. temperature and operating conditions (Figure 4). These mechanical measurements were used to investigate the effect of operating conditions by frequency analysis.
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227
3RLQW
3RLQW
3RLQW
0HDVXUHPHQWSRLQWVDUHORFDWHG DWPHWHURXWVLGHRIWKHURWRU GLDPHWHU 3RLQW
G% $ RFWDYHDQGRYHUDOO PHDVXUHPHQWVDQG))7 N+]DQGN+]
3RLQW
3RLQW
3RLQW 3RLQW
Figure 2: Location of acoustic measurement points
Figure 3: Generator’s rotor instrumentation
Stator core and stator frame accelerometers stator core and stator frame relative displacement laser sensor
Figure 4: Stator instrumentation
BookID 214574_ChapID 20_Proof# 1 - 23/04/2011
228 ACOUSTIC MEASUREMENTS RESULTS The overall noise level at a specific position vs. the operating conditions ranges from 80.7 dB(A) in SNL conditions to 85.5 dB(A) at 55 MW (100% of summer generator’s power specification (55 MW) (Figure 5). The noise level slightly exceeds the prescribed value of 85 dB(A) for the power generator. The maximum noise emission predominant frequency is 120 Hz (twice the line frequency) but includes a large frequency content, as illustrated in Figure 6, which shows the linear weighted noise level of the generator for the frequency range of 0-2 kHz under different operating conditions. This frequency content allows us to identify mechanical signals (rotation speed and harmonics, blade passing frequency, etc.) and electromagnetic excitation on the rotor and stator of the power generator (twice the line frequency and harmonics, slot passing frequency and harmonics, electromagnetic torque transmission between rotor and stator). 90,0 88,0 85,5
Noise level (dB(A))
86,0
84,5
84,7
84,0 82,0
80,7
81,3
80,0 78,0
Point 6 77,3
76,0 74,0 72,0 70,0 Bruit de fond
SNL
0%
70%
80%
100%
Operating conditions
Figure 5: Noise level vs. operating conditions
100
SNL
90
0% 70%
Noise level (dB)
80
80%
70
100%
60 50 40 30 20 0
500
1000
1500
2000
Frequency (Hz)
Figure 6: Noise level (dB) vs. frequency, 0-2 kHz
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229 MECHANICAL MEASUREMENTS RESULTS Rotor mechanical instrumentation The mechanical measurements taken on the rotor allowed us to evaluate the stress and vibration level of the rotor cross arms. The strain analysis in the SNL and other operating conditions shows that the strain level is low on the rotor structure. The average stress (for the rotor’s 16 cross arms) at 100% operating conditions is 94.0 MPa with a standard deviation of 3.2 MPa. This represents an increase of the stress from the SNL operating condition of 65.4 MPa cause by the mechanical torque transmission at maximum power. At the measurement points, for this type of material (carbon steel), the stress level is considered to be low and uniform over the rotor structure. These stress levels include the static and dynamic stress. Figure 7 shows the frequency spectrum of the dynamic part of the stress levels. The dominant frequencies are components of the electromagnetic torque between rotor and stator at 94.74 Hz and harmonics at 189.5 Hz but the signal also includes some of the low-frequency rotation speed (94.7 RPM or 1.5783 Hz) harmonics. Radial and axial acceleration levels were predominant on the rotor. The frequency analysis (0500 Hz) is shown in Figure 7 for the two signals at 100% operating condition in comparison with all strain gauge signals in the same operating condition. The specific frequency content is presented for the rotor measured signal, which represents the main electromagnetic torque excitation but, also, at a lower level, the mechanical excitation.
Figure 7: Accelerometers and strain-gauges frequency analysis Stator mechanical instrumentation The mechanical measurements on the stator allowed us to evaluate the relative distance between the stator and stator frame and to evaluate the vibration and displacement level of the magnetic core of the stator. The distance between the stator and its frame is constantly changing because of thermal expansion of the components. A new type of anchor for the stator core lamination to the stator frame was used to refurbish the generator. The core laminations are aligned with a tube in which a clamping stud is inserted: the theoretical gap between lamination and tube at room temperature is 0.1875 mm. The temperature difference between the warm core lamination and the colder stator frame in operation reduces this gap as the power is increased and creates interference at rated power. The average, minimal and maximal distances between the stator and its frame are given in Figure 8. The results show some interference between the core and the frame but only at the active rated power of 55 MW (100%). At the 85% operating condition level, the average value indicates interference but the minimum value shows that the contact is not
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230 steady. These results comply with temperature simulations of the thermal expansion between these two elements.
0,15
Interference
0,10
Distance (mm)
0,05 0,00 SNL
70%
85%
100%
-0,05 -0,10 -0,15
gap -0,20
Figure 8: Relative displacement between the core and the frame of the stator The displacement of the stator core in the radial direction was evaluated from double integration of the accelerometer signal. The values recorded at 120 Hz ranged from 18 to 22 μm, in compliance with the applicable limit of 30 μm for displacement of the stator core. Spectral analysis of the radial stator vibration under different operating conditions (Figure 9) shows that the predominant vibration level is 120 Hz and its harmonics. The 100% operatingcondition vibration presents a modulation of the 120 Hz at about 40 Hz. This low-frequency vibration causes a higher displacement than the 120 Hz. The origin and effect of this vibration need to be identified. 1 0,9
Acceleration (m/s2)
0,8 0,7 100%
0,6
80%
0,5
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0,4
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0,3 0,2 0,1 0 0
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Figure 9: Spectral analysis of the stator radial vibration
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231 RELATION OF ACOUSTIC AND MECHANICAL MEASUREMENTS TO EXITATION FORCES The forces on the rotor and the stator of the power generator are a combination of mechanical and electromagnetic excitation. Spectral analysis of the acoustic rotor and stator excitation shows that one signal alone cannot give a complete diagnosis of the status of the power generator. The frequency content at the stator shows a major contribution of the electrical excitation (120 Hz and harmonics). The rotor excitation shows a frequency content that is governed by mechanical and electromagnetic excitation (rotation frequency and electromagnetic frequencies). The acoustic excitation shows a richer frequency content. For example, the acoustic spectrum includes all the 120 Hz and harmonics (present at the stator) and also frequencies present at the rotor measurement points such as the electromagnetic torque between rotor and stator at 94.74 Hz rotation speed harmonics.
CONCLUSION A series of measurements were performed on a hydraulic power generator equipped with mechanical instrumentation on the rotor and stator. Acoustic measurements were also performed. Spectral analysis of the rotor and stator instrumentation signals shows that the different excitation forces are not present on the all sensors. The rotor instrumentation shows principally the rotation frequency and electromagnetic torque between the rotor and the stator. The results of the stator measurements illustrate mainly the line frequency (120 Hz) and its harmonics. The acoustic signal shows a richer frequency content that includes most of the electromagnetic and mechanical excitation, which paves the way for development of a generator diagnostic by non contact instrumentation. Further analysis of the complete set of mechanical and acoustic signals could lead to a more complete hydraulic power generator diagnostic based on non contact acoustic measurements.
ACKNOWLEDGMENTS The authors acknowledge the technical staff of the Hydro-Québec research institute (IREQ) for the quality of their work. This team includes Guillaume Chaput, Jean-Philippe Charest-Fournier, Calogero Guddemi, Luc Martell, Benedicto Navarette, Jean Picard and Mathieu Soares.
REFERENCES [1] IREQ-2009-0001-Méthode de caractérisation de l’écoulement à la sortie des évents du stator [2] Hydro-Québec Production, GEME – Guide des exigences de maintenabilité et d'exploitabilité, Module COM01- – Sécurité Technique, Mars 2009 [3] Hydro-Québec Production, GEME – Guide des exigences de maintenabilité et d'exploitabilité, Module GR07 – Alternateur, revision in progress
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Contact-less Wind Turbine Utilizing Piezoelectric Bimorphs with Magnetic Actuation Scott Bressers, Dragan Avirovik, Michael Lallart, D.J. Inman, and Shashank Priya CEHMS, Department of Mechanical Engineering, Virginia Tech, Blacksburg, VA 24061
Abstract The demand for efficient small-scale wind harvester is continually increasing in order to meet the local power needs for applications ranging from wireless sensor networks to charging of mobile devices. The efficiency of wind turbines is dependent upon several structural variables including frictional contacts. In order to overcome the problem of gearing and losses in mechanical contacts, we propose here a novel small-scale windmill design that utilizes magnetic attractive and repulsive force to create mechanical oscillation in piezoelectric bimorphs which is then converted into electric charge through direct piezoelectric effect. This contact-less wind turbine has several advantages including operation at much lower wind speeds and longer life span. The prototype was fabricated as a vertical-axis wind turbine featuring a modular Sarvonius rotor. Characterization was performed by utilizing several configurations for this modular rotor. Output power magnitude for steady-state operation in wind speeds of 2 – 10 mph was used to compare the performance of various configurations.
1
INTRODUCTION
In recent years, low-power electronics (i.e. microwatt microprocessors) have been deployed in variety of applications such as wireless sensors which opens the possibility of powering them with ambient energy sources. Currently, majority of wireless sensors operate on batteries which limits their lifetime, require disposal and adds extra expenses in maintenance. Finding an economical and environmentally friendly alternative to batteries has prompted research on small-scale energy harvesting devices. The outside solar energy has the capability of providing power density of 15000μ W/cm3 which is about two orders of magnitudes higher than other ambient sources. Definitely, solar energy is a very attractive source for powering the wireless sensor network and the solar technology has matured over the years. One of the major challenges in the implementation of solar technology on the “energy on demand” platform has been the requirement of bulky electronics. Further, the variation in the light intensity (cloudy vs. sunny day) can drop the efficiency significantly. As the power requirements for the sensor nodes have dropped down to a low level of 100 μW (and are decreasing continuously) an elegant solution to powering would be finding simpler and cheaper technology. Air flow is other most attractive alternative. The success in the development of technology for generating electricity from air/wind at the very small scale can lead to a revolution in the area microelectronics. Although wind energy harvesters (i.e. wind turbines) have traditionally relied on some form of electromagnetic generator to convert mechanical energy to electrical energy, piezoelectric materials have been shown to offer higher efficiency on small-scale. Several configurations of T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_21, © The Society for Experimental Mechanics, Inc. 2011
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small-scale wind turbines utilizing piezoelectric materials have been reported [1-2]. These designs rely on mechanical actuation of piezoelectric elements. The wind turbine concept proposed in this report features “contact-less” actuation of piezoelectric elements via magnetic forcing. A concept sketch is presented in Figure 1. The proposed design is a vertical axis wind turbine (VAWT) with a series of alternating polarity magnets mounted to the rotating shaft. Steady-state rotation of the vertical shaft induces a harmonic vibration in the piezoelectric elements via an alternating attractive/repulsive force between stationary magnets mounted at the tip of the piezoelectric elements and rotating, shaft-mounted magnets. The harmonic excitation is then converted to an AC voltage output via the direct piezoelectric effect. Not only will the magnetically induced actuation result in minimized frictional losses, but it should also extend the working life of each piezoelectric elements. Additionally, the reduced friction makes the design more applicable to low-wind-speed areas.
Figure 1. Concept sketch of “contact-less” vertical-axis wind turbine featuring magnetically actuated piezoelectric elements mounted onto the base in a cantilever configuration.
2
PROTOTYPE DESIGN
A prototype based on the contact-less concept was fabricated and all the important design metrics were identified through testing in home-built wind tunnel. During the course of prototype fabrication, the design shown in Fig. 1 was slightly modified to enhance the operation. The piezoelectric generator geometry was found to be better suited for a vertical-axis wind turbine than the traditional horizontal-axis wind turbine (HAWT) as the generator itself represents a significant obstruction to flow if placed along a horizontal axis. Furthermore, the VAWT configuration has ability to harvest multi-directional flow which will be of particular importance in “ground wind” applications (i.e. elevations of < 20 feet from relative ground). For these reasons, the VAWT configuration was maintained in the prototype as shown in Fig. 2. Additionally, the vertical rotor assembly was constructed to be modular to test multiple rotor configurations for a single piezoelectric generator base.
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(a)
(b)
9 in.
Figure 2. Contact-less wind turbine prototype featuring magnetically-actuated piezoelectric generator: (a) Two-blade, and (b) Four-blade Savonius rotor designs. Note: the three carbon fiber columns are used to stabilize axis of rotation.
2.1
Description of the piezoelectric generator
The foundation of contactless wind turbine presented in this study is piezoelectric generator (e.g. stator). The piezoelectric generator located at the base of the contactless wind turbine consists of five main components: piezoelectric elements, clamps, magnets, acrylic base, and shaft bearing. The complete generator assembly can be seen in Fig. 3.
Figure 3: Piezoelectric generator assembly featuring six cantilever piezoelectric elements with magnetic tip masses.
The assembly consists of six piezoelectric elements mounted in a cantilever configuration. As the rotor assembly rotates along the vertical axis, the shaft-mounted magnets interact via an
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alternating attractive/repulsive magnetic force with the tip-mounted magnets. This induces a pseudo-sinusoidal vibration in the piezoelectric elements resulting in a AC voltage. The power output of each piezoelectric beam is a function of the frequency of its vibration and consequently the frequency of shaft rotation as well as the several performance parameters. Piezoelectric bimorphs were acquired from APC International Ltd and their performance parameters are listed in Table I. Table I: Specifications for the piezoelectric bimorphs used in fabrication of contact-less windmill [3].
Dimensions Free (mm) Length Long Wide Thick (mm) 60.0 20 0.60 53
Total Deflection (mm) >2.5
Blocking Force (N) >0.25
Resonance Frequency (Hz) 65
Capacitance (pF) 170.000
The piezoelectric bimorphs were distributed in a circular pattern at 60 Ț intervals. The result is that opposing pairs of piezoelectric bimorphs are displaced equally but in reverse direction during each cycle. In this sense, the responses of opposite piezoelectric beams are 180 Ț out-ofphase with respect to opposing pair. Similarly, the response of adjacent piezoelectric beams are out-of-phase depending on the magnetic rotor configuration. This represents a significant problem for harvesting the combined output of all six bimorphs. 2.2
Vertical Rotor Assembly
The rotor assembly was responsible for converting the available energy in ambient airflow to rotational energy as well as transmitting this mechanical energy to the piezoelectric generator. The prototype’s vertical-axis rotor assembly has two primary components: the blade rotor and the magnet rotor. As modularity suggests, both the blade rotor and the magnet rotor can be easily swapped with alternate configurations having similar dimensions. 2.2.1 Blade Design The blade rotor was an important characteristic component of the fabricated vertical-axis wind turbine. As with conventional horizontal-axis airfoil design, significant research has been conducted on vertical-axis blade design. These VAWT blade designs can be categorized based on the fluid dynamic effect used to drive their rotations; principally, drag-based and lift-based designs. The most common designs are Savonius rotor, a drag-based design; and the Darrieus rotor, a lift-based design. For comparison purposes, the effectiveness of a given blade design is given by the coefficient of performance (i.e. aerodynamic power divided by total available power from airflow), CP , curve as a function of the wing-tip to wind speed ratio, λ . The theoretical performance of common blade designs is given by Fig. 3.
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Figure 4. Performance curves of common blade designs for wind turbines [5].
As can be seen in Fig. 4, the fast-running lift-based rotors offer higher performance coefficients than the slower-running drag-based rotors. However, Savonius rotors offer several design advantages that cannot be observed from their predicted power curve including, simple construction, high starting torque (e.g. low start-up wind speed), and structural resilience. It has also been shown that these rotors are able to continue to operate at high wind speeds where liftbased designs may suffer damage [5-6]. Principally, the starting torque and simplicity of the Savonius design provide justification for implementing it in our low-wind-speed prototype.
Figure 5. Top-down diagram of traditional Savonius designs with (a) joined blades (S-rotor), (b) gapped circular-arc blades, and (c) gapped circular-arc blades with tangential extensions. Note: U ∞ indicates the airflow velocity [7].
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One additional advantage that the Savonius rotor has over the Darrieus is its ability to self start. A pure lift-based, or Darrieus, design requires either human intervention or a small, startup motor to begin rotation. This represents a serious design flaw for a low-power turbine. While the Savonius rotor ordinarily has self-startup capability, there exists the special case where the Sarvonius blades are exactly parallel to a unidirectional flow (Fig. 5(a) bottom row). Although unlikely to occur in potentially multi-directional flow, this small design flaw can be overcome by increasing the number of blades in the rotor. To investigate the effect of additional blades, a fourblade Savonius rotor was constructed in addition to the more traditional two-blade design.
Figure 6. Both 2-blade (left) and 4-blade (right) Savonius rotors implemented in prototype. Note: both rotors feature gapped circular blades with the same physical dimensions and effective surface area of 30 in2.
2.2.2 Magnetic Rotor Configuration The magnetic rotor was responsible for converting the rotational energy in the vertical shaft to a harmonic excitation in the piezoelectric beams. This rotor consists of a series of alternating polarity magnets placed in a circular pattern on a Plexiglas disk. This disk is mounted just below the blade rotor along the rotating shaft and controls the dynamics of the entire harvesting scheme. By varying the physical parameters such as magnet size, strength, spacing, polarity, number, etc., the overall performance of the wind turbine can be affected. Keeping with the concept of modularity, two alternate magnetic rotor configurations were constructed: a 4-magnet configuration and a 6-magnet configuration. Both use the same highstrength, Neodymium-Boron magnets (1.00”x0.25”x0.10” thick). The radial spacing was also maintained as the position of beam-mounted magnets was not adjustable. One significant difference, however, is the polarity of adjacent magnets. Since the six-magnet rotor matches the number of corresponding piezoelectric bimorphss, the polarity is simply alternating. However, adjacent magnets in the four-magnet rotor may have alternating or the same polarity (Fig. 7). This introduces a particularly important principal of the contactless concept.
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Figure 7: Four-magnet rotor configuration (left) and six-magnet rotor configuration (right) with relative polarities indicated.
Opposite rotor magnets must be of alternating polarities. It was found that when the rotor is positioned such that a pair of opposite rotor magnets directly faces a pair of opposite piezoelectric bimorphs then the system maintains smooth operations. This creates inherent instability in the system and eliminates a potential static equilibrium position. Consequently, the overall “magnetic friction” or static torque load is reduced resulting in a lower startup wind speed. The magnetic rotor configuration also has implications towards the electrical output which will be explained in the magnetic rotor characterization section.
3
CHARACTERIZATION OF PROTOTYPE
Characterization of VAWT prototype was performed using a low-speed, open-channel, blow-down wind tunnel. For the purpose of characterization, only unidirectional airflow perpendicular to the surface area of the turbine blades was considered. For simplicity, data was initially collected from a single piezoelectric element. Ideally, the total output power would be the output power of single element multiplied by the number of piezoelectric elements. However, because the piezoelectric elements are excited out-of-phase from one another, a complex switching circuit or a series of rectifier bridge circuits are needed. The harmonic excitation of piezoelectric element generates an AC voltage output. This AC voltage was converted in to a rectified DC voltage by using a series of diodes and smoothing capacitors. This DC voltage was then measured across a simple shunt resistor used to simulate the impedance or electrical load. The steady-state output power at a given load resistance ( RL ) was calculated by using the simple power relation: 2 P = VDC (1) / RL 3.1
Blade Design Effects
To gain insight into the expected performance from prototype, steady-state DC voltage was recorded at various wind speeds. As with any energy harvesting system, the output power was a function of electrical load in addition to the physical parameters. Thus, a resistor sweep was performed for each wind speed to identify the optimal electrical load. This method of characterization was performed for four different rotor configurations: two-blade and foumagnet, two-blade and six-magnet, four-blade and four-magnet, and four-blade and six-magnet. In this section, the resistor sweep data for two-blade rotor and four-blade rotor are compared.
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Power Output (microW)
600
3 mph 4 mph 5 mph 6 mph 7 mph 8 mph 9 mph 10 mph
(a)
400
200
0 10
100
Power Output (microW)
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3.0 mph 4 mph 5 mph 6 mph 7.0 mph 8.0 mph 9 mph 10 mph
(b) 600
300
0
1000
10
Resistance (kOhms)
100
1000
Resistance (kOhms)
Figure 8. Resistor sweep data for both the (a) four-blade rotor, and the (b) two-blade rotor configurations for various wind speeds. Both resistor sweeps were performed using the 6-magnet rotor configuration
Max Power Output (microW)
Figure 8 shows the resistor sweep data. It is apparent from this figure that there is an optimal load resistance providing peak output power at a given wind speed. It should be noted that there is a slight shift in optimal resistance towards right at the lower wind speeds. Furthermore, this optimal load resistance is less defined at these lower wind speeds as the output power shows broad maxima with resistance. The broadness in the curve could be related to the smaller strain values obtained for piezoelectric bimorph at lower wind speeds. This represents an important design characteristic for the development of an accompanying electrical storage circuit. The peak output power at each wind speed was plotted for both the two-blade rotor and the four-blade rotor configurations are summarized in Fig. 9.
4-Blade, 6-Magnet 2-Blade, 6-Magnet
600
300
0 3
6
9
Wind Speed (mph) Figure 9. Power output from a single piezoelectric element at peak electrical loads for various wind speeds using both the two-blade and four-blade Savonius rotor configurations. Based on the output power values, it can be seen that the two-blade Savonius rotor out-performs the four-blade Savonius rotor. The two-blade rotor also had a lower startup wind speed of 2.2 mph as compared to 3.0 mph for four-blade rotor. This agrees with the theoretical models proposed in literature that show performance of Savonius rotors decreases with the addition of blades [9]. However, as
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mentioned in the section on blade design description, the two-blade rotor may not self-start in unidirectional flow. One proposed solution to the tradeoff between guaranteed self startup and loss of efficiency is a twisted, or helical Savonius blade design [9].
3.2
Effect of Magnetic Rotor Configuration
In addition to the number of blades, the prototype’s performance can be further explored using the different magnet configurations. As stated earlier, the only difference between the two rotors is the number of magnets used. However, the dynamic response of the system is greatly altered. This is apparent from the open-circuit voltage waveform recorded using an oscilloscope.
Figure 10. Voltage waveform from single piezoelectric element using the four-magnet rotor (top) and the six-magnet rotor (bottom). Waveform measured using a digital oscilloscope. Note: airflow characteristics were not identical for both cases.
As expected, the six-magnet configuration provides a far more sinusoidal signal than the four-magnet rotor. This is simply due to the difference in polarities between the two magnet rotors. Ideally, both the sinusoidal output from the six-magnet rotor and the “double-pulse” signal from the four-magnet rotor should be rectified a DC RMS value. However, the relatively low signal amplitude makes implementing a simple rectifier difficult, especially in the case of the four-magnet rotor where the signal has overlapping modes which results in added spurious. Thus, the six-magnet rotor would be proper choice for the signal conditioning. One would expect the six-magnet rotor to be more advantages from a performance standpoint as well since increasing the number of rotor magnets increases the frequency of vibration in the piezoelectric elements for a given angular velocity. However, at some threshold, increasing the frequency will reduce the maximum deflection of piezoelectric beam resulting in a reduction of the RMS voltage. This is the reason that six-magnet rotor produces higher power output at lower wind speeds (e.g. lower rotational frequencies) while the four-magnet rotor produces more power at higher wind speeds as shown in Fig. 11.
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1400
2-Blade, 4-Magnet 2-Blade, 6-magnet
1200 1000 800 600 400 200 0 2
4
6
8
10
Wind Speed (mph) Figure 11. Peak output power values for both the four-magnet and six-magnet rotor configurations as a function of wind speed.
4
CONCLUSOIN
In summary, this study reports the construction and performance of a novel vertical-axis wind turbine using piezoelectric bimorphs. The fabricated prototype shows the performance of the “contact-less” turbine that uses alternating attractive and repulsive magnet interactions to strain piezoelectric beams; replacing the conventional electromagnetic generator found in most traditional wind turbines. The two-blade rotor was found to outperform the four-blade rotor. The configuration using six magnets offers a lower startup wind speed and higher power below four mph. In subsequent studies, a deterministic experiment will be conducted to identify the best rotor configuration that combine all the advantages, namely, low start-up speed, wide operation range, and higher output power. Analytical and finite element models will be used to explore the system dynamics and investigate scaling of the physical parameters. The results from these studies will be reported elsewhere.
ACKNOWLEDGEMENT
The authors gratefully acknowledge the financial support from NIST and Pratt & Whitney.
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REFERENCES
1. Modeling of electric energy harvesting using piezoelectric windmill. Priya, S. s.l. : Applied Physics Letters, 2005. 87, 184101. 2. Small scale windmill. Myers, R., et al. s.l. : Applied Physics Letters, 2007. 90, 054106. 3. APC Incernational Ltd. Piezoelectric Ceramic Stripe Actuators. [Online] http://www.americanpiezo.com/products_services/stripe_actuators.html. Catalog No. 40-1010. 4. Wilson, R.E. and Lissaman, P.B.S. Applied Aerodynamics of Wind Power Machines. Oregon State University : s.n., 1974. Technical Report. 5. The Savonius rotor and its applications. Savonius, SJ. 5, New York, NY : Mechanical Engineering, 1919, Vol. 35. 0025-6501. 6. A double-step Savonius rotor for local production of electricity: a design study. Menet, J.L. 11, Valenciennes, France : Renewable Energy, 2002, Vol. 29. 1843-1862. 7. The Fundamentals of Wind-Driven Water Pumpers. Kentfield, John. Amsterdam : Gordon an Breach Science Publishers, 1996. 8. Optimum Design Configuration of Savonius Rotor Through Wind Tunnel Experiments. Saha, U.K., Thotla, S. and Maity, D. s.l. : Journal of Wind Engineering and Industrial Aerodynamics, 2008, Vol. 96. 1359-1375. 9. Performance Tests on Helical Savonius Rotors. Kamoji, M.A., Kedare, S.B. and Prabhu, S.V. 34, s.l. : Renewable Energy, 2009. 521-529.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
(IIHFWRIIXQFWLRQDOO\JUDGHGPDWHULDOVRQUHVRQDQFHVRIURWDWLQJEHDPV $UQDOGR-0D]]HL-U Department of Mechanical Engineering C. S. Mott Engineering and Science Center Kettering University 1700 West Third Avenue Flint MI, 48504, USA
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T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_22, © The Society for Experimental Mechanics, Inc. 2011
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W (t ) , X , M ,
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external moment Poisson’s ratio spatial function fundamental frequency of the beam angular velocity of the hub ( π = ݀ߠΤ݀) ݐ derivative with respect to time ( ݀Τ݀) ݐ derivative with respect to longitudinal coordinate ( ݀Τ݀) ݔ
$%675$&7 Radially rotating beams attached to a rigid stem occur in several important engineering applications, such as helicopter and turbine blades and certain aerospace applications. In most studies the beams have been treated as homogeneous. Here, with a goal of system improvement, non-homogeneous beams made of functionally graded materials are explored. Effects on natural frequency and coupling between rigid and elastic motions are investigated. Euler-Bernoulli theory, with Young’s modulus and density varying in a power law fashion, together with an axial stiffening effect, are employed. The equations of motion are derived using a variational method and an assumed mode approach. Results for the homogenous and non-homogeneous cases are treated and compared. Preliminary results show that allowing the Young’s modulus and the density to vary by approximately 2.15 and 1.15 times, respectively, gives an increase of 28% in the lowest bending natural frequency of the beam, an encouraging trend. ,1752'8&7,21 Rotating machinery form an important part of engineering and radially rotating beams constitute a major category of such systems. For instance, rotor blades, propellers and turbines fall into this category. For vibration control, it is important to identify possible system resonances and, if required and possible, change these values. Extensive work on these types of problems has been done in the aerospace literature. Comprehensive reviews can be found in the papers of Kane and Ryan [1] and Haering et al. [2]. They, and others, showed that at high speeds the rotating structure can be prone to instabilities. It is assumed here that the rotational speeds are small enough that no instabilities are encountered. There are numerous works on vibrations of radially rotating beams (uniform beams, beams including pre-twisted and tapered beams). Two classes of problems arise, namely, prescribed motions and prescribed torques. Earlier studies on the former type of problem can be found in the texts by Putter and Manor [3], Hoa [4], Hodges and Rutkwoski [5] and Hodges [6]. Putter and Manor used a finite element approach to obtain the natural frequencies and mode shapes of the beam, including shearing forces, rotary inertia and varying centrifugal forces. Hoa also utilized a finite element approach for the same objective, but effects of root radius, setting angle and tip mass were included. Hodges used asymptotic expansions to obtain an approximate value for the fundamental frequency of a uniform beam and Hodges and Rutkwoski used a finite element approach to calculate the eigenvalues and eigenvectors of the beam including different hub radii, tapered beams and beams with discontinuities. Kojima [7] investigated the transient flexural vibrations of a beam / mass system attached to a rotating rigid body. The prescribed torque problem has been studied by, for example, Yigit et al. [8], a work which the current closely follows. In that work the flexural motion of a rotating beam was investigated by using a specified torque profile to drive the rotating body (so that the rigid body motion was not known a priori). Lee et al. [9] presented experimental results confirming that centrifugal effects cannot be neglected, even at first order, when modeling these systems. Models utilizing a Timoshenko beam type approach (other than Euler-Bernoulli) are also numerous. See, for example, the work of Lin and Hsiao [10] which investigates the effect of Coriolis force on the natural frequencies of the rotating beam. More involved models including base excitation can be found in references [11] and [12].
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The fundamental frequency of rotating beams with pre-twist was investigated by, for example, Hu et al. ([13]). The role of Functionally Graded Materials (FGM - see for instance reference [14]) in radially rotating beams has not been fully investigated. Librescu et al. [15], in a prescribed motion problem, studied the effects of material variation through the beam thickness on the eigenfrequencies of the system. In the present work a prescribed torque problem is investigated with material properties varying along the length of the beam. The possibility of changing the fundamental frequency of vibration of a cantilever radially rotating beam is investigated. The approach includes changing the material of the beam from a homogeneous type to a FGM while keeping the physical dimensions of the beam constant. A one-term Galerkin approach is utilized and the investigation is conducted using numerical simulation. Preliminary results, using linearized versions of the equations of motion, indicate that, in the case of a beam made of Aluminum / Silicon Carbide, the fundamental frequency can be increased by approximately 28% when compared to a pure aluminum beam. This change was obtained by the use of a material that shows a Young’s modulus increase of about 2.15 times over the length of the beam. The density increase is about 1.15 times. 02'(/,1* Figure 1 shows a beam with length ܮattached to a rigid hub of radius ܾ . The hub rotates radially with angular YHORFLW\ȍ and is subjected to an external moment ߬()ݐ. A set of mutually perpendicular unit vectors ܽԦ1 , ܽԦ2 and ܽԦ3 is attached to the undeformed configuration of the beam. A second set of mutually perpendicular unit vectors, ݊ሬԦ1 , ݊ሬԦ2 and ݊ሬԦ3 is assumed to be the inertial reference frame. The vector ݊ሬԦ3 is the axis of rotation for the hub and remains parallel to ܽԦ3 during motion. ߐ is the angle between the vector ݊ሬԦ1 and ܽԦ1 and defines the angular position of the hub with respect to the inertial frame. Note that π = ݀ߠΤ݀ݐ. For rotation on the plane, the position, velocity and acceleration of point 3 on the deformed configuration of the beam are given, respectively, by: ሬሬሬሬԦ ܾ( = + ܽ)ݎԦ1 + ܽݓԦ2 ܴ (1) ሬሬሬԦ ܸ = െߠݓሶܽԦ1 + [(ܾ + ߠ)ݎሶ + ]ݓሶ ܽԦ2 (2) ሬሬሬሬԦ = െ[2ݓሶߠሶ +(ܾ + ߠ)ݎ2ሶ + ߠݓሷ ] ܽԦ1 + [ݓሷ + (ܾ + ߠ)ݎሷ െ ߠݓ2ሶ ] ܽԦ2 ܣ (3) The kinetic energy of the beam can be computed from (note ) ݔ = ݎ: 1 ܮ ߲ ݓ2 ߲ݓ 1 ܶ = න ߩ( ܣ)ݔቊ൬ ൰ + 2ߠሶ ൬ ൰ (ܾ + )ݔ+ ߠሶ 2 [(ܾ + )ݔ2 + ݓ2 ]ቋ ݀ ݔ+ ߠ ܬሶ 2 2 0 ߲ݐ ߲ݐ 2 (4) The potential energy of the system comes from two parts. The first part is caused by the bending elastic strain. Using Euler-Bernoulli beam theory, this can be calculated by: 1 ܮ ߲2 ݓ ܸ = ݏන ܫ)ݔ(ܧቊ( 2 )2 ቋ ݀ݔ 2 0 ߲ݔ (5)
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Rotating cantilever beam Y Y
3
Rigid hub
߬()ݐ
Figure 1 ±Rotating cantilever beam
The second part is due to the centrifugal force acting on the beam, which causes axial elongation (see, for example, reference [12]). It is given by: ܮ 1 ݓ߲ ܮ ߲ݓ ܸܽ = න ( )2 ቊන ߩ(ߠ[ܣ)ݔሶ 2 (ܾ + )ݔ+ 2ߠሶ + ߠݓሷ] ݀ݔቋ ݀ݔ 2 0 ߲ݔ ߲ݐ ݔ
(6) The total potential energy is obtained from: ܸ = ܸ ݏ+ ܸܽ . (7) To derive the equations of motion an assumed mode approach is adopted. The following form for the one-mode approximation is assumed for the elastic form. ݔ(ݓ, )ݐ(ߟ)ݔ(߮ = )ݐ (8) where ߮( )ݔis a spatial function that satisfies the geometric boundary conditions at the clamped end of the beam.
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The Lagrangian [16] can then be written as: 1 ܮ 1 ࣦ = ܶ െ ܸ = ቊන ߩ(ܣ)ݔ൛߮ 2 ߟሶ 2 + 2ߠሶ߮ߟሶ (ܾ + )ݔ+ ߠሶ 2 [(ܾ + )ݔ2 + ߮ 2 ߟ2 ]ൟ݀ ݔቋ + ߠܬሶ 2 2 0 2 ܮ ܮ 1 െ න ൝"߮ܫ)ݔ(ܧ2 ߟ 2 + ߮Ԣ2 ߟ 2 ቊන ߩ(ߠ[ܣ)ݔሶ 2 (ܾ + )ݔ+ 2ߠሶ ߮ߟሶ + ߮ߟߠሷ] ݀ݔቋൡ ݀ݔ 2 0 ݔ (9) In a prescribed motion problem, ߠሶ is known a priori and then one equation of motion is obtained from (9). When a torque is prescribed ߠ is a generalized coordinate and two coupled non-linear equations of motion are obtained. Solutions to these equations are computationally intensive and, to obtain preliminary results, simplified versions of the equations are utilized in the following. From the Lagrangian, linearized 1 equations of motion can be obtained as follows. ܮ
ܮ
ܮ
ቊන ߩ ( ߮ܣ)ݔ2 ݀ݔቋ ߟሷ + ቊන ߩ ( ܾ(߮ܣ)ݔ+ ݔ݀)ݔቋ ߠሷ + ቊන "߮ܫ)ݔ(ܧ2 ݀ݔቋ ߟ = 0 0
0
0
(10) ܮ
ܮ
ቊන ߩ ( ܾ(߮ܣ)ݔ+ ݔ݀)ݔቋ ߟሷ + ቊ ܬ+ න ߩ ( ܾ(ܣ)ݔ+ )ݔ2 ݀ݔቋ ߠሷ = ߬()ݐ 0
0
(11) Note that here, as done in reference [8], the rotation of the hub is not assumed to be prescribed. Next equation (11) is solved for ߠሷ and the result is substituted into equation (10). This leads to:
ߟܯሷ + )ݐ(ܨ = ߟܭ (12) where ܮ
2
= ܯන ߩ( ݔ݀ ߮ܣ)ݔെ 0
2
ܮ
ቀ0 ߩ( ܾ(߮ܣ)ݔ+ ݔ݀)ݔቁ ܮ
ܬ+ 0 ߩ( ܾ(ܣ)ݔ+ )ݔ2 ݀ݔ
ܮ
ܮ
,
= ܭන ݔ݀ "߮ܫ)ݔ(ܧ, 2
0
= )ݐ(ܨെ
ቀ0 ߩ( ܾ(߮ܣ)ݔ+ ݔ݀)ݔቁ ߬()ݐ ܮ
ܬ+ 0 ߩ( ܾ(ܣ)ݔ+ )ݔ2 ݀ݔ
It follows from equation (12) that the fundamental frequency is:
߱݊ = ඨ
ܭ ܯ (13)
1 Following Yigit et al [8], terms involving ߠሶ 2 are neglected. With this simplification equation (11) can be solved for ߠሷ and substituted into equation (10).
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In the preceding, no assumption was made concerning the material type in the derivation of the equations of motion. In the following both homogeneous and non-homogeneous material types are considered. The nonhomogeneous materials utilized here are FGMs. Two different models are employed, one involving power law variations (aluminum / silicon carbide) and another based on a volume fraction approach (aluminum / steel). Results for the fundamental frequency of the FGM beams are compared to those for a pure aluminum beam. )*0, The first FGM model is based on the one described by Chiu and Erdogan [17]. The material is assumed to be isotropic and non-homogeneous with properties given by:
E ( x)
E0 ( a
x x 1) m , U ( x) U0 ( a 1) n L L (14)
where a, m and density at x 0 .
n
are arbitrary real constants with a ! 1 .
E0 and U 0 are the Young’s modulus and mass
In the sequel the FGM utilized is a composite made from aluminum and silicon carbide. The properties of the material are given in Table 1 and are taken from reference [17]. ( E0 GPa )
$OXPLQXP6LOLFRQ&DUELGH 105.197
3 U0 ( kg / m )
2710.000
a m n X
1.14568 1.00000 0.17611 0.33
Table 1 – Material properties for Al / SiC FGM Note that in this model Poisson’s ratio is taken to be a constant. )*0,, The second FGM model is developed by assuming that its composition is derived from of a mixture of two materials, with the material variation given by a power-law gradient (see, for instance, reference [18]). The effective material properties of the beam are given by:
x x E ( x) Eb ( Et Eb )( )O , U ( x) Ub ( Ut Ub )( )O L L (15) where ߣ is a non-negative constant describing the volume fraction, which can be determined experimentally ([18]). The subscripts ܾ and ݐrefer to the value of the parameter at = ݔ0 and ܮ = ݔ, respectively. These values are the ones for the “pure” materials involved in the composition of the FGM and are obtained from tables or manufacturer’s specifications. Also, Poisson’s ratio varies slightly but it is usually taken to be constant.
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180(5,&$/5(68/76 For the numerical simulations, the beam is taken to have a circular cross section with a radius given by ܴ0 = 010127 ݉. The length of the beam is = ܮ0.896 ݉ and the radius of the hub is ܾ = 0.05 ݉. Also, initially, the mass moments of inertia of the rigid and flexible parts are taken to be related in the following manner: ܮ ( = ܬ0.5)(0 ߩ( ܾ(ܣ)ݔ+ )ݔ2 ݀( )ݔthis corresponds to an “inertia ratio” of 0.5). The spatial function chosen to be used in equation (12) is the first mode shape of a non-rotating cantilever beam (see, for example, [19]): (ܥcoshሺߚ1 )ݔെ cos(ߚ1 )ݔെ ߪ1 [sinh(ߚ1 )ݔെ sin(ߚ1 )])ݔ (16) where ܥis a constant, ߚ1 = ܮ1.87510407 and ߪ1 = (sinh(ߚ1 )ܮെ sin(ߚ1 ))ܮΤ (cosh(ߚ1 )ܮ+ cosሺߚ1 ) )ܮ. +RPRJHQHRXVPDWHULDOEHDP For the homogeneous material, aluminum was chosen. The properties are:
E
0.71e11 N/m 2 and
U 2710 Kg/m3 . In this case the fundamental frequency of the rotating beam, given by equation (13), is ߱݊ = 36.98 ݖܪ. Note that, if one assumes prescribed motion (ߠሶ = constant) the frequency would be ߱݊ = 22.50 ݖܪ. This is the fundamental frequency of a cantilever beam with the dimensions given in this example. The higher frequency obtained via equation (13) shows the stiffening effect that the rotation produces.
Figure 2 – Fundamental frequency variation of homogeneous beam as a function of the inertia ratio Figure 2 shows the variation of the fundamental frequency, for this case, as a function of the inertia ratio. It is seen that as the inertia ratio increases the fundamental frequency decreases substantially.
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)*0,EHDP For a beam made of Aluminum / Silicon Carbide, the fundamental frequency is ߱݊ = 47.40ݖܪ. When compared to the pure aluminum beam, this result shows a frequency increase of about 28%. The variation of this frequency as a function of the inertia ratio is shown in Figure 3. As in the previous case, inertia ratio increase causes a decrease in the frequency.
Figure 3 ±Fundamental frequency variation of FGM I beam as a function of inertia ratio
)*0,,EHDP Next the Aluminum / Steel case is considered. The beam is assumed to be aluminum “rich” at = ݔ0 and steel “rich” at ܮ = ݔ. Poisson’s ratio is taken to be constant: ߭ = 0.33 and ߣ = 1 is used. The fundamental frequency for this case is ߱݊ = 27.77 ݖܪ, which shows a frequency decrease of about 25% when compared to the pure aluminum beam. The frequency variation as a function of the inertia ratio is shown in Figure 4. Equation (12) also allows for a comparison of the generalized masses for each case. By computing the masses, the following observations can be made. The aluminum / steel beam is 2.5 times heavier than the aluminum one. On the other hand the aluminum / silicon carbide beam is only 11% heavier than its aluminum counterpart. Clearly these results show the advantages of using FGM I, when the objective sought is to increase the fundamental frequency of the system while minimizing overall weight gain.
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Figure 4 ±Fundamental frequency variation of FGM II beam as a function of inertia ratio &21&/86,216 The use of FGM for radially rotating cantilever beams can change the fundamental frequency of vibration significantly. For a beam made of aluminum / steel, the frequency decreases by approximately 25% when compared to an aluminum one. This is followed by an increase of 2.5 times in overall weight. The Young’s modulus increase is about 3 times over the length of the beam and the density increase is approximately 2.9 times. For this case the use of such FGM may not be practical, or desirable, for the application at hand. For a beam made of aluminum / silicon carbide, the frequency can be increased by approximately 28% when compared to an aluminum beam. This is achieved by the use of a material where the Young’s modulus increases about 2.15 times over the length of the beam, whereas density increases about 1.15 times. These changes are considered reasonable. Moreover, these results are achieved with only an 11% increase in weight when compared to an aluminum beam. Here these are considered encouraging results warranting further investigation. 5()(5(1&(6 [1] [2] [3] [4] [5] [6]
.DQH 7 5 5\DQ 5 5 DQG %DQHUMHH $ ., Dynamics of a cantilever beam attached to a moving base, Journal of Guidance, Control and Dynamics, vol. 10, pp. 139-151, 1987. +DHULQJ : - 5\DQ 5 5 DQG 6FRWW 5 $, New formulation for flexible beams undergoing large overall plane motion, Journal of Guidance, Control and Dynamics, vol. 17, pp. 76-83, 1994. 3XWWHU6DQG0DQRU+, Natural frequencies of radial rotating beams, Journal of Sound and Vibration, vol. 56, pp. 175-185, 1978. +RD69, Vibration of a rotating beam with tip mass, Journal of Sound and Vibration, vol. 67, pp. 369381, 1979. +RGJHV'+DQG5XWNRZVNL0-, Free-vibration analysis of rotating beams by a variable-order finiteelement method, AIAA Journal, vol. 19, pp. 1459-1466, 1981. +RGJHV ' +, An approximate formula for the fundamental frequency of a uniform rotating beam clamped off the axis of rotation, Journal of Sound and Vibration, vol. 77, pp. 11-18, 1981.
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[7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
.RMLPD+, Transient vibrations of a beam / mass system fixed to a rotating body, Journal of sound and vibration, vol. 107, pp. 149-154, 1986.
BookID 214574_ChapID 23_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
A Comparison of Gear Mesh Stiffness Modeling Strategies
Jim Meagher1, Xi Wu1, Dewen Kong2 , and Chun Hung Lee1 1-Mechanical Engineering Department, California Polytechnic State University, San Luis Obispo, CA93407, USA 2-College of Mechanical Science and Engineering, Jilin University, Changchun 130025, China
[email protected] Nomenclature: A E F G I C V
Cross Sectional Area Elastic Modulus Normal Contact Force Shear Modulus Area Moment of Inertia Compliance Shear Force
W r L Ng LCR,HCR
Face Width Gear Body Radius Poisson’s Ratio Tooth Length Number of teeth in gear Contact Ratio Low, High Contact Ratio
Abstract㧦 㧦 Gearboxes are prone to numerous faults that require vibration health monitoring to ensure proper operation. Monitoring methods typically employ casing mounted accelerometers. This data is difficult to interpret because of the many gear mesh frequencies present and is further complicated by the presence of significant amounts of noise. Several different dynamic system modeling strategies are currently being used by researchers to identify diagnostic indicators of gear health: a strength of materials based lumped parameter model, non-linear quasi-static finite element modeling, and rigid multi-body kinematic modeling with nonlinear contact stiffness. This study contrasts these methods of modeling gear dynamics by comparing their predicted stiffness cycle and its effect on dynamic response. A pair of ideal high contact ratio spur gears and a pair with a low contact ratio are considered. Data from experiments are shown for the high contact ratio pair. Results show that stiffness may be poorly matched either by exclusion of significant compliance, incomplete sensitivity and convergence studies, or by high frequency variations due to discretization. Keywords: gear stiffness, vibration health monitoring, finite element, multi-body kinematic model
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_23, © The Society for Experimental Mechanics, Inc. 2011
255
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256 Background: The approaches to modeling gear system dynamics are as varied as the goals of the study; to predict operational loads and life, to design system performance, or to study diagnostic indicators of gear defects. Three major approaches used are finite element modeling, lumped parameter modeling, and rigid body kinematics with nonlinear contact stiffness. This study is primarily concerned with the modeling of the nonlinear meshing stiffness for the purpose of condition monitoring using vibration signals. Two notable review papers that discuss the numerical modeling of gear dynamics are by Ozguven and Houser in 1988 [1] and by Parey and Tandon in 2003 [2]. Ozyguven categorized the models as dynamic factor models, models with tooth compliance, models for gear dynamics, those for rotor dynamics, and those for torsional vibration. The listed goals for the studies included reliability, life, stress, loading, noise, and vibratory motion. Curiously, condition monitoring was not included. Early work modeled the meshing stiffness as either an average or piecewise linear variation. Parey and Tandon’s review concentrated mostly on the modeling of defects but includes an extensive compilation of various lumped parameter models that have been used. Early work to model the meshing stiffness was based upon a mechanics of materials approach. Lin [3] has published extensive details regarding the estimation of the time varying stiffness using beam approximations. The contribution of bending, axial loading, shear deformation, and local contact deformation of the teeth are considered and delineated for points along the line of action of the teeth. Finite element approaches use a quasi-static method, whereby, after the stiffness is calculated in one position, the solid bodies are rotated to a new position and the FEA is performed again. Variations of the FEA method include partial gear body models [4], single teeth gear models [5], and whole tooth bodies [6]. Recent models include variations of the tooth profile [7]. Kuang and Yang [8] developed a closed form curve fit equation for stiffness based on modeling low contact ratio gears that included addendum modification. Multi-body dynamics software can also be used to model gear mesh stiffness using a rigid-elastic model [9]. Hertzian contact at the gear interface is used to represent gear elasticity as a compromise over fully elastic models; thereby reducing computational effort over FEA approaches. Lumped Parameter Model: A lumped parameter model for the gear pairs defined in table 1 below was created based upon the work of Lin [3] with modifications for torsion of the gear body and Hertzian contact. The compliance calculated is defined as the deflection along the line of action for a unit load of contact force. For the gear body compliance in torsion, the effective stiffness in the first torsional mode of vibration is used [10]. For the Hertzian contact stiffness, a linearization from Yang and Sun [11] is used. The compliance of a single tooth due to bending, axial loading, and shear, can be found using Castigliano’s Method [12]. The stiffness of a single tooth is the reciprocal of the sum of the compliances. The resulting tooth compliance is evaluated numerically from:
(1) The final two terms are from torsion of the gear body and Hertzian contact. For a tooth pair, the stiffnesses add in series. For multiple pairs in engagement, the stiffness adds in parallel.
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257 Number of Teeth
Pressure Angle
Face Width mm.
Diameter of Pitch Circle mm.
Diameter of Addendum Circle mm.
Diameter of Dedendum Circle mm.
Contact Ratio
High Contact Ratio - Two Stage Gearbox
Pinion Gear Pinion Gear
24 60 36 48
14.5⁰ 14.5⁰
19.0 19.0
50.8 127 76.2 101.6
45.5 121.7 80.4 105.8
55.0 131.2
Stage 1
2.023
96.3
Stage 2
63.5 83.8
57.0 77.4
1.626
2.065
Low Contact Ratio Pair
Pinion Gear
23 31
20⁰
25.4
58.4 78.7
Table 1: Gear Pair Parameters
Figure 1 represents all points along the line of contact for the LCR pair of gears with the pinion driving. The total gear roll angle corresponding to a gear tooth traversing the load zone equals the contact ratio, , times the angle a single gear tooth rotates before reaching the same relative position with respect to a pinion tooth:
Figure 1: Low Contact Ratio Gear Mesh Stiffness for Contact throughout the Line of Action.
. Along
the load zone, the sequence of number of tooth pairs in mesh is 2, 1, 2. This can be seen in figure 1 as higher relative stiffness over the first and last portions where multiple pairs of teeth engage. As the pair of gears rotates, the mesh stiffness repeats the pattern; two pairs engaged, one pair engaged, or the first
degrees of the figure.
Also shown is the contribution of a single gear and pinion tooth along the line of action. As the gear tooth enters the load zone it is loaded at the addendum so that its stiffness is low compared to being loaded near the root as occurs as it exits the load zone. The pinion demonstrates the reverse effect, stiff then compliant, since it is loaded near the root at the start of the load zone and exits the load zone with the contact force at the addendum. The individual contributions of bending, shear, gear body torsion, Hertzian contact, and axial compliance are shown in figures 2 and 3. These components act in series for an individual tooth. The numerical beam approximation shows that convergence of results occurs within 10 beam segments. These plots are for contact as the pinion tooth driver enters the load zone. Since the pinion is loaded near its root the torsional gear body compliance is the most dominant effect. The Hertzian Contact and tooth shear are the next most significant for the pinion tooth at this point. For the gear tooth at the corresponding position, the bending compliance dominates because the tooth is loaded at the tooth tip. This illustrates one advantage to lumped parameter modeling; the relative contribution to the stiffness for each component is available.
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Figure 2: Results of numerical integration
Figure 3: Results of numerical integration
approximation for tooth stiffness at the beginning of the load zone for the gear tooth.
approximation for tooth stiffness at the beginning of the load zone for the pinion tooth.
Finite Element Model: Finite element modeling has increasingly been used as part of a dynamic analysis of meshing gears. The meshing stiffness is typically obtained by loading one gear and restraining the other to calculate the stiffness for a given position and operating load. The gear bodies must be repeatedly rotated and the calculations repeated. Figure 4 shows a gear pair with two different mesh densities. The loading of two pairs of teeth is apparent for this position. The FEM is flexible, especially for non-standard tooth geometries, but can be time intensive for the user to define the geometry, and for the computer to solve the loading. Besides convergence studies to determine appropriate mesh densities, many other Figure 4: Finite Element Gear Models with factors must be considered to get accurate results. Different Mesh Densities. Plane stress, plane strain, and three dimensional modeling were performed using ABAQUS™. There were small differences in the results but computational time for a full 3-D study was considerable. The results from linear or quadratic planar elements are displayed in figure 5. Each data point on this plot is a separate FEA. The mesh density was not sufficient for linear elements to give accurate results but quadratic elements worked well as shown by the expected pattern and magnitude of the meshing stiffness.
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259 For a nonlinear contact analysis a tolerance must be set which defines whether contact occurs or not. The results as shown in figure 6 show the sensitivity of the results to this parameter. Over a relatively small range the results change dramatically. The middle three curves show the correct pattern of stiffness variation between two and one pair of teeth supporting the load. Even with proper tolerance, enough elements of high enough order, the results may not be accurate in a very important characteristic, frequency. This is displayed in figure 7. The beam model matches the theoretical meshing period exactly, it repeats every time a new pair of teeth comes into mesh (360⁰/Nteeth). This is important when the meshing stiffness is used for dynamic analysis where the vibration frequency is a primary consideration. Also, the point to point variation of stiffness for the FEA shows additional frequency content that will obscure the results for ideal gears or gears with defects. Figure 5: Results for various element types.
Figure 6: Sensitivity of FEA Results to Contact Tolerance Parameter
Figure 7: Comparison of FEA results to Lumped Parameter Model
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260 Multi-body kinematic model:
A rigid-elastic model of a pair of meshing gears is shown in Figure 8. The gears are rigid and the supports have lumped elasticity to model bearing support compliance. The contact surfaces between the gears are modeled as deformable flex-bodies. The nonlinear contact force is composed of an elastic and damping portion [13]. The damping force, Cv, is proportional to impact velocity, v. The stiffness coefficient, K, is taken to be the average value of stiffness over one tooth mesh cycle as calculated from an FEA model. The force exponent, e, was determined from trial simulations. The damping coefficient generally takes a numeric value between 0.1 %-1% of K. The computational time of this type of model is much less than a finite element model. The determination of force Figure 8: Multi-body exponents however is not obvious and must be based on experience. No direct Kinematic Model comparison of stiffness can made such as in figure 7 so the predicted responses are compared. In figure 9 is shown the Gear support acceleration.
Figure 9: Theoretical Gear Vertical Acceleration
The gear mesh frequency and harmonics are clearly evident. These frequencies are modulated by a 40Hz signal of unknown origin. With slight modifications of support stiffness and damping they are not prominent. A dynamics simulation using a lumped parameter approach predicts only the gear mesh frequencies and the harmonics for an ideal set of gears.
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261 Experiment: A parallel shaft gearbox was instrumented with accelerometers on the bearing housings.
Figure 10: Two Stage Parallel Shaft Gearbox
Figure 11: Accelerometer Mounting
The gearbox has high contact ratio pairs of spur gears as described in table 1. Since the gear shafts are supported by rolling element bearings, vibration signals contain ball passing frequencies and gear meshing frequencies from both stages, figure 12. These compared well to theoretical values [14], table 2.
-6
6
Power Spectrum(FFT): input rpm=1020(17Hz) Freq. Range:1-700Hz
x 10
fIRBP
5
fm2
Power Spectrum (Volt 2)
4
1/6Xfm1
fm1
3
2Xfm2 fIRBP-2Xf3 1/3Xfm1
2
fORBP 1/4Xfm1
fIRBP+2Xf3 fm2+2Xf3 fm2-f2
fm2+4Xf3
5XfORBP 5XfORBP-f1
fm2-2Xf2
1 4XfIRBP
5XfORBP-2Xf2 f1 0 0
100
200
300
Figure 12: FFT with Hanning Window.
400 Frequency (Hz)
500
600
700
800
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262
Item Shaft 1 (Input Shaft) Rotating Frequency Shaft 2 (N2 and N3 Shaft) Rotating Frequency Shaft 3 (N4 Shaft) Rotating Frequency First Stage Meshing Frequency Second Stage Meshing Frequency 2 times of Second Stage Meshing Frequency 1/6 times of First Stage Meshing Frequency 1/4 times of First Stage Meshing Frequency 1/3 times of First Stage Meshing Frequency 1/2 times of First Stage Meshing Frequency 2/3 times of First Stage Meshing Frequency
f1 f2 f3 f_m1 f_m2 2×f_m2 1/6×f_m1 1/4×f_m1 1/3×f_m1 1/2×f_m1 2/3×f_m1
Calc Value (Hz) 17 6.8 5.1 408 244.8 489.6 68 102 136 204 272
1/2 times of Second Stage Meshing Frequency 1/3 times of First Stage Meshing Frequency
1/2×f_m2
122.4
122.3
1/3×f_m1
136
136.2
Outer Race Ball Pass Frequencies
f_ORBP 2× f_ORBP 3× f_ORBP 4× f_ORBP 5× f_ORBP f_IRBP 2× f_IRBP 4× f_IRBP
116.22 232.44 348.66 464.88 581.1 155.78 311.56 623.12
119.4 238 357.4 476.8 595.5 153.1 306.2 612.3
Symbol Shaft Frequencies Gear Meshing Frequencies
Bearing Elements Frequencies
Inner Race Ball Pass Frequencies
Exp. Value (Hz) 16.85 408.7 245.4 490 68.12 101.8 136.2 204.3 272.5
Table 2: Identified Frequencies in the accelerometer response spectrum (0-700 Hz).
Conclusions: Three distinct approaches to modeling gears for the purpose of dynamic system modeling are presented. These methods each result in a model that predicts expected gear meshing frequencies for an ideal set of involute spur gears but the effort of modeling, flexibility, and the sensitivity of input parameters that may not be well known vary. The lumped parameter method calculates gear stiffness for involute gears with only a few parameters to specify the gears and calculates the stiffness more quickly than the other methods. The method also reveals the relative contribution of different parts of the compliance. However, for other types of gears or with addendum modifications, additional effort would be required. FEA and Multi-body kinematic approaches are flexible but take considerable user time to generate a model for each specific set of gears. The FEA approach is sensitive to tolerances, mesh density, and element choice. The calculated stiffness may cause errors in the response spectrum due to discretization errors. A Multi-body kinematic approach has a contact stiffness that depends on parameters that are not well known.
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263 References: [1] Ozguven H.N. and Houser D.R., Mathematical Models used in Gear Dynamics – A Review, J. Sound and Vibration, 121(3), 1988, 383-411 [2] Parey A. and N. Tandon N., Spur Gear Dynamic Models Including Defects: A Review, Shock and Vibration Digest 2003; 35; 465 [3] Lin H and Liou C.H., A Parametric Study of Spur Gear Dynamics, NASA/CR-1988-206598 [4] Barone S. , Borgianni L. , Forte P., Evaluation of the Effect of Misalignment and Profile Modification in Face Gear Drive by a Finite Element Meshing Simulation, Proc. DETC’03, ASME Design Eng. Conf., Chicago, Sept. 2003 [5] Howard H., Jia S., Wang J., The Dynamic Modeling of a Spur Gear in Mesh Including Friction and a Crack, Mechanical Systems and Signal Processing (2001) 15(5), 831-853 [6] Wyluda P. and Wolf D., Examination of Finite Element Analysis and Experimental Results of QuasiStatically Loaded Acetal Copolymer Gears, ANTEC Conf., Society of Plastics Eng., San Francisco, 2002 [7] Wang J. and Howard I., Finite Element Analysis of High Contact Ratio Spur Gears in Mesh, J. Tribology, Vol. 127, July 2005 [8] Kuang J.H. and Yang Y.T., An Estimate of Mesh Stiffness and Load Sharing Ratio of a Spur Gear Pair, Intl. Power Transmission and Gearing Conf., DE-Vol.43-1, ASME 1992 [9] Ebrahimi S. and Eberhard P., Rigid-elastic modeling of meshing gear wheels in multi-body systems, Multi-body System Dynamics (2006) 16:55–71 [10] Kang M.K., Huang R., Knowles T., Torsional Vibration of Circular Elastic Plates with Thickness Steps, IEEE Trans., Vol. 53, No. 2, Feb. 2006 [11] Yang D.C.H. and Sun Z.S., A Rotary Model for Spur Gear Dynamics, J. Mechanisms, Transmissions and Automation in Design, Vol. 107, Dec. 1985 [12] Juvinall R. and Marshek K., Fundamentals of Machine Component Design, 2 Sons,1991
nd
ed.,John Wiley and
[13] MSC Inc., MSC ADAMS reference manual [14] Taylor J. and Kirkland D., The Bearing Analysis Handbook, Vibration Consultants Inc., 2004
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BookID 214574_ChapID 24_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
8VLQJ2SHUDWLQJ'HIOHFWLRQ6KDSHVWR'HWHFW )DXOWVLQ5RWDWLQJ(TXLSPHQW Surendra N. Ganeriwala (Suri) Vetrivel Kanakasabai (Vetri) Arul Muthukumarasamy Spectra Quest, Inc 8201 Hermitage Road Richmond, VA 23228
Tom Wolff Mark Richardson Vibrant Technology, Inc 5 Erba Lane, Suite B Scotts Valley, CA 95066
$%675$&7 This paper is the third in a series where an operating deflection shape (ODS) is used as the means of detecting faults in rotating machinery [1] [2]. In this paper, ODS comparison is used as a means of detecting unbalance and misalignment in a rotating machine. Our purpose is to use significant changes in the ODS as an early warning indicator of rotating machine faults.
:KDWLVDQ2'6" An ODS is defined as the GHIOHFWLRQRIWZRRUPRUHSRLQWV RQ DP DFKLQH RUVW UXFWXUH. Stated differently, an ODS is the deflection of one point relative to all others. Deflection is a vector quantity, meaning that each of its components has both location and direction associated with it. Deflection measured at a point in a specific direction is called a DOF (Degree of Freedom).
For the examples presented in this paper, tests were performed on a machinery fault simulator on which various faults were introduced. Vibration data was simultaneously acquired from accelerometers and proximity probes on the simulator using a multi-channel spectrum analyzer system. ODS data was then extracted from the frequency spectra of the acceleration and displacement responses, and two different numerical measures were used to quantify changes in the ODS of the machine.
An ODS can be defined from any vibration data, either at a moment in time, or at a specific frequency. Different types of time domain data, e.g. random, impulsive, or sinusoidal, or different frequency domain functions, e.g. Linear spectra (FFTs), Auto & Cross spectra, FRFs (Frequency Response Functions), Transmissibility’s, ODS FRFs [4], can be used to define an ODS.
These changes can be used in an automated warning level (alert, alarm, and abort) detection scheme to give early warnings of machine faults. The results of this work provide a new simplified approach for implementation in an on-line machinery health continuous monitoring system. ,1752'8&7,21 Unscheduled maintenance of rotating equipment in a process plant can account for a significant percentage of the plant’s downtime. Not only is equipment downtime expensive because of lost production revenue, but most machine faults will result in increased costs of replacement parts, inventory, and energy consumption. Traditionally, vibration signatures (level profiling of singlepoint vibration spectra), and time domain based orbit plots have been the preferred tools for detecting and diagnosing machine faults. Although these tools may be effective when used by an expert, ODS analysis offers a simpler, more straightforward approach for fault detection. Many machine faults are more easily characterized by a visual as well as a numerical comparison of a machine’s ODS when compared with its Baseline ODS.
0HDVXULQJDQ2'6 In general, an ODS is defined with a magnitude & phase value for each DOF that is measured on a machine or structure. This requires that either all responses are measured simultaneously, or that they are measured under conditions which guarantee their correct magnitudes & phases relative to one another. Simultaneous measurement requires a multi-channel acquisition system that can simultaneously acquire all responses. Sequential acquisition requires that cross-channel measurements be calculated between a (fixed) reference response and all other roving responses. This ensures that each DOF of the resulting ODS has the correct magnitude & phase relative to all other DOFs. %DVHOLQHYHUVXV)DXOW2'6 The hypothesis of this paper is the following; 0DFKLQH )DXOW +\SRWKHVLV When an operating machine encounters a mechanical fault, its ODS will change. Many faults will cause a change in the vibration levels in many parts of a rotating machine. Therefore, an important question to ask is; “What constitutes a significant change in vibration level?” This will be answered by calculating a change in the ODS of the machine. In order to measure
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_24, © The Society for Experimental Mechanics, Inc. 2011
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Proximity probes (vert & horz) Tri-axial accelerometer
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a change in the ODS, the ODS acquired when the machine is operating properly (called the Baseline ODS), is compared with its ODS during ongoing operation. After a fault begins to occur (or has occurred), the ODS is referred to as the Fault ODS. %DVHOLQH2'6 The ODS of a machine when it is operating properly. )DXOW2'6 The ODS of a machine when a fault begins to occur or has occurred. 'DWD$FTXLVLWLRQ To verify our hypothesis, tests were performed using the machinery fault simulator shown in Figure 1. Accelerometers and proximity probes were used to measure the simulator’s vibration. The sensor locations are shown in Figure 2. The accelerometers provided 7 vibration signals and the proximity probes provided 6 vibration signals. These 13 channels of vibration data were simultaneously acquired with a multi-channel data acquisition system. Time domain records were acquired, using 77824 samples at a sample rate of 5000Hz, for a total of 15.565 seconds of data on each channel. This data was post-processed and cross channel frequency domain measurements were calculated between all channels and a single reference channel. 2'6)5)V To calculate ODS’s, first a set of ODSFRFs was calculated between each of the channels of data and a single reference channel. An ODSFRF is a “hybrid” cross-channel measurement, involving both an Auto spectrum and a Cross spectrum. It is formed by combining the phase of the Cross spectrum between a roving and the reference signal with the magnitude of the Auto spectrum of the roving response signal. The magnitude of an ODSFRF, provided by the Auto spectrum, is a true measure of the structural response. Data was acquired at various nominal operating speeds of 800, 1000, 2000 & 3000 RPM) under a variety of fault conditions. A typical ODSFRF is shown in Figure 3. It is clear that the ODSFRF is dominated by peaks at the machine running speed (first order) and its higher orders (multiples of the running speed).
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%DVHOLQH7HVWV With no faults, Baseline data was acquired from the simulator at operating speeds of 800, 1000, 2000 & 3000 RPM. 0DFKLQH)DXOW&RQGLWLRQV The following faults were then introduced on the rotating machine simulator: 8QEDODQFH)DXOWV )DXOW One 4 gram unbalance screw on inboard rotor )DXOW Two 4 gram unbalance screws, one on each rotor $QJXODU0LVDOLJQPHQW)DXOWV )DXOW 10 mils angular misalignment on the outboard bearing )DXOW 20 mils angular misalignment on the outboard bearing 3DUDOOHO0LVDOLJQPHQW)DXOWV )DXOW: 10 mils parallel misalignment on both bearings )DXOW: 20 mils parallel misalignment on both bearings
ODS’s were obtained by saving the peak cursor values at the running speed, or one of its orders, as shown in Figure 3. Each peak cursor value is a DOF of the ODS. 7KH 2'6 LV DFRO OHFWLRQ RI SHDN YDOXHV IURP D VHW RI 2'6)5)VDWRQHRIWKHRUGHUVRIWKHPDFKLQH
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ODSF ODSB ODSB
1XPHULFDO&RPSDULVRQ2I2'6¶V Two different numerical methods were used to compare ODS’s from before (Baseline ODS) and after a machine fault was introduced (Fault ODS). One method is called the SCC (Shape Correlation Coefficient), and the other is the SPD (Shape Percent Difference). Both of these calculations yield a percentage value. The SCC measures the co-linearity of the two ODS’s, and the SPD is the percent difference between the Baseline and the Fault ODS.
where: ODS B
6&& 6KDSH&RUUHODWLRQ&RHIILFLHQW An ODS is a complex vector with two or more components, each component having a magnitude & phase. Each component of the ODS is obtained from a vibration signal measured at a single DOF on the machine.
The SPD measures the percentage change relative to the Baseline ODS. A value of 0 indicates no change in the ODS, and a value of 1 means a 100% change in the ODS.
The SCC is a calculation which measures the similarity between two complex vectors. When this coefficient is used to compare two mode shapes, it is called a MAC (Modal Assurance Criterion) [3]. The SCC is defined as;
SCC
ODSF $ ODS*B ODSF ODSB
where: ODS B
ODSF
ODS*B
Baseline ODS Fault ODS complex conjugate of ODSB
indicates the magnitude squared $ indicates the DOT product between two vectors
The SCC is a normalized DOT product between two complex ODS vectors. It has values between 0 and 1. A value of 1 indicates that the ODS has not changed. As a “rule of thumb”, an SCC value greater than 0.90 indicates a small change in the ODS. A value less than 0.90 indicates a substantial change in the ODS. The SCC provides a single numerical measure of a change in the ODS of an operating machine. The ODS can have as many DOFs as are necessary for detecting machine faults. Many DOFs may be required in order to detect certain kinds of faults. The location and direction of the sensors is subjective and will vary from machine to machine.
SPD
ODSF
Baseline ODS Fault ODS
indicates the magnitude of the vector If ODSF ODSB then the SPD is negative
To summarize, when a machine is operating properly, the SCC will be at or near 1, and the SPD will be at or near 0. As a fault condition begins to occur, the SCC will decrease toward 0, and the SPD will increase or decrease depending on the change in machine vibration levels. &RPSDULQJ%DVHOLQH2'6¶V Even under a no fault condition, the ODS of a machine can change with operating speed, so it is important to compare ODS’s that were acquired at the same operating speed in order to detect faults. First, Baseline ODS’s from four different operating speeds will be compared to quantify their differences. Baseline ODSs were compared in the following cases; Case 1: 800 versus 1000 RPM Case 2: 800 versus 2000 RPM Case 3: 800 versus 3000 RPM Case 4: 1000 versus 2000 RPM Case 5: 1000 versus 3000 RPM Case 6: 2000 versus 3000 RPM Both the SCC and SPD values for these 6 cases are shown in Figure 5. For Case 1, the SCC value (0.95) indicates that the Baseline ODS at 800 RPM is essentially the same as at 1000 RPM. However, the SPD indicates that the vibration level has grown by 58% from 800 to 1000 RPM.
One difficulty with the SCC is that it only measures a difference in the “shape” of two vectors. In other words, two vectors can be co-linear, meaning that they lie along the same line, but they can still have different magnitudes. If the vibration levels increase in a machine but the “shape” of the ODS does not change, the SCC will still have a value of 1, indicating no change. 63' 6KDSH3HUFHQW'LIIHUHQFH A different measure of change in an ODS is the SPD (Shape Percent Difference). The SPD measures both a change in level and in shape. )LJXUH%DVHOLQH2'6&RPSDULVRQV
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Cases 2, 3, and 4 indicate the greatest changes in the ODS between two different operating speeds. The first critical speed of the machine shaft is 530, so these large changes in the ODS are expected. In Case 6, the SCC value (0.90) indicates that the ODS’s are co-linear (similar in shape), but the SPD value (-1.47) indicates that the ODS has decreased in magnitude from 2000 to 3000 RPM. This decrease in vibration level occurred because at 3000 RPM, the machine is operating beyond its critical operating speed. 8QEDODQFH&DVHV Two unbalance faults were simulated in the rotating machine simulator by adding unbalance screws to the (gold) rotors on the shaft. Fault 1 was created by addingan unbalance weight of 4 grams to the inboard rotor. Fault 2 was created by addingan unbalance weight of 4 grams to each of the rotors. ODS data was then obtained for the following 8 cases. Case 1: Fault 1 (One 4 gram unbalance) at 800 RPM Case 2: Fault 1 at 1000 RPM Case 3: Fault 1 at 2000 RPM Case 4: Fault 1 at 3000 RPM
$QJXODU0LVDOLJQPHQW&DVHV Two angular misalignment faults were simulated in the rotating machine simulator by turning screws at the base of each shaft bearing block to move it out of alignment with the motor shaft. Fault 3 was created by addingPLOV of misalignment to the outboard bearing block. This is depicted in Figure 7. Fault 4 was created by addingPLOV of misalignment to the outboard bearing block. ODS data was then obtained for the following 8 cases. Case 1: Fault 3 (10 mil angular misalign) at 800 RPM Case 2: Fault 3 at 1000 RPM Case 3: Fault 3 at 2000 RPM Case 4: Fault 3 at 3000 RPM Case 5: Fault 4 (20 mil angular misalign) at 800 RPM Case 6: Fault 4 at 1000 RPM Case 7: Fault 4 at 2000 RPM Case 8: Fault 4 at 3000 RPM The SCC and SPD values for the 8 angular misalignment cases are plotted in Figure 7. In each case the Baseline ODS for a different machine speed is compared with the Fault ODS at the same speed.
Case 5: Fault 2 (Two 4 gram unbalances) at 800 RPM Case 6: Fault 2 at 1000 RPM Case 7: Fault 2 at 2000 RPM Case 8: Fault 2 at 3000 RPM The SCC and SPD values for these 8 cases are plotted in Figure 6. In each case the Baseline ODS for a different machine speed is compared with the Fault ODS at the same speed.
)LJXUH$QJXODU0LVDOLJQPHQW2'6&RPSDULVRQV
)LJXUH8QEDODQFH2'6&RPSDULVRQV The values in Figure 6 clearly indicate that Fault 1 is less severe than Fault 2. The SCC is on the borderline of no change in the OD for Cases 1 to 4, but indicates a significant change for Cases 5 to 8. The SPD indicates a significant change in all cases, with the smallest change being 62% difference in the ODS for Case 2.
The values in Figure 7 give a mixed result. The SCC is almost 1.0 for all cases, meaning that the Baseline ODS and Fault ODS are co-linear at all speeds. However, the SPD indicates a change of the ODS in all cases, and the percent difference between the Baseline ODS and Fault ODS increases with machine speed for all Cases, except the last one. In Case 8, the difference becomes negative with less magnitude than Case 7, but still indicates a 15% change in the ODS. 3DUDOOHO0LVDOLJQPHQW)DXOWV Two parallel misalignment faults were simulated in the rotating machine simulator by turning screws at the base of each shaft bearing block to move it out of alignment with the motor shaft. Fault 5 was created by addingPLOV of misalignment to both bearing blocks. This is depicted in
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Figure 8. Fault 6 was created by adding PLOV of misalignment to both bearing blocks. ODS data was then obtained for the following 8 cases. Case 1: Fault 5 (10 mil parallel misalign) at 800 RPM Case 2: Fault 5 at 1000 RPM Case 3: Fault 5 at 2000 RPM Case 4: Fault 5 at 3000 RPM Case 5: Fault 6 (20 mil parallel misalign) at 800 RPM Case 6: Fault 6 at 1000 RPM Case 7: Fault 6 at 2000 RPM Case 8: Fault 6 at 3000 RPM The SCC and SPD values for the 8 parallel misalignment cases are plotted in Figure 8. In each case the Baseline ODS for a different machine speed is compared with the Fault ODS at the same speed.
(running speed) of the machine from each set of ODSFRF functions. Three different types of common rotating machinery faults were simulated; unbalance, angular misalignment, and parallel misalignment. Two different unbalance faults and four misalignment faults were simulated. Operating data was acquired at 4 different operating speeds, 800, 1000, 2000 & 3000 RPM. The first critical speed of the machine was at 1590 RPM. By comparing Baseline (no fault) ODS’s for the different operating speeds, it was clear that the ODS’s for speeds below the critical speed were quite different from the ODS’s above the operating speed. This is expected whenever a resonance is excited. In each test case, the Baseline ODS was numerically compared with the ODS acquired after a fault was simulated (the Fault ODS). A total of 24 different fault cases were evaluated. Two different numerical measures of the difference between the Baseline and Fault ODS were calculated; the SCC (Shape Correlation Coefficient) and SPD (Shape Percent Difference). The SCC indicates whether or not the two shapes are co-linear. The SPD measures the difference between the two ODS’s, hence it also measures the severity of the machine fault. In all cases, the SPD indicated a “significant” change in the ODS when faults were introduced. This confirmed our original hypothesis; “When an operating machine encounters a mechanical fault, its ODS will change”.
)LJXUH3DUDOOHO0LVDOLJQPHQW2'6&RPSDULVRQV The values in Figure 8 also give a mixed result. The SCC is close to 1.0 for all cases, meaning that the Baseline ODS and Fault ODS are co-linear at all speeds. But, the SPD indicates a change of the ODS in all cases, and the percent difference between the Baseline ODS and Fault ODS increases with increased machine speed in all cases. Cases 4 and 8 indicate large changes in the ODS. &21&/86,216 Operating Deflection Shapes were used in this paper to detect faults in a rotating machine fault simulator. In two previous papers [1] [2], only accelerometers were used to sense vibration signals. In this paper, proximity probes, which measured displacement of the shaft relative to its bearing housings in two directions, were also used. Seven DOFs of vibration data were acquired from the motor & bearings using accelerometers, and 6 DOFs were acquired from the shaft bearings using proximity probes. These 13 channels of acquired data were post-processed, and a set of ODSFRFs was calculated for each fault condition. ODS’s were then created by using the peak values at the first order
The high values of the SPD shown in most cases indicated that it has a strong sensitivity to changes in the ODS. Therefore, the SPD could be used to detect lesser changes in machine responses, providing early detection of impending fault conditions. This is the third in a series of technical papers investigating the use of ODS comparisons as a means of detecting machine faults. In these papers, only unbalance and misalignment faults have been simulated. Other machine faults such as bearing oil whirl, loose connections, gear tooth faults, soft foot (improper foundation) might also be detected by ODS comparisons. In addition to vibration, an “operating shape” could also contain other monitored data such as temperatures, pressures, voltages, currents, and flow rates. These parameters could also be correlated with certain kinds of machine faults. The simplicity of this approach to machinery fault detection makes it a strong candidate for implementation in an online fault detection system. The machinery fault simulator used to obtain these results is a product of Spectra Quest, Inc. The ODS analysis software is part of a MechaniCom Machine Surveillance System™, a product of Vibrant Technology, Inc.
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5()(5(1&(6 1. Surendra N. Ganeriwala (Suri), Zhuang Li and Richardson, Mark “Using Operating Deflection Shapes to Detect Shaft Misalignment in Rotating Equipment” Proceedings of International Modal Analysis Conference (IMAC XXVI), February, 2008. Surendra N. Ganeriwala (Suri, Schwarz, Brian and Richardson, Mark “Using Operating Deflection Shapes to Detect Unbalance in Rotating Equipment” Proceedings of International Modal Analysis Conference (IMAC XXVII), February, 2009. 3. R.J. Allemang, D.L. Brown "A Correlation Coefficient for Modal Vector Analysis", Proceedings of the International Modal Analysis Conference pp.110-116, 1982. 4. M.H. Richardson, “Is It a Mode Shape or an Operating Deflection Shape?” Sound and Vibration magazine, March, 1997. 5. B. Schwarz, M.H. Richardson, “Measurements Required for Displaying Operating Deflection Shapes” Proceedings of IMAC XXII, January 26, 2004.
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BookID 214574_ChapID 25_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII
February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc. 'DPSLQJ0HWKRGRORJ\IRU&RQGHQVHG6ROLG5RFNHW0RWRU6WUXFWXUDO 0RGHOV
S. Fransen, H. Fischer, S. Kiryenko, D. Levesque, T. Henriksen European Space Agency, ESTEC, P.O. Box 299, 2200 AG Noordwijk, The Netherlands, EU
$%675$&7 ESA’s new small launcher – VEGA – has been designed as a single body launcher with three solid rocket motor stages and an additional liquid propulsion upper module used for attitude and orbit control, and satellite release. Part of the mission analysis is the so-called launcher-satellite coupled loads analysis which aims at computing the dynamic environment of the satellite for the most severe load cases in flight. To allow such analyses to be processed in short time, all stages of the launcher finite element model are condensed. The condensed launcher mathematical model can subsequently be coupled to a condensed satellite mathematical model. To obtain accurate predictions of the satellite dynamic environment it is evident that the damping of the entire system has to be defined in a representative way. This paper explains a methodology to compute the modal damping matrix of a superelement on the basis of the structural damping ratios assigned to the various materials in the associated finite element model and the associated complex strain energy of the modeshapes. The methodology turns out to be well suited for the computation of the modal damping matrix of condensed solid rocket motor structural models, as evidenced by correlation with firing tests conducted for the first stage motor of the VEGA launcher. ,QWURGXFWLRQ ESA’s new small launcher – Vega – has been designed as a single body launcher with three solid rocket motor stages and an additional liquid propulsion upper module used for attitude and orbit control, and satellite release. The main launcher components are depicted in figure 1. The three solid rocket motor stages are the P80 (first stage, 88 tonnes of propellant), the Z23 (second stage, 24 tonnes of propellant) and the Z9 (third stage, 10 tonnes of propellant). The lift-off mass of the launcher is about 137 tonnes.
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An important step in the design and verification process of spacecraft structures is the coupled dynamic analysis with the launch vehicle in the low frequency domain and is also referred to as coupled loads analysis (CLA) [1,2]. The low-frequency domain, typically from 0 to 100 Hz, corresponds to the frequency content of forcing functions used in the CLA. The objective of CLA is the computation of the mechanical environment of the spacecraft (payload) induced by the launcher dynamic loads at particular instants of time during flight. The excitation may be of aerodynamic origin (wind, gust, buffeting at transonic or maximum dynamic speed) or may be induced by the propulsion system (thrust build-up or tail-off transients, acoustic pressure oscillations in the combustion chamber of solid rocket motor, or acoustic loads impinging on the exterior surface of the launcher at lift-off). It is evident that the damping characteristics of the launcher-payload assembly are of utmost importance in order to predict the payload dynamic environment with sufficient confidence. The CLA process as embedded into the design cycle of the spacecraft structure is depicted in figure 2.
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)LJXUH &RXSOHG/RDGV$QDO\VLV/RRS Usually the damping is defined separately for each condensed model. In this way the damping of the launcherpayload system depends on the damping properties of its substructures, provided that the dynamic response analysis is performed by solving the equations of motions without modal decomposition. It is evident that in modal dynamic response analyses, a modal damping profile applicable to the entire system has to be defined based on good engineering judgement. Obviously this is time consuming and could potentially lead to a loss of accuracy of the flight limit loads as the measured or identified modal damping of the launch vehicle substructures is not used at all. In addition it is very unlikely that the modal damping characteristics for the entire launcher-payload system are known from operational modal analysis performed on the basis of flight data. 6XEVWUXFWXUH0RGDO'DPSLQJ One way to define the substructure modal damping matrix is by the identification of the natural frequencies and associated modal damping ratios from sine vibration tests (spacecraft) or operational modal analysis (solid rocket motors). However, this information is seldom available during the design phase of a spacecraft or launcher. Consequently, the damping factor is often estimated as a constant within certain frequency bands, or is taken as a progressively increasing factor towards higher frequencies, as depicted in figure 3. The latter will avoid strong oscillations from unrealistic high frequency modes. If test data is missing – as in the early phases of spacecraft and launcher design projects – one could construct a damping profile from statistical data. However, such approach could easily lead to either under or overestimation of the loads on the satellite structure. This can be
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seen from the research conducted by Carrington and Ottens [3]. They tested several spacecraft and found that the modal damping of spacecraft varied between 3% and 7%. For spacecraft structures one usually chooses a low damping ratio, i.e 1% or 2%, to be conservative on the predicted loads. For launcher solid rocket motors such approach with a conservative and flat modal damping profile cannot be used. The reason for this is that the damping depends even stronger on the mode shape. The high damping ratio of the solid propellant will increase the modal damping for those axial modes that exhibit a high deformation of the solid propellant.
)LJXUH 0RGDOGDPSLQJSURILOHFRQVWUXFWHGIURPH[SHULPHQWDOGDWD In order to estimate the modal damping factors for solid rocket motors in the early stage of the design process, one could derive them from structural damping values assigned to the various materials used in the finite element model of the SRM. Suppose the structural damping has been defined for all elements of a substructure FE model, either globally or element wise or both. Then the structural damping matrix K S can be computed as follows: E
J g K ¦J e Ke
KS
(1)
e 1
For the equation of motion of a system excited by harmonic forces we can then write:
:
2
M i K S K xˆ
Fˆ
(2)
By computation of the modal subspace \ composed of N mass-normalized eigenmodes, we can define the coordinate transformation x \ q which leads to the following equation:
: or,
2
\ T M \ i \ T K S \ \ T K \ qˆ \ T Fˆ
(3)
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:
2
fˆ
I i DS / qˆ
(4)
In eq.(4) I is the identity matrix and DS is the fully populated structural damping matrix of the modal model. Note that i DS could be regarded as a complex modal strain energy matrix associated to the complex stiffness i K S . The diagonal matrix / is the modal stiffness matrix defined by:
diag (Z 2 )
/
(5)
Solving the modal dynamic response qˆ from eq.(4), we find:
qˆ
:
where
=
2
I i DS /
1
fˆ
= 1 fˆ
(6)
= is the complex impedance matrix defined by:
:
2
I i DS /
(7)
)LJXUH ([DPSOHRIHTXLYDOHQWPRGDOGDPSLQJFRPSXWHGIRUDVROLGURFNHWPRWRU
Q can now be computed as the ratio of the dynamic response over the static response at each resonance frequency : Z of the modal model, i.e.: The quality factor or amplification factor
Q :
Z
= 1 :
Z fˆ
/ fˆ 1
= 1 : 1
/
Z
|
4 = 1 : 1/ Z
Z 2
(8)
Note that the function 4 takes the diagonal term at : Z . Hence we find for each natural frequency an equivalent viscous damping (or equivalent modal damping) as follows:
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9 : Z
1 2 Q :
Z
1/ Z2 2 4 = 1 :
Z
(9)
In a similar fashion one can work out a relationship for equivalent modal damping for all modal co-ordinates of a condensed model defined by Craig-Bampton formulation. As an example for this approach, the equivalent modal damping of a solid rocket motor is plotted in figure 4. Damping for System of Condensed Models The damping matrix of a system of reduced models is now simply assembled from the substructure damping matrices conform the assembly of the mass and stiffness matrix of the condensed system. For expanded matrices one can write: C
¦M c 1
c
C
C
c 1
c 1
q ¦ Bc q ¦ K c q
M sys q Bsys q K sys q
C
¦f
(10)
c
c 1
0RWRU)LULQJ7HVW&RUUHODWLRQ In order to derive a correlated motor FE-model, correlation with accelerometer data from firing tests was performed. At first instance the correlation of the acceleration equivalent sine spectra was exercised and by means of sensitivity analyses the most dominant parameters were identified and selected. It is evident that for this correlation activity the firing test shall be simulated and hence the forcing functions are needed as an input. The forcing functions – basically the pressure fluctuations in the combustion chamber - were derived from computational fluid dynamics (CFD) computations that simulated the internal flow in the combustion chamber of the motor. The firing test configuration is sketched in figure 5. Forcing functions were assumed to be correct and for that reason were not considered as parameters in the correlation activity, hence only the FE-model was subject to the correlation activity.
0RWRU)(PRGHO )RUFLQJ IXQFWLRQV
$GDSWHUV
7HVW VWDQG
)LJXUH )LULQJWHVWVLPXODWLRQPRGHO One of the dominant correlation parameters was the structural damping ratio of the solid propellant that is used to compute the equivalent modal damping of the motor modes. These modes constitute the condensed model of the motor FE-model. In figure 6 one can see how well the equivalent modal damping model can be used to tune the amplification at the dominant excitation frequency. At most accelerometer stations the sine spectra found by simulation did match reasonably well with the test spectra. The structural damping ratio of the solid propellant that provided the best match was in line with the damping ratio found by dynamic tests on propellant samples. As can be observed from figure 6, the modal density of the model should be increased to find better agreement at higher frequencies.
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'HFUHDVLQJVWUXFWXUDO GDPSLQJUDWLRRIVROLG SURSHOODQW
)LJXUH 9DULDWLRQRIVWUXFWXUDOGDPSLQJUDWLRRIVROLGSURSHOODQW /LIWRII$QDO\VLV Having chosen – amongst other parameters – the correct structural damping ratio for the solid propellant by means of correlation, the correlated motor FE-models were used in a VEGA CLA. Here we will only describe one load case – lift-off – as the information described in the previous section is obviously directly linked to this load case. At ignition of the solid rocket motors the internal pressure builds up rapidly, which leads to axial vibrations transmitted to the spacecraft interface. This transient is a function of the size and ignition characteristics of the solid motor, but typically excites the main axial modes of the vehicle up to 100 Hz. Besides the main axial ignition loads, see figure 7a, also the less dominant lateral blast wave loads are taken into account, see figure 7b.
D
E )XG
)PU
)OG )Q]
. )LJXUH /RDGVLQOLIWRIIVLPXODWLRQ
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As highlighted in figure 1, one of the primary outputs of the CLA are the sine spectra at the spacecraft interface, which should provide levels below the sine test spectra defined in the user manual of the launcher. In addition one can verify, whether the primary notches computed to not exceed the max COG loads of the spacecraft, as defined in the user manual, are feasible. In figure 8, a graph is shown with the sine spectrum of lift-off (as well as spectra for other load cases and the envelope) which shows that the simulated payload interface loads are lower than the dynamic design loads across the entire frequency spectrum of interest. The primary notches are well above the envelope which allows the use of a notched spectrum during sine vibration testing of the spacecraft.
)LJXUH &RPSXWHGVLQHVSHFWUDDWVSDFHFUDIWLQWHUIDFH±ORQJLWXGLQDOGLUHFWLRQ 0RGHVKDSHVDQG0RGDO'DPSLQJ5DWLRV When adopting (condensed) 3D-models for the launcher stack, it is usually difficult to detect the system modes that drive the payload response, especially for loadcases that excite the axial modes. In order to find those modes, as well as their damping rates, it is useful to decompose the response – computed from an output transformation matrix [4] and the generalized response – into its modal participation factors or modal gains:
Vi
P
¦ ) ip q p 1
P
¦V
p
(11)
1
where the modally converted output transformation matrix ) ip of the spacecraft is given by:
) ip
) iq M sys
(12)
and where the modally converted generalized response is given by:
qp
T M sys M sys q
(13)
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Note that in eq.(13) the generalized response q is computed by direct time integration and that the modal conversion requires a normal modes analysis for the whole system. By filtering out the main modal contributions, one can immediately find the corresponding modeshapes and obtain a better physical insight into the coupling between launcher and spacecraft at resonance peaks. Besides the frequencies of the main modes, one can also find the corresponding modal damping ratios from the following system damping approximation:
9 sys
T diag (M sys Bsys M sys )
Note that
(12)
Bsys has been defined previously in eq.(10). In figure 9 the modal gain of the longitudinal spacecraft
interface response is plotted for all system modes up to 200 Hz. One can clearly identify the main modes. By using only the main modes with a modal gain higher than 5% one can reconstruct the spacecraft interface response quite well, as can be observed in figure 10.
)LJXUH 0RGDOJDLQDVDIXQFWLRQRIPRGHQXPEHUDQGPDLQV\VWHPPRGH
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)LJXUH 6SDFHFUDIWLQWHUIDFHUHVSRQVHUHFRQVWUXFWHGIURPPRGDOJDLQV &RQFOXVLRQV In this paper a damping methodology has been discussed which allows to compute a modal damping matrix that is based on structural damping ratios assigned to the materials present in a substructure FE-model, and the associated complex strain energy of the modeshapes. Comparison with firing test data has shown that the method provides good damping models for solid rocket motors that are comprised of materials with low and high structural damping ratios. The damping model was implemented in the motor models used for Vega CLA activities. The lift-off load case has been briefly discussed in terms of loads and responses. It has been shown that the responses obtained by direct time integration schemes can be decomposed into their modal participation factors or modal gains (after having performed a normal modes analysis), which allows the detection of the driving system modes as well as the evaluation of their modal damping ratios. 5HIHUHQFHV [1] Fransen, S., Methodologies for Launcher-Payload Coupled Dynamic Analysis, European Conference on Spacecraft Structures, Materials & Mechanical Testing, Noordwijk, The Netherlands, 2005. [2] Fischer, H., et al., A Dynamic Analysis Tool for Europe’s Small Launcher Vega, European Conference on Spacecraft Structures, Materials & Mechanical Testing, Noordwijk, The Netherlands, 2005. [3] Carrington, H. and Ottens, H., A Survey of Data on Damping in Spacecraft Structures. Technical Report ESRO CR-539, ESTEC contract No. 2142173, Fokker Space and NLR, Noordwijk, The Netherlands, 2005. [4] Fransen, S., A Comparison of Recovery Methods for Substructure Models with Internal Loads. AIAA Journal, 42(10):2130-2142, 2004.
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BookID 214574_ChapID 26_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII
February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc. $QDO\VLVDQG2SWLPL]DWLRQRIWKH&XUUHQW)ORZLQJ7HFKQLTXHIRU6HPL SDVVLYH0XOWL0RGDO9LEUDWLRQ5HGXFWLRQ
S. Manzoni1, M. Redaelli2, M. Vanali1 1
Politecnico di Milano, Department of Mechanics Via La Masa, 1 – 20156 Milan (Italy) email:
[email protected] 2 AgustaWestland, Research & Development Group Via G. Agusta 520 - Cascina Costa di Samarate (VA) (Italy) $%675$&7 This article deals with semi-passive multi-modal vibration control by means of piezo-benders, particularly on an already available method: the Current Flowing technique, which has a number of advantages with respect to the other methods presented in literature. Such a technique relies on the link between the bender and an electrical impedance, which allows mechanical energy dissipation. Though its analysis has already been discussed in the state of the art, many points are still open. The paper gives a deep analysis of this control technique and then presents an algorithm for the optimization of the electrical network linked to the bender. ,QWURGXFWLRQ The use of smart-materials to vibration attenuation purposes is widely extended, especially thanks to a number of advantages with respect to other actuators (i.e. low weight, the possibility to exploit them for energy harvesting purposes, etc.). The piezo-actuators are one of the most used type of smart-materials. They are adopted both in active [1] and in semi-passive [2] vibration attenuation. This latter approach mainly consists in the link between the piezo-actuator and a proper electric network ([3], Figure 1) able to partially convert mechanical energy, associated to structure vibration, into electrical energy, which is then dissipated. The electrical network can have different layouts, depending on the required kind of vibration attenuation. When a mono-modal attenuation is needed, the electrical network can be made up by the series of a resistance and an inductance, which constitute a single degree of freedom system together with the capacitance of the piezo-material. Once the electrical parameter values are fixed so that the electrical eigenfrequency is tuned on the mechanical eigenfrequency to be attenuated, the vibration energy associated to the mechanical resonance frequency is partially transformed into electrical energy and then dissipated [3]. In other words the electrical impedance works like a classic mechanical tuned mass damper system [4]. Other impedance kinds can be adopted: as an example, a time-variant passive impedance [5] or a time-invariant one based one operational amplifiers [6] can be adopted for broad-band vibration attenuation. In the present paper attention is focused on strategies based on time-invariant passive impedances aimed at multi-modal control using a single piezo-actuator. Different control techniques are already available in the state of the art. The mostly known are the Current-Blocking method [7], the Hollkamp method [8] and the Current-Flowing method. The first two approaches (Current-Blocking and Hollkamp) are less effective than the third one, especially when the number of mechanical resonances to be controlled increases [9]. Their first drawback concerns the complication of the electrical network because of the increase of modes to be attenuated. A further important problem is that the value of each electrical network variable (i.e. the values of resistances, inductances and capacitances constituting the Z impedance of Figure 1) influences the attenuation performances on different modes, therefore it is difficult to have a simple and effective tuning of the each variable value. Otherwise the third mentioned approach (the Current-Flowing, CF in the following) lowly suffers the mentioned drawbacks. Before discussing the advantages of this technique, its working principle has to be explained [9] (refer
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_26, © The Society for Experimental Mechanics, Inc. 2011
283
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to Figure 2). The electrical impedance is made up by as many branches as the number of modes to be attenuated. Each branch corresponds to a single mechanical mode. In each branch two different parts can be evidenced. The first is named Shunting Branch (Figure 2), which is a series of a resistance and an inductance. This series constitutes the resonant circuit, together with the piezo-material capacitance as explained previously in this section, and it is dedicated to the vibration attenuation of the corresponding mechanical mode (the electrical resonance frequency in tuned on a mechanical resonance frequency). The second branch part is named Current Flowing Branch (Figure 2) and it is constituted by a series of an inductance and a capacitance. A proper choice of the latter element values allows the Current Flowing Branch to behave like a short-circuit at the mechanical frequency to be attenuated. At the same time the branch tends to become an open-circuit as long as the frequency is different from that of the mechanical mode (the Current Flowing Branch practically acts as a band-pass filter). This means that each branch is devoted to vibration attenuation of a single mechanical eigenmode and does not affect the attenuation performance at the other modes. Comparing this method to those previously mentioned (Current-Blocking and Hollkamp), it shows two important advantages. First it has a moderate network complication when the considered number of modes increases. Furthermore the impedance Z is constituted by different branches (Figure 2), one for each mode to be attenuated, and the value of a single electrical variable affects the performances just on a single mechanical mode, depending in which branch the electrical element is inserted in [9]. For example, looking at Figure 2, the inductance L2 affects the attenuation performances only on the second mode to be controlled, at least till when the mechanical modes are well separated in frequency [9].
Figure 1: Shunted piezo-bender linked to the Z impedance (a) and the corresponding electrical model (b) Reference [9] gives the following formulas for a proper choice of the Z impedance variables when the vibration has to be attenuated at frequencies Z1 , Z 2 ,..., Z n :
Lˆ1 ~ L1
1 ,..., Lˆ n Z C1 1 ~ ,..., Ln 2 Z1 C p 2 1
1 Z Cn 1 2 Zn C p
(1)
2 n
(2)
where Cp is the bender capacitance. The expression of the Z impedance in the Laplace domain is:
Z ( s)
1 (1 / Li ) s ¦ 2 i 1 s ( Ri / Li ) s (1 /( Li C i ))
(3)
N
where N is the number of modes to be attenuated and Li
~ Lˆ i Li .
Though some analyses on this method are already available in the state of the art ([9],[10]), some points still require a deeper investigation. In fact a small amount of numeric data is available in literature about the independence of the various modes to be controlled. In other words few data are given to explain how much the frequency span among the various resonances has to be in order to have mode independence. Furthermore no analyses are available concerning Equation 1, where two variables are linked by a single equation and thus there is still a degree of freedom. As an example it is not discussed in literature what is the effect on the choice of the Ci values (Equation 1) on the whole system behavior. This paper is therefore aimed at offering a detailed description and analysis of the CF method, paying particular attention to the applicability conditions and to its optimization. A numerical modal model simulating the behavior of a vibrating structure (a cantilever beam in this case) coupled to a piezo-bender, used to implement the CF technique, has been developed and validated by experimental data. This validated model has allowed to carry out
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many simulations in order to satisfy all the paper aims. Section 2 describes the experimental set-up described by the model and used to validate it. Section 3 gives details on this numerical model and its experimental validation. Section 4 discusses the method applied to the attenuation of a single mode, which is a preliminary task; then Section 5 explains how the CF technique behaves when a pair of modes have to be controlled, giving details on the way to choose the electrical variable values of the impedance Z. Finally, Section 6 describes how to optimize such a choice by means of a recursive approach.
Figure 2: Layout of the shunting impedance Z(s) used with the CF method ([SHULPHQWDOVHWXS The experimental set-up is constituted by a cantilever beam, which is deeply analyzed in the state of the art. This simple mechanical case allows to focus the study on the electro-mechanical interactions, developing a numerical modal model without introducing any undesired complication on the mechanical side. Nevertheless such a choice does not compromise the generality of the results. The steel cantilever beam has a length of 0.5 m, a width of 40 mm and a thickness of 3 mm. Two piezo-benders are bonded to the beam at 75 mm from the constraint (they are in a co-located configuration, as evidenced by Figure 3). One bender is used to excite the beam, while the other is exploited to apply the CF method. The main set-up data are given in Table 1. The actuator position has been chosen in order to be able to attenuate the mechanical vibration in correspondence of the third and fourth eigenmodes (Table 2) as they have a significant curvature in correspondence of the bender position.
Figure 3: Experimental set-up (cantilever beam) Beam Young’s modulus 210 GPa Beam density 7800 Kg/m3 Overall length of the piezo-actuator 0.101 m Overall thickness of the piezo-actuator 0.5 mm Overall width of the piezo-actuator 0.025 m Length of a single patch (active element) 0.092 m Thickness of a single patch (active element) 0.13 mm Width of a single patch (active element) 0.021 m Overall Young’s modulus of the piezo-actuator 69 GPa Bender capacitance 0.165 ȝF Piezoelectric constant d31 -179 pm/V Electromechanical coupling coefficient 0.63 Table 1: Nominal data of the experimental set-up
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Mode number
Frequency [Hz]
Non-dimensional damping ratio [%] 9.0 0.26 55.4 0.61 157.3 0.09 309.6 0.53 Table 2: Identified modal data for the first four cantilever beam eigenmodes
1 2 3 4
The next section describes the modal model developed to simulate such a system. 0RGDOPRGHORIWKHFDQWLOHYHUEHDPZLWKSLH]REHQGHUV This section is intended to describe the system model, composed by the cantilever beam and the piezo-benders. The model development starts from that proposed by Moheimani and Fleming in [10], although others are present in literature ([2],[11]). A model description is given below. Suppose to have a pair of co-located piezo-benders (Figure 3), one used as sensor and the other as actuator. The dynamics of the structure is described by Equation 4:
Eb I
w 4 z ( x, t ) w 2 z ( x, t ) U A b wx 4 wx 2
w 2 M x ( x, t ) wx 2
(4)
where ȡ, Ab, Eb and I represent density, cross-sectional area, Young’s modulus of elasticity and moment of inertia about the neutral axis of the beam respectively. x and z are defined in Figure 3; t is the time. The moment Mx(x,t) exerted on the beam by the actuator is given by Equation 5:
M x ( x, t )
k V (t )^u ( x x1 ) u ( x x 2 )`
with u(x) defined as the unit step function (u(x)=0 for x<0 and u(x)=1 for
(5)
x t 0) and V as the voltage applied to the
actuator. Finally k is given by Equation 6 [10]:
k
Eb IDd 31 tp
(6)
tb t t p )2 ( b )2 ) 2 2 tb t t 2[ E p (( t p ) 3 ( b ) 3 ) E b ( b ) 3 ] 2 2 2
(7)
with
D
3E p ((
where Ep is the Young’s modulus of elasticity of the piezoelectric material, tp is the thickness of the piezo-bender and tb that of the beam. Then, I is again the moment of inertia about the neutral axis of the beam, d31 is the piezoelectric constant of the bender. Note that k depends on the mechanical, geometrical and electrical features of the system composed by the beam and the benders. Equations 5 and 7 allow to obtain the moment Mx acting on the structure. The solution of Equation 4 then passes through modal coordinates qk(t) This leads to a set of uncoupled ordinary differential equations with the form of Equation 8:
qk (t ) 2] k Z k q k (t ) Z k2 q k (t )
k \ V (t ) UAb k
(8)
where Ȧk is the kth mechanical eigenfrequency, ȗk is the associated non-dimensional damping ratio and ȥk is a term proportional to the curvature associated to the kth eigenmode at the extreme points of the piezo-bender (Figure 3). Particularly
\k
wk' ( x 2 ) wk' ( x1 ) where f ' is generically the first derivative of the function f with
respect to x and wk(x) is the kth eigenvector at position x. Assuming the zero initial condition, it is possible to obtain the transfer function between the voltage applied to the actuator and the consequent displacement z(x,t). Its Laplace domain expression is shown by Equation 9:
G ( x, s )
k UAb
wk ( x )\ kT ¦ 2 2 k 1 s 2] k Z k s Z k f
(9)
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where T indicates the transpose operator. The voltage Vp induced to the co-located sensor bender due to the structural deflection is given by Equation 10 [10]:
V p (t )
d 31 E pW p (
tb tp) 2
Cp
f
¦\
k
(10)
q k (t )
k 1
where Cp is the capacitance value of the piezo-sensor and Wp is its width. Therefore the transfer function Gvv(s) in the Laplace domain between the voltage V applied to the actuator and the voltage Vp induced on the sensor is (Equation 11):
Gvv ( s )
f
\ k2 2 2 1 s 2] k Z k s Z k
J¦ k
(11)
with
J
tb t p )k 2 C p UAb
d 31 E pW p (
(12)
This discussion has been focused on a co-located actuator-sensor pair up to now. Actually this a different situation from that of a semi-passive vibration attenuation, where just a single bender is present. Nevertheless the abovementioned discussion is useful to comprehend the CF approach situation. In the case of a semi-passive vibration attenuation, Reference [10] shows that an underlying feed-back structure associated to the piezo-electric shunt damping exists. When the beam is deflected by the mechanical vibration, the piezo-bender is deflected as well. Therefore a voltage V2 (Figure 1) rises. Consequently a voltage V1 across the bender electrodes (Figure 1) appears too and its value depends on the piezo-bender features and on the amount of deflection, but also on the kind of the adopted shunt impedance Z. Then this voltage V1 across the electrodes causes a further bender deflection and thus an actuation to the beam. Starting from the electrical model of the piezo-element in Figure 1, it is possible to describe this feed-back effect with the scheme of Figure 4 [10], where Gvv(s) has the same form of Equation 11 and represents the transfer function between the voltage applied across the conducting electrodes of the transducer (V1 in Figure 1) and the induced piezoelectric voltage (V2 in Figure 1). Gvw(s) is the transfer function between the disturbance w (in this case the input voltage to the actuation piezo-bender of Figure 3) and the voltage V2 (Figure 1). As the piezobender used to excite the structure is co-located to the piezo-bender used for the shunt damping (Figure 3), thus in this case Gvw=Gvv [10]. Finally:
K ( s)
sC p Z ( s )
(13)
1 sC p Z ( s )
(K(s) represents a sort of controller), where Z(s) is again the impedance linked to the piezo-bender. Relying on the block diagram of Figure 4 and on the fact that the disturbance bender is co-located to the shunted one, the transfer function Tvw between the disturbance w and the voltage V2 is defined by Equation 14 [10]:
Tvw ( s )
Gvw ( s) 1 K ( s )Gvv ( s )
Gvv ( s) 1 K ( s )Gvv ( s )
(14)
Many numerical simulations have been performed to validate this modal model. The results show that the model gives a detailed description of the experimental set-up and the agreement between the simulated and experimental data is good. Before going on, it has to be underlined that some of the results presented in the following are given in terms of the transfer function Tzw between the disturbance w and the displacement z at x = 495 mm; this transfer function can be easily calculated combining Equations 9, 11 and 14 as Tzw ( s )
Tvw ( s ) x G ( x, s ) . Gvv ( s)
The further step is the adoption of such a validated model to deeply investigate the CF method and to understand how to optimize it. The next section starts this discussion analyzing the CF method adoption to the purpose of vibration attenuation of a single structure eigenmode. This is a preliminary step towards a full comprehension of the CF technique.
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9LEUDWLRQDWWHQXDWLRQIRUDVLQJOHPHFKDQLFDOHLJHQPRGH In this section the vibration attenuation in correspondence of the third beam eigenmode (157.3 Hz, see Table 2) is implemented. The disturbance is a broad band white noise and it is given by a piezo-bender (Figure 3). Looking at Figure 2 it comes that a single branch constitutes the shunting impedance Z (Figure 1). The value of calculated by Equation 2, where
Z1
2S x 157.3 rad/s.
~ L1 can be
However it is not possible to determine the values of C1 and Lˆ1 as the only constraint is given by Equation 1, which is not enough to calculate the two variable values. Thus the system behavior values has been analysed for different C1 (and thus
Lˆ1 ).
Figure 4: Shunted feed-back interpretation of the semi-passive shunt damping The value of R1 has been fixed each time imposing the electrical non-dimensional damping ratio
hel (this value
choice also depends on the mechanical mode damping value; typically values between 0.1 % and 0.6 % have been used here for control purposes). Then the value of R1 can be obtained by Equation 15:
R1
~ 2hel ( L1 Lˆ1 )Z1
(15)
Before showing the results, a fact has to be pointed out as it will be useful for the further discussion. The form of the Z impedance in this case is given by Equation 16:
s 2 ( R1 / L1 ) s (1 /( L1C1 )) (1 / L1 ) s ~ where Li Lˆ i Li . Therefore it comes (Equation 13) that: Z ( s)
K ( s)
C p s 3 (C p R1 / L1 ) s 2 (C p /( L1C1 )) s C p s 3 (C p R1 / L1 ) s 2 (C p /( L1C1 ) 1 / L1 ) s
(16)
(17)
The transfer function of Equation 17 is represented in the frequency domain (Frequency Response Function, FRF in the following) in Figure 5 for different values of the pair C1 - Lˆ1 . When the C1 value increases, the frequencies of the zeros decrease while those of the poles do not change (the pole and zero in (0,0) are not affected by the capacitance increase). When the C1 value tends to be null (and thus Lˆ1 and L1 tend to infinite) the numerator and denominator of Equation 17 tend to be equal and thus with similar pole and zero frequencies. Figure 6 shows the transfer function Tzw for the same C1 values of Figure 5. The minimization of Tzw involves the minimization of the Tvw (minimizing V2 means minimizing the displacement z of Figure 3). This figure shows that the higher is the C1 value, the broader is the controller frequency band, as well as in Figure 5. The results obtained in this section will be recalled in next one where the CF technique is applied to two eigenmodes.
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Figure 5: FRF of the electrical circuit in correspondence of the third mode of Table 2 for different C1 values (hel = 0.1 %)
Figure 6: Tzw trend in correspondence of the third mode of Table 2 for different C1 values 9LEUDWLRQDWWHQXDWLRQDWWZRPHFKDQLFDOPRGHV In this case it has been chosen to attenuate the vibration in correspondence of the third and fourth eigenmodes (Table 2). As the considered modes are well separated in frequency (the span is nearly 152 Hz), it is expected from literature [9] that the CF method can work on the two modes separately, without any cross influence. Now the impedance Z has the form given by Equation 18:
Z ( s)
§R § 1 § R1 R · RR · R2 · 1 1 ¸¸ s s 4 ¨¨ 1 2 ¸¸ s 3 ¨¨ 1 2 ¸¸ s 2 ¨¨ L1 L2 C1C 2 © L1 L2 ¹ © L1C1 L2 C 2 L1 L2 ¹ © L1 L2 C 2 L1 L2 C1 ¹ §1 1 · 3 § R1 R2 · 2 § C1 C 2 · ¨¨ ¸¸ s ¨¨ ¸¸ s ¨¨ ¸¸ s © L1 L2 ¹ © L1 L2 ¹ © L1 L2 C1C 2 ¹
(18)
Suppose to tune the variable values of the two branches (the modes to be controlled are two) independently, as previously done (Section 4). The values of C1 and C2 are chosen to be small (10 nF in this case) in order to tune the controller frequency band just in correspondence of the two mechanical eigenfrequencies (Figure 5). This should guarantee mutual independence between the two considered modes. Figure 7 presents two zooms of the
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electrical transfer function K of Equation 13 in the frequency domain in correspondence of the two considered eigenmode frequencies. In both plots this transfer function is compared to that obtained by using a Z impedance composed by a single branch (this branch is devoted to vibration attenuation of the third mode in Figure 7a and of the fourth mode in Figure 7b). A slight shift of the transfer function peaks is appreciable, due to a slight interdependence of the two modes. Figure 8 shows the zooms of the mechanical transfer function Tzw (defined in Section 3). It is evident that the values of C1 and C2 are too small. The frequency bandwidth of the controller in correspondence of the two eigenmodes is too narrow to obtain a satisfactory vibration attenuation. Moreover it has already been highlighted in the previous section (Figure 6) that the higher is C1, the higher is the vibration attenuation in the case of a single mode under control. These facts show that the Ci values has to be greater to reach a satisfactory vibration reduction, as the electrical transfer function bandwidth is too narrow in correspondence of the two mechanical modes, which is also due to the low electrical damping ratio values. One could think that the low adopted Ci values would be enough in case of mechanical damping ratios lower than those considered in this case,. This is wrong too as the lower is the mechanical damping ratio, the more critical are the slight transfer function shifts evidenced in Figure 7. Thus the values of Ci usually have to be large to guarantee satisfactory attenuation performances. A further problem arises if the values of Ci are increased and this is explained in Figure 9, where the electrical transfer function K (Equation 13) for C1 C 2 150 nF is shown (therefore for capacitance values much higher than those of Figures 7 and 8). In Figure 9 the electrical transfer function is also compared to those obtained by using a Z impedance composed by a single branch (this branch is devoted to vibration attenuation of the third mode in the case of the dashed line and of the fourth mode for the solid line). When a single branch is used, the resonance of the transfer function K is almost at the same frequency value of the mechanical eigenfrequency. The method to tune the variable values of each branch for multi-modal control is the same adopted previously, which means that the variable values of each branch have been chosen independently from those of the other branch. It is evident that the electrical transfer function resonances have significantly shifted from the mechanical resonances (which prevents obtaining satisfactory attenuation performances), when both modes are controlled. The presence of two branches with high capacitance values causes a frequency shift of the zeros (Figure 9) as well as in the case of a single branch (Section 4), but also a frequency shift of the poles (Figures 9). This is due to the high capacitance values, which increase each branch frequency response bandwidth. Finally, this effect generates an interaction of the two modes and the pole shifts. Thus the higher are the capacitance values, the higher are the distances between the zeros and the corresponding poles and the higher are the pole shifts from the original positions.
Figure 7: FRF of the electrical circuit in correspondence of the third mode (a) and the fourth one (b) with
C1
C2
10 nF
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The outcome is that when different branches are used to the purposes of CF vibration attenuation, different problems exist both with low and high branch capacitance values: x when the capacitance values are low, there is mode independence but also a worsening of the vibration attenuation performances; x when the capacitance values are high, there is mode interdependence but also a potential improvement of the vibration attenuation performances. Therefore a new method for CF technique optimization is needed, which should allow to choose high capacitance values, without frequency shifts between mechanical and electrical eigenfrequencies. This method is described in the next section.
Figure 8: Tzw in correspondence of the third mode (a) and the fourth one (b) with C1
C2
Figure 9: FRF of the electrical circuit in correspondence of the third and fourth modes with C1
10 nF
C2
150 nF
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A remark has to be pointed out. It has been highlighted that a large electrical transfer function bandwidth in correspondence of the modes to be controlled is needed. Here this is obtained by increasing the capacitance values. However another approach is possible and it has already been presented in literature [12]. It relies on using Equations 1 and 2 maintaining low Ci values and in a proper choice of the resistance values; nevertheless this method can be very complicated and that is the reason why a different solution is approached (this point is considered again in the next section). (OHFWULFDOSDUDPHWHUWXQLQJE\DQLWHUDWLYHWHFKQLTXH This new proposed method gives the possibility to adopt high capacitance values without significant frequency shifts between mechanical and electrical eigenfrequencies. The inputs to this algorithm are the piezo-bender capacitance value Cp, the mechanical eigenfrequencies to be controlled and the values of Ci. These latter values have to be high enough to have high vibration attenuation and huge control bandwidths (see Sections 4 and 5).
~ The values of Lˆ i and Li are then calculated by means of Equations 1 and 2 and Ri by means of Equation 15. ~
Then the Li values are
changed step by step through an iterative minimization algorithm till when the
frequencies associated to the electrical circuit poles are close enough to those of the mechanical poles. This minimization may be carried out by means of common minimization algorithms. Equation 19 explains which is the variable to be minimized: N
¦
f ( Pi ) f (Z i )
H d tolerance
(19)
i 1
where i is the counter on the N modes to be controlled, f(Pi) is the frequency (in Hertz) of the ith electrical pole and f(Ȧi) is the frequency value (in Hertz) of the ith mechanical mode to be controlled. A minimum threshold on the overall distance has to be fixed by the user (the authors of the paper have used in this case 0.02 Hz for two modes to be controlled). Figure 10 gives an example of the performances obtained with this new method (with
~
~
C1=245 nF, C2=35 nF, L1 =4.72 H and L2 =2 H). Here the capacitance value of the third eigenmode branch (245 nF) has been chosen to be higher than that of the fourth mode branch (35 nF). This comes from the will to have a higher attenuation of the third mode, due to the fact that it has a higher dynamic amplification under uncontrolled conditions.
Figure 10: Tzw trend in correspondence of the third and fourth modes
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Here the traditional approach using Equations 1 and 2 completely fails if a complicated and laborious choice of the resistance values is not carried out (see the end of Section 5) ([9],[12]). Furthermore in this case the attenuation performances with the new proposed method are better than those of the traditional one, especially on the third mode (Figure 10).
~
~
A further aspect is worthy of attention. This is that the adopted values of L1 and L2 (see above in this section) are very different from those obtainable by means of Equation 2 (about 6.2 H for the third mode and 1.6 H for the fourth one), once the capacitance values have been fixed (C1=245 nF, C2=35 nF in this case). )LQDOUHPDUNV The present paper has dealt with the Current Flowing technique, which is a semi-passive method for multi-modal vibration attenuation by means of piezo-actuators. Although the method is already known in literature, a deeper technique description has been given thanks to a modal model able to accurately simulate a cantilever beam coupled to a piezo-bender used to the purpose of vibration attenuation. This analysis has allowed to comprehend the role of each parameter composing the electrical impedance linked to the piezo-actuator. One of the main outcome is that the impedance capacitance values have to be high enough to guarantee an effective vibration reduction. On the other hand this makes the various modes to be controlled dependent each other, loosing the main advantage of the Current Flowing method. This has pulled towards looking for a new method to fix the values of the other electrical parameters, with the final aim to optimise the electrical parameter value choice. Such a method is iterative and allows attenuation performances equal or better than those offered by the traditional approach. Furthermore the new method is easier to be implemented. 5HIHUHQFHV [1] Lallart M, Guyomar D, Jayet Y, Petit L, Lefeuvre E, Monnier T, Guy P, Richard C, Synchronized switch harvesting applied to self-powered smart systems: Piezoactive microgenerators for autonomous wireless receivers, Sensors and Actuators A: Physical, Vol. 147, No. 1, pp 263-272, 2008. [2] Preumont A, Mechatronics: dynamics of electromechanical and piezoelectric systems, Springer, Dordrecht, 2006. [3] Hagood N W, von Flotow A, Damping of structural vibrations with piezoelectric materials and passive electrical networks, Journal of Sound and Vibration, Vol. 146, pp. 243-268, 1991. [4] Krenk S, Hogsberg J, Tuned mass absorbers on damped structures under random load, Probabilistic Engineering Mechanics, pp. 408-415, 2008. [5] Guyomar D, Badel A, Nonlinear semi-passive multimodal vibration damping: An efficient probabilistic approach, Journal of Sound and Vibration, Vol. 294, pp. 249-268, 2006. [6] de Marneffe B, Preumont A, Vibration damping with negative capacitance shunts: theory and experiment, Smart Materials and Structures, Vol. 17, article number 035015, 2008. [7] Wu S Y, Method for multiple mode piezoelectric shunting with single PZT transducer for vibration control, Journal of Intelligent Material Systems and Structures, Vol. 9, No. 12, pp. 991-998, 1998. [8] Hollkamp J, Multimodal passive vibration suppression with piezoelectric materials and resonant shunts, Journal of Intelligent Material Systems and Structures, Vol. 5, No. 1, pp. 49-57, 1994. [9] Behrens S, Moheimani S O R, Fleming A J, Multiple mode current flowing passive piezoelectric shunt controller, Journal of Sound and Vibration, Vol. 266, No. 5, pp. 929-942, 2003. [10] Moheimani S O R, Fleming A J, Piezoelectric transducers for vibration control and damping, Springer-Verlag, London, 2006. [11] Agneni A, Balis Crema L, Sgubini S, Damping by piezoceramic devices with passive loads, Mechanical Systems and Signal Processing, Vol. 17, pp. 1097-1114, 2003. [12] Behrens S, Moheimani S O R, Optimal resistive elements for multiple mode shunt-damping of a piezoelectric laminate beam, in Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, Australia, December 12-15, pp. 4018-4023, 2000.
BookID 214574_ChapID FM_Proof# 1 - 23/04/2011
BookID 214574_ChapID 27_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII
February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc. 6RPH3DVVLYH'DPSLQJ6RXUFHVRQ)ORRULQJ6\VWHPVEHVLGHVWKH70'
Lars Pedersen Aalborg University Department of Civil Engineering Sohngaardsholmsvej 57 DK-9000 Aalborg $%675$&7 Impulsive loads and walking loads can generate problematic structural vibrations in flooring-systems. Measures that may be taken to mitigate the problem would often be to consider the implementation of a tuned mass damper or even more advanced vibration control technologies; this in order to add damping to the structure. Basically also passive humans on a floor act as a damping source, but it also turns out from doing system identification tests with a floor strip that a quite simple set-up installed on the floor (cheap and readily at hand) might do a good job in terms of reducing vertical floor vibrations for some floors. The paper describes the tests with the floor strip, and the results, in terms of dynamic floor behaviour, are compared with what would be expected had the floor instead been equipped with a tuned mass damper. 120(1&/$785( f
] L
Floor frequency Floor damping ratio Span length
f1
]1 M
TMD frequency TMD damping ratio Modal mass
P x m
Mass ratio Distance from support Mass
,1752'8&7,21 Damping characteristics of flooring systems have a profound effect on how floors perform when excited by dynamic loads. For instance, walking loads may generate problematic resonant phenomena [1], but also vibration sources outside a building may induce vibrations in floors which are problematic. Guidelines are available suggesting damping characteristics for different types of flooring systems, but still floor vibration problems emerge from time to time. Flooring systems of today are generally more slender than those erected several years ago, and thus more prone to react likely to walking loads. That fact that modern open-space office environments do not involve many partitions is also likely to have a bearing on the dynamic behaviour of floors, as partitions add damping to the floor. Likewise, the tendency to store information on a pc instead of in bookcases probably also has an effect on magnitudes of floor response as bookcases would contribute with damping. So a number of non-structural components can have an influence on the dynamic behaviour of floor slabs [2]. This paper is not dealing with the passive damping sources mentioned above. Instead it places focus on other types of passive damping sources on flooring systems; damping sources which are often not addressed or considered when focus is on floor vibrations. When floor vibrations are problematic it is often because the vibrations are perceived as annoying by floor users. Often it would be a stationary person (not the walking person generating the vibrations) that would perceive
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_27, © The Society for Experimental Mechanics, Inc. 2011
295
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296 vibrations as annoying. Hence, at least one stationary person (for instance a sitting person) would need to be present on the floor for someone to perceive vibrations as annoying. Measurements made on grandstands [3-4] and on test floors in laboratory environments [5-7] have proven that stationary humans act as passive damping sources, and the present paper will quantify the amount of damping added to a floor strip by humans in sitting posture. It also turns out (from the experimental investigations reported in this paper) that there are other passive damping sources on floors that may come into play. Specifically, the paper considers how different types of chairs perform in terms of mitigating floor vibrations. The chairs themselves do not add much damping to the floor, but the paper investigates what happens when sandbags are placed in the seats of chairs. The tested chairs are rigid office chairs and swivel chairs. In tests, the sandbag mass, and the number of chairs on the floor, is gradually increased so as to explore what effect this has on the damping added to the floor. When floor vibration problems occur, a tuned mass damper (TMD) may be useful [8], but even more advanced remedial measures (vibration control devices) have also been considered for flooring systems [9]. In order to give some perspective to the amount of damping found to be added by humans and by chairs carrying sandbags, results are compared with the expected performance of a TMD fitted to the floor strip. Experiments and test procedures are outlined in section 2, and section 3 presents the results in terms of the damping added to the floor strip. 7+((;3(5,0(176 7KHWHVWIORRUDQGWKHLQVWUXPHQWDWLRQ The test floor is a hollow-core concrete element pin supported at both ends. The distance between the supports of the one-way spanning element is about 11 m, and the width of the element is about 1.2 m. The weight of the element amounts to more than 5,000 kg. As the floor-strip is pin-supported, its fundamental mode (the first vertical bending mode) is well separated from other modes of vibration, and in tests this mode is excited. It is the damping ratio of the fundamental mode which is determined in tests. This is done by bringing the element into free decaying vibrations by applying an impact load at midspan, and from recordings of floor vertical displacement response (by LVDT’s positioned at floor midspan), the damping ratio of the floor is identified using the logarithmic decrement method. Without any chairs or humans atop the test floor its undamped frequency was found to be 5.8 Hz and the damping ratio was found to be around 0.25 %cr. 7HVWVZLWKFKDLUVDQGKXPDQVDWRSWKHIORRU Six test sequences were carried out, and they are denoted A, B, C, D, E, and F. The first two test sequences are described below: A. One rigid office chair atop the floor strip at midspan B. One swivel chair atop the floor strip at midspan In these tests, floor damping was determined with different numbers of sandbags placed in the seat of the chairs. Each sandbag had a weight of 40 kg, and after doing a test without a sandbag, first one, then two, three and finally four sandbags were placed in the seat. In the presentation of results, the sandbag weight is denoted m, and floor damping was thus determined for values of m of 0, 40, 80, 120, and 160 kg. This provides insight into how the value of m influences floor damping for the two different types of chairs. In tests with the swivel chair, a number of different swivel chairs were used so as to investigate variability in results (for m = 80 kg) from chair to chair. All swivel chairs were of the same type of construction. The swivel chairs employed in tests are perhaps 15 years old and are not provided with modern damping devices between seat and wheel frame. The wheel frames of the chairs carry five wheels. The rigid office chair is a standard four-legged office chair used at Aalborg
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297 University in meeting rooms, whereas the swivel chair is the type used by students at their desks. A principle sketch of the two chairs is shown in figure 1.
)LJXUH
The swivel chair (left) and the rigid office chair (right) used in tests. The chairs are shown carrying two sandbags (m = 80 kg).
In test sequences C and D not only a single chair was used. These tests involved: C. Up to four office chairs atop the floor strip at midspan D. Up to four swivel chairs atop the floor strip at midspan In these tests, three or four chairs were positioned on the floor strip at the same time, and the chairs were each carrying either 80 kg of sandbag (2 sandbags) or no sandbags. Figure 2 shows an example. The tests were made in the way that first one chair carried 80 kg (m = 80 kg), then two chairs carried 80 kg each (m = 160 kg), and then three chairs carried 80 kg each (m = 240 kg), etc. This procedure allows for investigating how floor damping is influenced when the sandbag mass is split (carried by more than one chair), which also accommodates a higher total sandbag mass than what is possible to carry by a single chair.
)LJXUH Side view of floor strip at midspan. Three swivel chairs each carrying 80 kg (m = 240 kg). Test sequence E: E. Single swivel chair atop the floor strip at various positions In this test sequence a randomly selected swivel chair was placed at different positions on the floor strip carrying a sandbag mass of 80 kg (see figure 3), and for each position floor damping was identified. The different positions were chosen such that floor damping could be mapped as a function of the distance from floor support.
)LJXUH
The swivel chair placed at a distance of x from floor support.
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298 Test sequence F involved humans, so as to establish a reference for the damping introduced by chairs: F. Humans sitting atop the floor strip at midspan In this test, human sat on the floor strip with legs hanging down over the side of the floor strip as shown in figure 4. Floor damping was determined in situations were one, two and three persons sat on the floor strip.
)LJXUH Side view of floor strip. Three humans sitting at floor midspan. The individuals were asked to assume this posture during the entire phase of decaying vibrations. Each individual was weighted prior to the tests, such that the total mass (m) of the crowd was known. Thereby, it was possible to relate estimates of floor damping to the total mass of the crowd of people. For all test conditions, a series of free decay tests were made allowing a series of estimates of floor damping to be produced. For simplicity, the result section only presents mean values of floor damping (obtained under similar conditions). 5(68/76 From the floor decays, the mean value of estimates of floor damping, ], was calculated for different values of the mass (m), representing either sandbag or human mass. 6LQJOHFKDLURQWKHIORRU WHVWV$DQG% Figure 5 shows estimates of floor damping obtained with different swivel chairs (solid lines) and with the rigid office chair (dashed line). For m = 0 kg the chair does not carry a sandbag. As can be seen, the positioning of sandbags on the rigid office chair does not result in a change in floor damping. However, this is not the case for the swivel chair. When placing sandbags in the seat of this type of chair, a significant increase in floor damping is observed. With a sandbag weight of 80 kg in the seat of one of the tested swivel chairs, the damping increases from 0.25 %cr to a value above 3 %cr (corresponding to an increase in damping of more than a factor 12). This was the observation that initiated the investigations of this paper.
4
] [%cr]
3 2 1 0
)LJXUH
0
50
100 m [kg]
150
200
Variations of floor damping with sandbag mass when in seat of swivel and office chair. (solid and dashed lines, respectively)
As can be seen not all swivel chairs turned out to perform as well in terms of adding damping to the floor, but it is noticeable that all tested swivel chairs carrying 80 kg of sandbag increased floor damping by a factor 6 or more.
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299 This is quite significant considering that the concrete element, atop of which the swivel chair is placed, weights more than 5,000 kg. It can be seen that in most cases floor damping increases when more sandbag weight is placed in the seat. 6HYHUDOFKDLUVRQWKHIORRU WHVWV&DQG' Figure 6 shows results of floor damping obtained with sandbag weights of 80 kg placed in the seat of one chair (m = 80 kg), two chairs (m = 160 kg), three chairs (m = 240 kg), etc. Results obtained for the rigid office chair are not shown as basically no difference in floor damping could be noticed compared with the reference case (m = 0 kg).
5
] [%cr]
4
For the tests that involved swivel chairs, eight chairs were available. Among these chairs, three different combinations of chairs were used in tests (3 to 4 chairs were randomly selected for each combination/test) This explains why three ]-m relationships are obtained.
3 2 1 0
0
0
100
200 m [kg]
300
400
Variations of floor damping with sandbag mass when placed in seat of swivel chairs.
)LJXUH
As can be seen, generally the floor damping increases when an additional swivel chair is equipped with a sandbag mass of 80 kg. There are exceptions as in two tests a slight decline is noticed when placing 80 kg in the seat of an rd th additional chair (3 or 4 chair). It is not readily possible to explain why this occurs.
Besides from the slightly deviating results there is the tendency that when more chairs are equipped with sandbags, more damping is added to the floor. 9DULRXVSRVLWLRQVRIVZLYHOFKDLU WHVW( For an attached damping source it would be expected that the amount of damping that it adds to the floor depends on the location of the damping source. Whether this feature is also valid for the “swivel chair with sandbag”–damper was examined for a randomly selected swivel chair and the results are shown in figure 7.
2
] [%cr]
1.5
It can be seen that when the swivel chair is located right over the pin support of the floor strip, the floor damping corresponds to that of the floor without the chair present (0.25 %cr).
1 0.5 0
)LJXUH
0
0.1
0.2 0.3 x/L [-]
0.4
Variations of floor damping with location of swivel chair carrying a sandbag weight of 80 kg. (x/L = 0: pin support, x/L = 0.5: midspan).
As the chair is moved towards midspan, floor damping increases and the chair adds most damping when positioned at midspan of the floor strip (x/L = 0.5). These are meaningful results.
BookID 214574_ChapID 27_Proof# 1 - 23/04/2011
300 +XPDQVRQIORRUVWULSDQGWKHLUGDPSLQJFDSDFLW\ WHVW) Floor damping was also identified when humans sat on the floor strip, and figure 8 shows the results (+). For reference, figure 8 also shows floor damping as measured with sandbags in the seat of swivel chairs (o) and rigid office chairs (x).
5
] [%cr]
4
It is noticeable that humans add much damping to the floor. With three persons sitting at midspan (m | 230 kg), floor damping is above 4 %cr, which is more than 16 times the damping of the empty floor.
3 2 1 0
0
100
200
The damping values shown for the swivel chairs are the averages of the results shown in figure 7 (for m = 80, 160, and 240 kg).
300
m [kg]
)LJXUH
Definitely, passive damping sources such as humans and swivel chairs with sandbags contribute with much damping.
Floor damping measured with humans sitting on the floor (+), with sandbags in the seat of swivel chairs (o) and rigid office chairs (x).
In order to provide some perspective to the damping added by humans and swivel chairs, numerical calculations were made predicting decaying floor vibrations with a TMD installed on the floor strip (instead of humans and swivel chairs). For the TMD design its frequency (f1) and damping ratio (]1) was determined from eq. 1, as suggested in [10]. f1 = f /(1+P)
]1 = (3/8·P /(1+P ))0.5
P = m/M
(1)
In eq. 1, M is the modal mass of the first bending mode of the floor strip, and f is the frequency of this mode (as measured for the empty floor). For the calculations, the TMD mass (m) was assumed to be 80 kg, which is equal to the weight of two sandbags and approximately equal to the weight of a person. Using a Newmark time integration scheme, decaying floor response to an impact load was computed. Figure 9 compares the calculated decay with those measured in tests. 1
1
1
1
0
0
0
0
-1
0
10 time [sec]
Empty floor
20
-1
0
10 time [sec]
20
Floor with 1 person
-1
0
10 time [sec]
20
Floor with swivel chair
-1
0
10 time [sec]
20
Floor with TMD
)LJXUH Normalised floor decays recorded for the empty floor, the floor with 1 person, the floor with a swivel chair carrying a sandbag weight of 80 kg, and simulated for the floor strip with a TMD. As can be seen, the TMD adds more damping to the floor than the swivel chair carrying a sandbag weight of 80 kg. However, it can be recognised that both the swivel chair and a person sitting on the floor perform quite well in reducing floor vibrations. It should be noted that when deriving damping estimates from decays with swivel chairs on the floor, the initial oscillations were discarded (the first 5 oscillations right after the impact). In the discarded time window, oscillations decayed more rapidly (for some chairs) than in the remaining part of the decay in which damping was fairly constant.
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301 &21&/86,21$1'',6&866,21 The investigations of the paper quantified how chairs carrying sandbags influenced damping characteristics of a floor strip with a frequency of 5.8 Hz. It was found that swivel chairs with sandbags in their seats added much damping to the floor strip (to its fundamental mode, first vertical bending mode), and that the “swivel chair with sandbag”-damper added most damping when located at midspan of the floor strip. It was also shown that persons sitting on the floor strip added much damping, and that floor damping depends on the size of the crowd of people present atop the floor. The damping added by an optimally tuned TMD with a mass of 80 kg was shown to be higher than the damping added by a single person sitting at midspan and higher than the damping added by the swivel chair with 80 kg of sandbag in the seat. Nevertheless, the results indicate that passive damping sources such as humans and swivel chairs carrying humans can add much damping to the floor. The TMD has the advantage that it can be tuned and targeted to solving a specific vibration problem, basically for any floor frequency. This is not the case for the swivel chair carrying sandbags. Its performance in mitigating floor vibrations for a specific floor is by default dictated by the mechanical characteristics of the chair having fixed characteristics. As long as the mechanical characteristics of the chair are unknown (and they are unknown to the author of this paper) it is quite difficult to predict how the swivel chair would perform in mitigating vibrations on a floor with a natural frequency different from that used in the present tests. It might perform even better or it might perform worse on other floors. Empty floor modal mass and damping would also be parameters influencing the damping capacity of swivel chair(s), but this is also the case for the TMD. In any case it is not expected that swivel chairs will be used as a permanent solution for solving vibration problems in flooring-systems (for a number of reasons, although the chairs are cheap and readily at hand), but they might, for some floors, be considered for use as a temporary remedial measure taking the top of excessive vibrations until permanent and reliable solutions are found. At least the results of the investigations suggest that the swivel chair has an inherent damping capacity that can be brought into play when loaded by sandbags, which might be useful to have in mind. Also it is considered useful to have in mind that floor damping (and floor frequency) is not a constant, but a value which will change over time depending on the number of stationary people present on the floor. 5()(5(1&(6 [1]
Ellis, B.R.,On the response of long-span floors to walking loads generated by individuals and crowds, The Structural Engineer, Vol. 78 (10), pp. 1-25, 2000
[2]
Falati, S.,The contribution of non-structural components to the overall dynamic behaviour of concrete floor slabs, Ph.D.-thesis, University of Oxford, Oxford, UK, 1999
[3]
Ellis, B.R., and Ji. T., Human-structure interaction in vertical vibrations, Proceedings of the ICE: Structures and Buildings, Vol. 122, pp.1-9, 1997
[4]
Reynolds, P., Pavic, A., and Ibrahim, Z., Changes of modal properties of a stadium structure occupied by a crowd, 22nd International Modal Analysis Conference (IMAC XXII), Dearborn, Detroit, USA, 2004
[5]
Pedersen, L. Updating of the dynamic model of floors carrying stationary humans, Proceedings of the 1st IOMAC, edited by Rune Brincker and Nis Møller, Copenhagen, pp. 421-428, 2005
[6]
Sachse, R., Pavic, A., and Prichard, S., The Influence of a Group of Humans on Modal Properties of a Structure, Structural Dynamics – EURODYN2002, Grundmann & Schuëllers (eds), Swets & Zeitlinger, Lisse, ISBN 90 5809 510 X, pp.1241-1246, 2002
[7]
Pedersen, L. Damping added to floors by seated crowds of people, SPIE’s Proceeding of 13 Annual Conference on Smart Structures and Materials, Damping and Isolation, San Diego, California, USA, 2006
th
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302 [8]
Bachmann, H., and Weber, B.,Tuned mass absorbers for slivelys structures, Structural Engineering International, Journal of IABSE, Vol. 5, No. 1, pp. 31-36, 1995
[9]
Reynolds, P., Diaz, I.M., and Nyawako, D., Vibration testing and active control of an office floor, 27 International Modal Analysis Conference (IMAC XXVII), Orlando, Florida, USA, 2009
[10]
Hartog, J.P.,Mechanical Vibration, 4th edn, McGraw-Hill, New York, 1956
th
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Constrained Layer Damping Test Results for Aircraft Landing Gear
Trevor Collins, Graduate Research Assistant Dr. Kevin Kochersberger, Research Associate Professor Virginia Tech, Mechanical Engineering Blacksburg, VA 24061 NOMENCLATURE
E * - Complex Elastic Modulus G * - Complex Shear Modulus η - Loss Factor D – Distance between neutral axis of host structure and composite system
Es , Ev* , Ec - Elastic modulus of structure, viscoelastic material, and constraining layer, respectively hs , hv , hc - Thickness of structure, viscoelastic layer, and constraining layer, respectively g v* - Shear parameter
Cn - Rao correction factor ξ - Beam eigenvalue Yr (x ) - Cantilever beam mode shape Wd - Energy dissipated by viscoelastic material U - Max strain energy of viscoelastic material
ABSTRACT In aircraft, weight reduction represents one of the principal design goals, and landing gear design is no exception. Accounting for 3 – 7% of an aircraft’s weight, the landing gear is essentially dead weight after takeoff, and so reducing this weight becomes a priority of aircraft design. In addition to keeping the weight low, fixed gear designs can add significant drag if the design has not been optimized. The ideal landing gear should be low weight and low drag, but these criteria are typically at odds with a requirement for absorbing landing loads and preventing rebound. The use of constrained layer visco-elastic damping on landing gear structural members is a new application since historic use of constrained layer damping has been found on thin plate-like structures. Benefits of low weight and low drag are achievable using the conformal treatment, and this paper investigates specific constrained layer damping applications for cantilever-loaded spring steel landing gear. The design of the damped system considers the high stiffness and low surface area typical on a cantilever landing gear leg. Damping levels are examined for a 163 kg. aircraft with and without a Dyad 606 constrained layer damping treatment on the main and nose gear members. A 29% increase in damping was observed on the main landing gear, and a 25% increase in damping was observed on the nose gear when the treatment was applied. A full aircraft drop test is performed that showed inconclusive results in damping.
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_28, © The Society for Experimental Mechanics, Inc. 2011
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304 INTRODUCTION Traditional small aircraft landing gear systems employ energy damping devices such as springs or dashpots, bushings, or shock absorbers. It is the purpose of this paper to examine a constrained layer damping treatment that represents a lower weight, more aerodynamic, and more cost effective solution to providing damping. Constrained layer damping treatments are more commonly found on plate-like structures and are not typically found on stiff, structural members such as a cantilever spring landing gear (Figure 1). This application represents a unique use of the constrained layer treatment which is well suited to aircraft landing gear design, in general. Viscoelasic materials, when used in a constrained layer damping application, provide a material damping substitute for more massive, high drag damping components. Viscoelastic material properties are defined in the complex domain, having a real part, associated with the elastic behavior of the material, and an imaginary part, associated with viscous material behavior. Complex moduli are generally modeled as
E* = E '+iE" = E ' (1 + iη ) G* = G '+iG" = G ' (1 + iη )
(1)
where the real part of the modulus is called the storage modulus and the imaginary part called the loss modulus. The loss factor, η , is a measurement of the ability of a viscoelastic polymer to dissipate energy and is generally the parameter of choice when describing damping characteristics of viscoelastic polymers. Customer requirements such as weight, cost, aerodynamics, and operational environment limit the range and type of viscoelastic materials which can be used in a landing gear application. For this reason, common rubbers, namely butyl, silicone, buna-n/nitrile, vinyl, and SBR rubbers were cyclically tested, as well as Dyad 601 and 606, manufactured by SoundCoat, to determine the best materials for the application. The highest performing material, Dyad 606, was applied to a 163 kg. aircraft and bounce tests were used to determine the effects of the treatment independently on the nose and main gears. A drop test was also performed to determine rebound, maximum acceleration and damping.
Figure 1. Cantilever type, leaf spring main landing gear leg.
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305 GENERAL VISCOELASTIC THEORY Ross, Kerwin, and Ungar developed one of the earliest damping models for three-layered sandwich beams based on damping of flexural waves by a constrained viscoelastic layer [1,2]. They employed several major assumptions, including: - For the entire composite structure cross section, there is a neutral axis whose location varies with frequency - There is no slipping between the elastic and viscoelastic layers at their interfaces - The major part of the damping is due to the shearing of the viscoelastic material, whose shear modulus is represented by complex quantities in terms of real shear moduli and loss factors - The elastic layers displaced laterally the same amount - The beam is simply supported and vibrating at a natural frequency, or the beam is infinitely long so that the end effects may be neglected These assumptions apply to any constrained layer damping treatment applied to a rectangular beam. Figure 2 illustrates an example system which the Ross, Kerwin, and Ungar (RKU) equations could be applied to. This laminate beam system is also the lay-up for the cantilever beam used to test materials in subsequent sections of this paper.
Figure 2. Three layer cantilever beam with host beam, viscoelastic layer, and constraining layer clearly defined. Comparison between experimental data and RKU theory have shown that results from theory correlate well to experiment. The model is represented by a complex flexural rigidity, (EI)*, where the ‘*’ denotes a complex quantity, given by
Ev hc2 (d − D) Eh Eh Eh ( EI )* = + + − 12 + E s hs D 2 * 12 12 12 (1 + g v ) 3 s s
* 3 v v
3 c c
+ Ev* hv (hvs − D) 2 + Ec hc (d − D) 2
(2)
ª E h (hvs − D) º ª (d − D) º −« + Ec hc (d − D)» « * » 2 ¬ ¼ ¬ (1 + g v ) ¼ * v v
where D is the distance from the neutral axis of the three layer system to the neutral axis of the host beam,
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306
d E v* hv (hvs − ) + g v* ( E v* hv hvs + E c hc d ) 2 D= * E h E s hs + v v + g v* ( E s hs + E v* hs + E c hc ) 2 h + hv hvs = s 2 * Gv * gv = E c hc hv p12
(3)
hs + hc 2 * In these equations Es , Ev , Ec and hs , hv , hc are the elastic moduli and thicknesses of the host structure, d = hv +
viscoelastic layer, and constraining layer, respectively. varies from very low when Gv
*
The term
g v* is known as the ‘shear parameter’ which
is small to a large number when Gv
*
is large. The term ‘p’ within the shear
th
parameter is the wave number, namely the n eigenvalue divided by the beam length. The shear parameter can also be expressed in terms of modal frequencies by:
g v* =
ξ n4 = where
ω n is the n th
Gv* L2 Ec hv hcξ n2 C n
(4)
ρ s bhsωn2 L4
modal frequency and
Es I s
Cn are correction factors determined by Rao [3] and are given in Table
1. Table 1. Rao correction factors for shear parameter in RKU equations (Jones, 2001). Boundary Conditions
Correction Factor Mode 1 1 1.4 1 0.9 1
Pinned-Pinned Clamped-Clamped Clamped-Pinned Clamped-Free Free-Free
Mode 2+ 1 1 1 1 1
For a rectangular beam with a constrained layer damping treatment, the loss factor of the system is simply the ratio of imaginary to real parts of the complex flexural rigidity [4], namely:
ηcomp =
Im( EI * ) Re( EI * )
(5)
The RKU method of analysis is simple and computationally inexpensive. It will be the primary method of analysis for loss factor predictions of cantilever beam test later in this report.
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307 MATERIAL TESTING Cyclic load testing was conducted with an ElectroPuls E1000 cyclic loading machine (Figure 3) for 10 materials. Each material was cycled at 10 Hz and a 0 - P amplitude of 0.5 mm, and the loss factors and storage moduli were determined from the hysteresis plots similar to the one shown in Figure 4 (Sun and Lu [5]). The material property results are shown in Table 2.
Figure 3. Specimen loaded in the ElectroPuls E1000 tensile testing machine
Figure 4. Hysteresis plot for Dyad 606
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308 Table 2. Complex modulus data found from cyclic load testing.
Material
Loss factor
Storage Modulus [MPa]
Dyad 606
0.898
72.0
Loss Factor * Storage -6 Modulus * 10 64
Nitrile 60
0.274
19.8
5.43
Vinyl 70
0.266
13.3
3.54
Butyl 60
0.319
8.98
2.86
Dyad 601
0.356
4.98
1.77
SBR70
0.171
10.04
1.72
Silicone 50
0.133
3.427
.46
Nitrile 40
0.131
2.822
.37
SORB
0.263
0.839
.22
Silicone 30
0.079
2.26
.18
These numbers were generated by employing two major assumptions during the cyclic load testing: 1. The stress is uniform throughout the cross section of the material and the approximation that stress is equal to the load divided by the original cross sectional area hold. 2. The strain is uniform throughout the length of the specimen and is equal to the change in position of the grips divided by the original length of the specimen. These assumptions are not entirely accurate, mostly due to lateral stresses produced by the grips which hold the material sample in place during loading. However, they are sufficient to establish relative results and allow comparison between materials to achieve a hierarchy of most effective damping material. The RKU equations indicate that the product of (loss factor) * (storage modulus) represents the damping advantage provided by the constrained layer treatment; this product is shown in Table 2. The Dyad 606 is a superior material when compared to the other materials tested, and as a result, this material was applied to the aircraft for in-situ testing. Winch
TESTING PROCEDURES The end goal of testing was to accurately compare dynamic response of the aircraft’s front and rear landing gear with and without a viscoelastic damping treatment. The aircraft was weighed in multiple orientations to determine not only the mass, but the center of gravity location. A swing test was employed to determine the pitch moment of inertia which would be used in bounce tests to extract the nose and main gear damping coefficient. Figure 5 shows the swing test configuration, and Eqns 6 and 7 are used to extract the pitching MOI.
Accel: ߠݎሷ
Figure 5. Swing test configuration
T=
2π
ωn
=
2π mgr Jo
(6)
J o = J cg + mr 2
(7)
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309 For the dynamic tests, two configurations were examined for every test: a damped and an undamped configuration. For the damped configurations, a constrained layer damping treatment was applied to the main and nose gear legs consisting of a 0.049” layer of Dyad 606 and a 0.028” aluminum constraining layer. The treatment was applied to both sides of the gear legs. The bounce tests were executed by supporting the aircraft at either the front or rear on a hard point, and exciting the aircraft in a pitching motion where the nose or main gear would provide a direct rotational stiffness and damping to the system. Figure 6 shows the accelerometer locations used to measure the response and Table 3 summarizes the equipment used in the measurements. The aircraft was harmonically excited then released, and the under-damped free response was measured. Figure 7 shows the front gear bounce test configuration, with a lateral sliding plate used to allow horizontal motion so that vertical stiffness or damping would not be influenced by other motion-induced stiffness components. Also note that the front tire (and rear tires for the rear gear test) had been removed to eliminate variability in the results due to tire pressure, and to focus on the characteristics of the gear leg only. Symmetry Plane
F(t) Left Accel
Right Accel
Left & Right Accels
Front Accel
Figure 6. Accelerometer Placement for Main and Front Gear Testing
Table 3: Testing equipment list Equipment Accelerometer DC Accel Power Supply Data Acquisition Unit
Model # Y3801D1FB20G/M001 PCB 478A01 National Instruments ENET-9234
Notes
Used with LabVIEW software
Periodic Force Pivot on Two Bolts
Fixed or Pivot Point
Figure 7. Testing Configuration for Front Gear Bounce Test
The main gear configuration was constructed with the same methodology, giving freedom of movement to the left and right gear legs (Figure 8). For both the nose and main configurations, 20 runs were completed, collecting data through the Data Acquisition Unit and LabVIEW software. Data recorded included raw, unfiltered voltages from the accelerometers as well as voltages passed through a 20 Hz Butterworth Lowpass Filter. The filter allowed unwanted noise and insignificant higher resonances in the structure to be cut out. Again, note that for all
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310 the tests a baseline was established for the original gear without damping, and a second set of tests were conducted for the gear with the viscoelastic damping layer bonded to the top and bottom gear surfaces.
Close-up of Pivot Point
Full View of Configuration
Close-up of ‘Frictionless’ Platforms
Figure 8. Configuration of Bounce Test for Main Landing Gear
DROP TEST A drop test was performed to simulate actual landing loads experienced from a 5.7 ft/s impact. Video footage captured maximum rebound heights for the main and front landing gear and accelerometers under the wings measured the response. For the drop test to be successful, the aircraft had to be released from its drop height cleanly so there was only environmental resistance as it accelerated to the ground. It was also important for the main landing gear to touch down a split second before the front gear. The reasoning behind this requirement was twofold. Firstly, the main will always touch down before the front when the aircraft is landing in actual operation. Secondly, the center of gravity was found to be set back in the body of the aircraft, so it is intuitive for the bulk of the force to dissipate its energy into the ground first through the main gear. With these requirements fulfilled, consistent and clean responses were expected. The quick release configuration used is shown in Figure 9.
Close-up of Quick Release
Figure 9. Configuration for Drop Test
The drop tests were performed at heights of 3, 4.5, and 6 inches. Six runs were completed for each height, while the DAQ was collecting data from the right and left accels under the wings. The following steps were completed before each run: • The left and right tires were inflated to their full 30 psi. • Heights of the left and right tires were checked to be equal, to ensure minimal swaying motion about the symmetry plane.
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311 • •
Height of the nose wheel was checked to be in the range of 0.5” to 2” above the main tires’ altitude. The aircraft was steadied and resting with minimal motion before the quick release was pulled.
RESULTS AND ANALYSIS The bounce and drop tests provided both useful qualitative and quantitative data. In the proceeding results, comparisons will be made between the baseline case and the viscoelastic damping layer case. In Table 4, aircraft parameters are listed:
Table 4. Parameters Used in Derivation of Damping Coefficents Parameter
Test From Which it Was Found
Moment of Inertia About C.G.
Swing test
Stiffness of Single Main Gear Arm
Given from Pre-Existing Tests
Stiffness of Front Gear Arm
Given from Pre-Existing Tests
Each landing gear was treated as a linear spring and damper that could be characterized by a second order equation of motion. This simplifying assumption was made after observing how well the response behaved utilizing the ‘frictionless’ platforms in the bounce test. The damping ratio was determined from the decaying exponential envelope of the accelerometer data. In Figure 10, a typical damped response plot shows the response for the main landing gear with the viscoelastic damping layer. Also seen is the scheme to fit the exponential envelope. With Dyad 606 Layer: Main Landing Gear, Run 19 With Dyad 606 Layer: Main Landing Gear, Run 19
Fitting the Exponential Envelope: Min Points Used For Coefficients
0.08
0.08 R Accel L Accel Min Peaks Max Peaks
0.06
0.04
0.02
0.02
0
0
y
Voltage
0.04
-0.02
-0.02
-0.04
-0.04
-0.06
-0.06
-0.08
0
0.5
1
1.5
2 Time
2.5
3
3.5
data fitted curve data fitted curve
0.06
4
-0.08
0
0.5
1
1.5
x
Figure 10. Main Gear Response and Exponential Fitting Scheme
Damping coefficients for all configurations were calculated and listed in Tables 5 and 6. Note that a 29% increase in damping is realized for the main gear, and a 25% increase is realized for the nose gear with the damping treatment applied. This is a significant result, and lends credibility to the damping treatment concept.
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312 Table 5. Main Landing Gear Damping
Baseline
With Dyad 606 Damping Layer
Run 4 Run 15 Run 20
21.09 20.60 20.67
0.039 0.039 0.035
146.61 146.61 131.58
--------------------------------
---------------------------------
------------------------------------
Average:
20.79
0.038
141.60
Run 7 Run 18 Run 19
23.66 24.00 25.53
0.054 0.044 0.049
203.00 165.41 184.21
----------------------------
-------------------------------
--------------------------------
24.40
0.049
184.21
Average:
Table 6. Front Landing Gear Damping
Baseline
With Dyad 606
Run 23 Run 25
25.58 24.70
0.016 0.014
25.96 22.72
--------------------------------
---------------------------------
------------------------------------
Average:
25.14
0.015
24.34
Run 20 Run 21
28.86 28.04
0.020 0.019
32.45 30.83
----------------------------
-------------------------------
--------------------------------
28.45
0.020
31.64
Average:
The drop test uncovered both quantitative and qualitative data. The data was especially informative for the 6 inch height, where the rebounding characteristics could more easily be seen. The frames of video footage that proved useful occurred when the tires of the aircraft were fully compressed or at their maximum rebound height. In the frames of Figure 11, the baseline case of the main landing gear is shown rebounding. The board behind is a reference to measure inches. In this figure, it appears each consecutive maximum height is about half of the one before it. Note in proceeding drop test Figures, the first frame is the aircraft resting before the drop.
Figure 11. Main Landing Gear 6” Drop Test ~ Baseline Case
In Figure 12 with the viscoelastic damping layer applied, the maximum rebound heights closely match the baseline case above. If the damping layer is in fact lessening the consecutive maximum heights, the change is difficult to observe through the frames. Figure 13 shows the accelerometer traces for these drop tests – again, the difference in responses is negligible. A reason for this may be the increased stiffness resulting from the damping treatment – the system did not just gain damping but stiffness as well, resulting in an a livelier response.
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Figure 12. Main Landing Gear 6” Drop Test ~ Dyad 606 Layer Case
Figure 13. Accel plots for main gear without damping (left) and with damping treatment (right)
For the front landing gear (Figures 14 and 15), it was observed that the third rebound in the damped case was considerably smaller than in the undamped case – a very positive result if the aircraft is to maintain directional stability on rollout during a hard landing. Acceleration data for the nose was not acquired during the drop tests, but it is expected to confirm that energy was dissipated at a higher rate when the treatment was applied.
Figure 14. Front Landing Gear 6” Drop Test ~ Baseline Case
Figure 15. Front Landing Gear 6” Drop Test ~ Dyad 606 Layer Case
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314 CONCLUSIONS Many aircraft equipped with non-retractable landing gear use a cantilever beam system to provide stiffness to the gear and very limited damping. These types of gear are popular due to their minimal weight and low aerodynamic drag, however their simplicity comes with a disadvantage: damping is very low due to the method of construction and the choice of material the gear is fabricated from. An evaluation of a low cost, low drag and low weight damping treatment has been performed that is easily applied to existing cantilever spring-type landing gear systems, and provides a measurable increase in damping. Constrained layer damping treatments are not typically used for stiff, load bearing structures. However, this paper shows that a constrained layer treatment is effective for increasing the damping of a landing gear system on a small aircraft that utilizes cantilever spring landing gear. In this test, the landing gear of a 163 kg. aircraft was modified with a constrained layer damping treatment. A treatment was applied consisting of a 0.049” layer of Dyad 606 and a 0.028” aluminum constraining layer, bonded to both sides of the gear legs. A bounce test that individually excited the nose and main gear legs with and without the treatment showed a substantial increase in damping with the treatment, from 25% for the nose gear to 29% for the main gear. These encouraging results were tempered by the drop test results that showed an inconclusive benefit for the main gear. However, the nose gear exhibited a substantially better response after two rebound cycles, with less rebound and more ground contact that could impact the stability of the vehicle in a hard landing. The added stiffness to the gear resulting from the constraining layer and damping material is suspected of changing the baseline dynamics, and a better test would keep the stiffness constant while adding only damping to the system. A dedicated damped gear system should take this into consideration, altering the design of the baseline gear to accommodate the constrained layer system.
REFERENCES 1) Ross, D., Ungar, E.E., Kerwin, E.M. Jr. 1959. Damping of plate flexural vibrations by means of viscoelastic laminate. In ASME (Ed.). Structural Damping (pp. 49-88). New York: ASME. 2) Jones, D. I. G. 2001. Handbook of Viscoelastic Vibration Damping. West Sussex, England: John Wiley and Sons, LTD. 3) Rao, Y., V.K. Sadasiva, and Nakra, B.C. 1974. Vibrations of unsymmetrical sandwich beams and plates with viscoelastic plates. Journal of Sound and Vibration, 34(3), 309-326. 4) Hao, M. and M. D. Rao. 2005. Vibration and Damping Analysis of a Sandwich Beam Containing a Viscoelastic Constraining Layer. Journal of Composite Materials. 39: 18:1621-1643. 5) Sun, C. T. and Y. P. Lu. 1995. Vibration Damping of Structural Elements. Englewood Cliffs, NJ: Prentice Hall
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
$6ORVKLQJ$EVRUEHUZLWKD)OH[LEOH&RQWDLQHU M. Gradinscak, S.E. Semercigil and Ö.F. Turan Victoria University, School of Engineering and Science Footscray Park Campus, PO Box 14428 MC Melbourne, Victoria 8001 AUSTRALIA
$%675$&7 Liquid sloshing may be employed for vibration control of resonant structures, similar to that of a classical tuned vibration absorber. For such a case, the sloshing frequency is tuned at a critical frequency of the structure in order to gain the benefits of the pressure forces as control forces. Such an absorber is practically free of maintenance. The work presented in this paper utilizes a flexible container partially filled with water, as the sloshing absorber. Numerical predictions are presented where a “tuned” flexible container can be advantageous over a rigid container for effective control. ,1752'8&7,21 The concept of using sloshing forces for control of light and flexible structures, has been a subject of interest in the literature. Abe et al. [1] reported effective control of structural vibration using a U-tube with a variable orifice passage. Seto and Modi [2] used fluid-structure interaction to control wind-induced instabilities. Reed et al. [3] investigated tuned liquid dampers under large amplitude excitation. Nomura [4] and Yamamoto and Kawahara [5] suggested finite elements with moving grids, based on the arbitrary Lagrangian-Eulerian formulation. Anderson et al. [6] proposed a sloshing absorber of standing-wave type. Sakamoto et al. [7] proposed a tuned sloshing damper using an electro-rheological fluid, utilizing both analytical and experimental methods. All preceding works cited here deal with rigid containers as the sloshing absorber. There are no reported attempts in the literature to explore the possibility of employing a flexible container. This paper summarises the current research at Victoria University on tuning flexible containers for structural control. Container flexibility introduces an additional tuning effect to that of an already existing tuning issue between the sloshing liquid and the structure to be controlled. The additional tuning is required to account for the energetic behaviour of the flexible container, in response to the dynamic forces from the sloshing liquid and the oscillations of the structure to be controlled. Extensive numerical predictions have been completed to investigate the problem of tuning. Selective cases are discussed below. Full details may be found in References 8 and 9. No experiments are presented here. However, the design of a flexible container to achieve sloshing suppression through tuning, has been verified with experimental observations earlier [9]. 180(5,&$/352&('85( A standard finite element analysis package, ANSYS [10], was used to model the dynamics of the container, sloshing liquid and the structure to be controlled. A schematic view of the model with a grid size of 50mm x 50mm, is given in Figure 1(a). The sloshing absorber consisted of a rectangular aluminium container of 1 mm wall thickness and 1.6m x 0.4m x 0.4m in length, width and height, respectively. Two-dimensional rectangular shell elements were used for the container walls with 1.5% critical damping in the fundamental mode (effects of different damping values will be presented at the end of the discussion in Section 3). The container was filled with water to a depth of 0.3 m, corresponding to a mass of approximately 192 kg. The liquid was modelled using threedimensional brick elements. Liquid had no viscous dissipation.
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_29, © The Society for Experimental Mechanics, Inc. 2011
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Fluid-container interaction was achieved by coupling the displacement of the liquid and container walls in the direction normal to the container walls. A solid element was used to model the mass of the structure. The structure was attached with four springs to a rigid wall. The container was attached rigidly on the structure. The mass of the structure was 2000 kg which resulted in a mass ratio of slightly under 10% between the sloshing absorber and the structure to be controlled. The sloshing absorber was orientated such that liquid sloshing was induced in the Z-direction in response to a 5mm initial displacement given to the structure in the Y-direction. A transient solution was then obtained (with a time step of 0.01s and for a total duration of 20 s) by numerically integrating the resulting differential equations. The concept of using liquid sloshing in flexible container to control structural vibration is similar to that of using a tuned absorber. The fundamental sloshing frequency of a liquid in a rigid container of the same dimensions is approximately 1.34 Hz [6]. The natural frequency of the structure was also set (tuned) to this value to achieve strong interaction, similar to that of a classical tuned absorber. In the case of a flexible container, there is a second level of tuning between the structure to be controlled and the container filled with liquid. As mentioned earlier, the primary objective of the presented work is to demonstrate the effect of this particular secondary tuning on structural response. Hence, case runs include a range of structural critical frequencies of the flexible container. Different frequencies of the flexible container were obtained by symmetrically adding two lumped masses in the middle of the 1.6 m length, at the free top edge.
0.006
',63/$&(0(17>P@
0.004
0.002
0
-0.002
-0.004
-0.006 0.0
2.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
18.0
20.0
7,0(>V@
(a)
(b)
Figure 1. Showing (a) the computational model and (b) the displacement history y(t) of the structure after an initial displacement. 180(5,&$/35(',&7,216 In Figure 1(b), the displacement history of the structure in Y-direction is given without the sloshing absorber. Since no damping is included in the structure’s model, the induced initial displacement of 5 mm remains unchanged indefinitely. Figure 1(b) is included here as the comparison base for all cases with the sloshing absorber. The numerical predictions presented in Figure 2 represent the displacement histories of four cases. Frames (a) to (d) correspond to the cases of rigid container, flexible container with no added mass, with 9 kg and with 13 kg added mass, in descending order. As mentioned in the preceding section, additional masses are used to vary the structural critical frequencies of the flexible container. The 9 kg mass case represents the tuned flexible container for most effective suppression of sloshing [8,9]. No mass and 13 kg mass cases are outside this tuning, at varying severities. With the exception of the rigid container case in Figure 2(a), each frame has the displacement histories of four nodes, two of the flexible container and two of the liquid. The histories of the two container nodes are the top and
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317
bottom ones (dark blue and red), whereas those of the liquid nodes (light blue and magenta) are in the middle. All nodes are selected to be at the level of the free surface, in the middle of the long side of the container. The container displacement is in Y-direction, whereas the liquid motion takes place in Z-direction. In Figure 3, the displacement histories of the structure are presented, by following the same order of Figure 2. It is important to remember that the uncontrolled displacement history in Figure 1(b), is the comparison base for all cases in Figure 3. Again, following the same order, the frequency spectra are given in Figure 4. In this figure, liquid sloshing is marked with red, flexible container displacement is marked with blue, and the structural displacement is marked with green. Sloshing (red) corresponds to the difference of the displacements of the two free surface liquid nodes, to present the most detrimental out-of-phase surface motion. In Figure 2(a), displacement of the two liquid nodes are given which experience a perfectly out-of-phase motion, in response to the initial displacement of the structure. The peak-to-peak magnitude at the wall is 60 mm. The clear beat in the envelope is the result of “tuning” the sloshing frequency and the natural frequency of the structure at 1.34 Hz. The beat indicates strong interaction, and the back-and-forth flow of energy between the liquid and the structure. The beat envelope is at a peak when most of the kinetic energy is with the sloshing liquid. The beat envelope diminishes to approximately zero when the kinetic energy is transferred to the structure. The same clear beat is also apparent in the displacement history of the structure in Figure 3(a), but out-of-phase with that of Figure 2(a). The peak displacement of the structure is 5 mm, the same as in Figure 1(b), since neither the structure nor the liquid has any means of dissipating energy. In Figure 4(a), the tuning is clearly marked with the sharp trough at 1.34 Hz, along with the two spectral peaks at approximately 1.2 Hz and 1.45 Hz (which are responsible for the beat envelope with an approximately 4-second period). In Figure 2(b), the flexible container case is shown with no added mass. The container is clearly off-tuned.The top and bottom histories (dark blue and red), corresponding to the displacement of the container walls, show large deflections at frequencies clearly different than those of the liquid nodes (light blue and magenta). Peak-to-peak liquid displacements reach values up to 120 mm, decreasing to about 80 mm, towards the end of the 20 s simulation period. These values are clearly larger than the 60 mm displacement of the rigid container case. In Figure 3(b), the displacement history of the structure, the beat is much less apparent than the case discussed for the rigid container. The longer beat period of about 10 s, is the result of the double peaks at approximately 1.4 Hz and 1.5 Hz in Figure 4(b). A weaker beat is also discernable due to the interaction of spectral peaks at about 1 Hz and 1.5Hz, with a period of about 2 s. As a result of container oscillations, the peak displacement of the structure is attenuated to 4 mm, from the initial displacement of 5 mm, by the end of the 20-second simulation. Hence, the amplification of the sloshing amplitude is not necessarily detrimental to the control action on the structure. Despite the fact that the container is clearly off-tuned, it is interesting to note that the oscillation frequency of the structure (green) within the envelope is quite comparable to that of sloshing liquid (red), in Figure 4(b). The offtuned container has spectral peaks at the same frequencies forced by the sloshing liquid and the structure, and additional spectral peaks quite outside of those of the liquid and the structure at 0.8 Hz, 1.7 Hz, 2.4 Hz and 2.7 Hz. The spectral distributions in Figure 4(b) are rather complicated, as compared to those in Figure 4(a), due to the presence of multiple modes of the flexible container in this off-tuned case. Multiple spectral peaks make it difficult to comment on their relative importance, with the exception of the original sloshing frequency of approximately 1.34 Hz. As a result of lack of any coherent interaction with its flexible container, sloshing largely reverts to its original frequency, also forcing a response from both the structure and the container at this frequency. Any addition of mass to the container has an immediate effect of lowering its critical frequencies quite dramatically. The double peak, pointed out earlier in the response of the structure, is further split apart and continue to be responsible for the beat in the histories of the structural displacement. Observations with the 9-kg added mass, correspond to the most effective tuning reported earlier [8, 9] for the suppression of the liquid motion. In Figure 3(c), the liquid surface moves in almost perfect phase, resulting in a virtual elimination of the most damaging liquid sloshing in the fundamental mode. In Figure 3(c), the beat period is about 5.5 s, and the peak of the last beat around 18 s, is reduced to smaller than 1.5 mm. The two dominant peaks are now clearly separated for both the structure and the liquid in Figure 4(c). In a way, these two peaks around 1.3 Hz and 1.45 Hz are quite similar to those of the rigid container in Figure 4(a). The difference, of
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course, is the drastically reduced magnitudes as a result of the container flexibility. It is also interesting to note that the spectral distribution of the flexible container now favours lower frequencies. With further increases of the added mass from 9 kg, the flexible container starts to de-tune clearly from the dynamics of sloshing - 13 kg is given in Figures 2(d), 3(d) and 4(d). However, the beat in the structure’s response is still present in Figure 3(d), with a further reduction of the peak response to about 1 mm, around 17 seconds. Hence, the liquid in the container and the structure seem to interact quite strongly, almost ignoring the loss of tuning of the container. This observation is also supported by the spectral distribution given in Figure 4(d). A collective summary of relevant simulations is given in Figure 5. In Figure 5(a), the root mean square (rms) averages of sloshing amplitudes are given for different values of added mass, whereas the rms averages are shown in Figure 5(b) for the structural displacement. The various values used equivalent viscous damping ratio of flexible container, are 0 (), 1.5% (), 5% (S), and 10% (z). It should be mentioned here that rms averages are presented here as a means of rank ordering effectiveness. In Figure 5(a), except for the no added mass case, a flexible container gives a smaller sloshing magnitude than the rigid container(ŷŷ). The smallest sloshing is around 7kg to 9 kg which is in close agreement with the tuning suggested earlier to suppress sloshing [9]. The tuning effect is most apparent for the undamped container (), and it diminishes with increasing damping. For the 10% critical damping case (z), little difference could be observed within the range from 5 kg to 13 kg. In Figure 5(b), all sloshing absorbers, including the rigid container, suggest a smaller structural displacement as compared to the uncontrolled case (ŷŷ) which retains the 5-mm initial displacement indefinitely. A negative slope can be observed in all flexible container cases, with the exception of the undamped container (), for increasing added mass. Again, except for the change from no mass to 3 kg, the change is gradual with diminishing sensitivity as the container’s damping increases. &21&/86,216 Numerical predictions with standard finite element analysis are presented to show the effect of the container flexibility of a sloshing absorber in suppressing the transient oscillations of a resonant structure. The mass of the sloshing absorber is limited to about 10% of that of the structure. Flexible container has the only damping in the system. Although, there are trends in the simulated response which could not yet be reasoned with confidence, attenuations in the order of 80% are suggested in the transient response of the structure (with 1.5% structural damping of the container) with the use of container flexibility as a design parameter. Reported observations are certainly encouraging for further investigation. Part of this further investigation is to attempt to validate the numerical predictions with prototype testing. 5()(5(1&(6 1. ABE, M., KIMURA, S. and FUJINO, Y. (1996), “Semi-active Tuned Column Damper with Variable Orifice Openings”, Third International Conference on Motion and Vibration Controls, Chiba, 7-11. 2. SETO, M.L. and MODI, V.J. (1997), “A Numerical Approach to Liquid Sloshing Dynamics and Control of FluidStructure Interaction Instabilities”, The American Society of Mechanical Engineers, Fluid Eng. Div. Summer Meeting, Paper no. FEDSM97-3302. 3. REED, D., YEH, H., YU, J. and GARDARSSON, S. (1998), “Tuned Liquid Dampers Under Large Amplitude Excitation”, Journal of Wind Engineering and Industrial Aerodynamics 74-76, 923-930. 4. NOMURA, T. (1994), “ALE Finite Element Computations of Fluid-Structure Interaction Problems, Computer Methods and Applied Mechanics and Engineering 112, 291-308. 5. YAMAMOTO, K. and KAWAHARA, M. (1999), “Structural Oscillation Control Using Tuned Liquid Dampers, Computers and Structures 71, 435-446. 6. ANDERSON, J.G., SEMERCIGIL, S.E. and TURAN, Ö.F. (2000a), ”A Standing-Wave Type Sloshing Absorber to Control Transient Oscillations”, Journal of Sound and Vibration, 232(5), 839-856. 7. SAKAMOTO, D., OSHIMA, N. and FUKUDA, T., (2001), “Tuned Sloshing Damper Using Electro-Rheological Fluid”, Smart Materials and Structures.10, PII:SO964-1726(01)27860-4, 963-969. 8. GRADINSCAK, M., SEMERCIGIL, S.E. and TURAN, Ö.F. (2006), “Liquid Sloshing in Flexible Containers, Part 1: Tuning Container Flexibility for Sloshing Control”, Fifth International Conference on CFD in the Process Industries, CSIRO, Melbourne, Australia, 13-15 December 2006. 9. GRADINSCAK, M., A study of sloshing with container flexibility”, Ph.D., Victoria University, Australia 2009 10. ANSYS 6.1 Users Manual, (2002), ANSYS Inc. Houston, Texas, USA.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
(IIHFWLYH9LEUDWLRQ6XSSUHVVLRQRID0DQHXYHULQJ7ZR/LQN)OH[LEOH$UP ZLWKDQ(YHQW%DVHG6WLIIQHVV&RQWUROOHU Abdullah Özer1,2 and S. Eren Semercigil2 1
Gyeongsang National University, School of Mechanical and Aerospace Engineering, Jinju, Gyeongnam 660-701,KOREA 2 Victoria University, School of Engineering and Science Footscray Park Campus, PO Box 14428 MC Melbourne, Victoria 8001 AUSTRALIA
$%675$&7 Vibration control of a maneuvering flexible robotic arm is a challenging task in the presence of changing structural dynamics which has to deal with measurement inaccuracies and complex modeling efforts. This paper presents an effective and versatile controller for a maneuvering flexible arm. Proposed Variable Stiffness Control (VSC) is stable, due to its being dissipative in nature. The technique is suitable to be implemented as an add-on controller to existing robots, and it requires no additional hardware. Control is based on the detection of a kinematic event, peak relative displacement, rather than an accurate knowledge of structural dynamics. Hence, although there may not be a claim for the suggested control to be the most effective, it certainly represents significant practical advantages for cases where there may be structural uncertainties. ,1752'8&7,21 For increased productivity, it is important to use lightweight manipulators with high payload-to-weight ratios. Lightweight, and therefore flexible, robots can maneuver at higher speeds, consume less energy and have safer interaction with their environments. However, the flexibility of these lightweight manipulators inevitably induces undesirable vibrations at the end effector. On the other hand, demanding tasks such as plasma-welding, lasercutting, or high-speed operations in the presence of obstacles, require advanced trajectory-tracking control capabilities. Thus, vibration control of flexible manipulators represents a major importance in lightweight robotic applications. The flexibility of lightweight robots can be inherent at their joints (electro-mechanical drives, in series with other units such as harmonic drives, couplings and belt drives) and in their slender linkages. Oscillations at these components can be triggered either by external disturbances or by the motion of the robot itself. If not controlled effectively, these oscillations not only lead to end point positioning and tracking inaccuracies but also cause long idle waiting periods between tasks, to perform the intended operation safely and accurately.Vibration control of a moving arm is a more challenging task than that of a stationary arm. The challenge is the result of the sensitivity of the control techniques to the changes in structural parameters, and modelling and measurement inaccuracies. The techniques to control flexible manipulators can be grouped in two main categories: passive control techniques and active control techniques. A passive controller designates the mass, stiffness and energy dissipation properties of the structure to minimize dynamic response. The primary advantage of passive controllers is their simplicity. However, the effectiveness of a passive controller may deteriorate drastically for varying design conditions [1-4]. Active control may be more feasible for flexible manipulators, where the control action is implemented by sensing the structural response and producing the required corrective forces using actuators. As a result, stability of the control technique becomes critically important, since energy is added for control purposes. While inverse dynamics techniques can overcome the stability issues, they assume the existence of an exact structural model. Parametric uncertainties, unmodelled structural dynamics and neglected time delays in actuators can all affect the accuracy of the required model.
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_30, © The Society for Experimental Mechanics, Inc. 2011
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Most reported research in literature concentrate on the one-link flexible robots and stationary arms [5-10]. The reason for such limitation is the need for demanding computational and hardware requirements for multi-link arms in manuever. Therefore, there is room to have a control technique which is relatively insensitive of the structural dynamics, stable and simple to implement. In this paper, such a control is reported. A two-link flexible manipulator is considered here. The structural dynamics has been derived using standard finite element method [11]. The flexibility of the manipulator is assumed to be coming from the slender structural components (beams) and joints. The proposed vibration controller accounts for the excessive oscillations by employing the Variable Stiffness Control (VSC) technique in co-operation with a resident motion controller. The VSC is stable and its performance is relatively independent of system parameters. It is also suitable to be implemented as an add-on controller. It does not require any additional hardware. &21752/$1'180(5,&$/,03/(0(17$7,21 In this section, a brief description of the numerical model and the simulation procedure is given, as they form the base for developing the control scheme for prototype implementation. A more detailed discussion is available in Reference 13. A single degree of-freedom undamped oscillator is used to illustrate the control. In Figure 1(a), the oscillator has two parallel springs, one passive (K-'K) and one active ('K). The system has a total stiffness of “K” when the clamp of the active spring is on, and “K-'K” when the clamp is off. M represents the mass of the oscillator while coordinate x indicates its absolute displacement. The Variable Stiffness Control (VSC) is based on changing the effective stiffness between the two states, at instances when it is most beneficial for control. The active spring is kept clamped until the instant of peak displacement, Xo, following a transient disturbance. At this instant, the active spring is unclamped and re-clamped instantaneously without giving a chance of any displacement of mass M. Through this instantaneous unclamping-clamping action, the active spring releases all the potential energy it had gained. Along with this dissipated energy, the active spring is now in an undeformed state when the clamp is re-applied while the mass M is still displaced by Xo from its original equilibrium position. Thus, the control action produces a second effect with its shifted equilibrium of the active spring. The shifted equilibrium position imposes a constant restoring force of amplitude “'KXo” to oppose the velocity of oscillations, similar to a case of Coulomb friction. It can be analytically shown that the system oscillations for transient oscillations can be eliminated with two actuations if the stiffness ratio, ǻK/K, is chosen to be 0.5 for a single degree-of-freedom oscillator [13]. The control action may also be shown in the force-displacement diagram in Figure 1(b). Oscillations start at the origin and follow State 1, with full stiffness K, until the maximum displacement Xo. At Xo, the active spring is unclamped, and the oscillator’s response jumps from “a” to “b”, into State 2 with an effective stiffness of K-'K. When the active spring recovers, the effective stiffness again becomes K and oscillations resume in State 3. Potential energy dissipated by the control action is represented by the area of the trapezoid bounded by States a, b and the two parallel lines representing the full stiffness K (States 1 and 3) in the first quadrant. The equilibrium shift that produces the constant restoring force, is the result of shift in the state of the system from State 1 to State 3. The shifted equilibrium position will be at a smaller displacement for each subsequent actuation, eventually leading to zero displacement when oscillations diminish (unlike that of a Coulomb damping). The two-link flexible robotic arm model under investigation, is shown in Figure 1(c). The model consists of two flexible beams and joints. The passive spring constants, Kelbow-'Kelbow and Kbase-'Kbase, represent compliant elbow and base joints while, 'Kelbow and 'Kbase model the active components of the controller. Finite element method has been used (with 10 elements for each link) to model the dynamics of the manipulator [11]. Torsional viscous dampers to account for frictional dissipation and structural damping (0.5% critical damping for the first two modes) are also included in the moving arm model to represent inherent energy dissipation Bathe[14]. The arm is assumed to move in the horizontal plane. Lumped masses located at the tip, Mtip, and at the elbow, Mmotor, model the payload and motor mass. The rotary inertia effects at both base and elbow are neglected in the model, and the mass of the base actuator is assumed to be a part of the fixed base. The flexible arms are assumed to be Euler-Bernoulli beams.
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Figure 1 Showing (a). Undamped oscillator with variable stiffness, (b) force-displacement graph and (c) the o o model and desired path ( ) of the two-link flexible robot arm used for simulations (D varies from 0 to 120 in 3s, with a constant speed of ʌ/3 rad/s).
Numerical simulations have been obtained using a custom coded program in MATLAB [15]. First, the program produces the global matrices. Then direct numerical numerical integration is performed using the Newmark-E scheme [16]. As the arm moves, the elbow angle, D in Figure 1(c), changes in time. As a result, the system matrices vary in time. Hence, the system matrices are updated continually at time increments of 0.0167 s (which corresponds to an approximately 1o of elbow angle rotation in the prescribed maneuver). Figure 1(c) indicates the motion profile used for the numerical tests. The interaction between the control torques and the driving torque is neglected during the numerical simulations. At zero time, it is assumed that the arm is passing smoothly through zero degree elbow angle, and at this instant the arm is disturbed with a tip impact
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represented with a half-sinusoid tip disturbance at the fundamental natural frequency, in order to produce largest transient oscillations for the uncontrolled system. The arm parameters used in the simulations are listed in Table 1. As the arm folds in, its first natural frequency changes from 12.52 rad/s, for 0o elbow angle, to 20.08 rad/s, for 120o elbow angle [13]. An integration step of 0.001 s has been found to be small enough for all elbow angles. The stiffness ratio, ǻK/K of 0.5 is used based on earlier observations [13]. Table 1. Structural parameters used for simulation where both beams (arms) are identical. Length : 0.5 m Width of beam : 0.05 m Thickness of beam : 0.00625 m Bending stiffness of beam (EI) : 73 Nm2 Mass/length of beam : 0.85 kg/m Tip mass, Mtip : 0.5 kg Actuator mass, Mmotor : 0.5 kg Passive elbow stiffness, Kelbow-'Kelbow : 100 Nm/rad Active elbow stiffness, 'Kelbow : 100 Nm/rad Passive base stiffness, Kbase-'Kbase : 200 Nm/rad Active base stiffness, 'Kbase: : 200 Nm/rad Equivalent viscous damping ratio : 0.005 Elbow viscous damping coefficient : 1 Nm/rad/s Base viscous damping coefficient : 3 Nm/rad/s 180(5,&$/35(',&7,216 Numerical simulation results are presented for uncontrolled and dual-controlled cases where both of the joint actuators are used for control. As indicated in Figure 1(c), the tip arm is assumed to move smoothly from an o o elbow angle D of 0 to 120 in three seconds with a constant speed of ʌ/3 rad/sec. When the arm passes through the fully open orientation, the tip of Link 2 is struck with a half-sinusoid normal force of 5 N at the fundamental frequency. Tip displacement histories are given in Figure 2 for the uncontrolled (frames a and b) and controlled (frames c and d), respectively. The two columns indicate the histories of the X and Y coordinates, along with the desired displacements. In all figures, the dashed lines (- - - -) represent the desired path while the solid lines ( _____ ) representing the position of the tip. The deviation from the desired path is much clearer in Y displacement, in Figure 2(b), reaching about 6 cm amplitude at the beginning of motion as this coordinate coincides with the direction of the impact. In X displacement plot in Figure 2(a), the deviations are clearer through the middle of the path, as the transverse direction starts to coincide with this axis. At the end of the motion, the oscillations clearly exist but at smaller amplitudes due to structural damping and dissipation at the joints. The improvement is clear for the controlled cases in Figures 2(c) and 2(d). After about one and a half cycles, deviations from the desired path virtually diminish.
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To show the insensitivity of the proposed control technique to the uncertainties such as magnitude of the external excitation, simulations have been repeated by increasing the magnitude of the tip impact by ten times to 50 N. Uncontrolled and controlled cases are again presented in Figure 3 in an identical format to that of Figure 2. In Figure 3, despite the fact that the tip of the arm receives a ten times larger impact than the earlier case, the controlled case starts to follow the desired path in a comparable time period to that with 5 N excitation. The initial deviations, of course, are now much larger due to higher magnitude of the impact.
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In order to show the insensitivity of the proposed controller to the changes in structural parameters, this time the length of the second link length is doubled from 0.5 m to 1 m. Space based robots can be an example such slender manipulators as they are necessarily long and of light weight. As a consequence, significant structural flexibility in the manipulator links is unavoidable along with the joint flexibility. Alternatively, this case can be considered to correspond to a manipulator with a telescopically extending tip link. Results of controlled and uncontrolled displacement histories of each global coordinate are given in Figure 4, with an identical format of the earlier figures. Again, simulations suggest that considerable suppression of the tip oscillations, is possible. As the structural flexibility of the two-link arm increased with an elongated link-2, the deviations from the desired path are larger naturally, and over prolonged periods. Uncontrolled tip follows an undesirable motion profile with continuous deviations all along the path. Controlled case also presents an inevitable initial overshoot. Nonetheless, oscillations diminish in approximately one and a half cycles, tracking the desired path afterwards.
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Figure 4. Same as in Figure 2 but with twice as long tip link. &21&/86,216 Reported attempts are limited in the literature, to control the vibrations of robotic arms during maneuver, due to either the sensitivity of the control techniques to the dynamic parameter changes or highly complex modelling and hardware requirements. To overcome these difficulties, an effective and versatile vibration control scheme is suggested in this paper. Control is applied at the joints through existing actuators. Since the implementation of the control is a software rather than a hardware issue, no additional hardware is needed. The technique is adoptable to significant parameter changes. It is stable, due to being dissipative in nature. The presented numerical predictions demonstrate the effectiveness of the control, and its relative insensitivity to significant parameter changes. The simulations suggest that significant reduction in deviations from the desired path of the tip, are possible. Although not presented due to space limitations, the control has also been implemented in a prototype laboratory model with close correspondence to the predicted performance. Experiments confirm that the VSC technique can be simply merged with existing structures, as an add-on controller. This last feature is a significant design advantage. These results are currently being compiled for a separate publication.
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5()(5(1&(6 1. Book, W. J., Controlled Motion in an Elastic World, ASME Journal of Dynamic Systems,Measurement and Control, (113) 252-261, 1993. 2. Andeen, G. B. editor in chief, Robot Design Handbook, McGraw-Hill Book Company, New York 1988. 3. Wang, D. and Vidyasagar, M., Passive Control of A Stiff Flexible Link, The International Journal of Robotics Research,11(12) 572-578, 1992. 4. Hong, S. and Park, Y., Vibration Reduction for Flexibel Manipulatros using Torque Wheel Mechanism, Proceedings of the Third International Conference on Motion and Vibration Control, Chiba, Japan, 364-369, 1996. 5. Lopez-Linares, S., Konno, A. and Uchiyama, M., Vibration Suppression Control of 3D Flexible Robots Using Velocity Inputs, Journal of Robotic Systems 14(12), pp.823-837, 1997. 6. Zhu, G., Ge, S. S. and Lee, T.H., Simulation Studies of Tip Tracking Control of a Single-Link Flexible Robot Based on a Lumped Model, Robotica, (17) 71-78, 1999. 7. Bernzen, W., Active Vibration Control of Flexible Robots Using Virtual Spring-Damper Systems, Journal of Intelligent and Robotic Systems, 24 69-88, 1999. 8. Khulief, Y. A., Vibration Suppression in Rotating Beams using Active Modal Control, Journal of Sound Vibration, 242(4), 681-699, 2006. 9. Romano, M., Agrawal, B. N. and Bernelli-Zazzera, F., Experiments on Command Shaping Control of a Manipulator with Flexible Links, Journal of Guidance, Control and Dynamics, 25(2) 231-239, 2002. 10. Dadfarnia, M., Jalili, N., Liu, Z. and Dawson, M.D., An Observer-Based Piezoelectric Control Of Flexible Cartesian Robot Arms : Theory and Experiment, Journal of Control Engineering Practice, 12 1041-1053, 2004. 11. Thomson, W.T. Theory of Vibrations with Applications, Prentice Hall, Englewood Cliffs, NJ, 4th Edition, 1993. 12. Özer A. and Semercigil, S. E., An Event-based Vibration Control for a Two-Link Flexible Robotic Arm; Numerical and Experimental Observations, Journal of Sound Vibration, 313, 375-394, 2008. 13. Özer, A., Vibration Control of Flexible Robot Manipulators, Ph.D. dissertation, Victoria University of Technology, Melbourne-Australia, 2006. 14. Bathe, K.J., Finite Element Procedures in Engineering Analysis, Prentice Hall, Englewood Cliffs, NJ, 1982. 15. Matlab, MATLAB, The Language of Technical Computing, Using MATLAB Version 5” The MathWorks, Inc, MA, 1997. 16. Craig, R.R., Structural Dynamics, An Introduction to Computer Methods, John Wiley and Sons, Inc. USA, 1981. 17. DMC-18x2 Manual Rev. 1.0d, Galil Motion Control, Inc., California.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
VibrationTestingofBridgeStayCablestoObtainDampingValues MartinE.Turek TVPEngineeringLtd.,Vancouver,Canada
CarlosE.Ventura UniversityofBritishColumbia,Vancouver,Canada
KhaledShawwaf DywidagSystemsInternational,Bolingbrook,IL,USA
ThispaperdiscussesaseriesoftestsperformedonacablestayedbridgeinBritishColumbia,Canadato obtain the cable damping properties. Free vibration tests were performed on selected cables, in two phases:oneduringastateofconstruction,withthecablesataloadlessthan100%,andthesecondafter thecableswerecompleteandatthefinalload,andwithadditionalneopreneinsertsatthedeckendof thecable.Inacomparisonof8cablesonasingletower,thetypicalresultsshowedanincreaseinmodal frequencyandincreaseindampingvalue.Dampingvaluestypicallyrangedbetween0.2and0.4%.
Introduction The Pitt River Bridge is a cableͲstayed bridge connecting Pitt Meadows and Maple Ridge in British Columbia,Canada.ThebridgeopenedtotrafficinOctober,2009.Thebridgehasasinglespan,withthree 60mconcretetowersoneithersideofthe190mmainspan.Anelevationschematicisshowninelevation inFigure1.AphotoshowingtheeasttowersisshowninFigure2.
Figure1:ElevationSchematicofBridge
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_31, © The Society for Experimental Mechanics, Inc. 2011
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Figure2:Easttowersofbridge
Thepurposeofthestudypresentedinthispaperwastoobtainthedampingpropertiesofselectedstay cables; before and after neoprene inserts were installed at the deck end of the cables. Each cable was fabricatedonsitebymeansofassemblingasetof15.7mmdiametersteelstrands,pulledthroughasteel plateandanchoredwitha wedge.Thecentrefanof cableshas amaximum of61 strands,and thetwo outerfanshaveamaximumof31strands.Thecablesaretensionedatthetopend,andfixedatthedeck end(showninFigure3).Thestrandsarethenclampedtogetherapproximately1malongitslength,and theassemblyofstrandsiscoveredbyaplasticpipe(crosssectionisshowninFigure4).Asteelexitpipeis boltedtotheanchorassemblycompletingthecable.Twoadditionaldampingmechanismsareaddedto thecables: 1) Forthesixshortestcables,roundneopreneinsertsareplacedbetweenthecableandthesteel exitpipeatthedeckend 2) Forthetwolongestcables,externaldampersareaddedatthedeckend(theneopreneinserts areinstalled,butwitha10mmgapastonotinterferewiththedampers) Note:forthetestsdescribedinthispaper,theexternaldampersonthelongestcableswerenotyetfully assembledandinoperation.
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Figure3:PittRiverBridgeTypicalCableDeckEndConnection
Figure4:CrossͲSectionof61ͲStrandCable This study focuses on a comparison of damping estimates for eight of the cables, between two sets of tests,definedasfollows: Test1 x Duringconstruction x Noneopreneinserts x Cablesatload10Ͳ15%lowerthanfinal x Exitpipenotbolted Test2 x Justpriortobridgeopening x Neopreneinsertsinstalled x Cablesat100%finalload x Exitpipebolted x Externaldampersnotinoperation
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FreeͲVibrationTests Inordertoobtainestimatesofthemodaldampingvaluesforthecables,aseriesoffreeͲvibrationtests wereperformed.Twosetsoftestswereperformed,referredtoasaboveasTest1andTest2.Atotalof8 cablesweretested,showninFigure5.Thisincludescables8,6,4and2onboththemainspan(MS)and backspan (BS). The cables tested typically had 58 or 59 strands in a standard 61 cable, and 28 or 29 strandsinastandard31cable. Testingwasdonebymeansofpulling asling andropewrappedaroundthecableapproximatelyatthe cable midͲpoint (shown in Figure 6). The cables were manually excited until sufficient amplitude was achieved; theropewasreleased allowingthecable togo intofreeͲvibration.MeasurementofthefreeͲ vibrationwasdoneusinganaccelerometerontothecablenearthedeckend.
8
6
4
2
MS
BS
Figure5:TestNotations
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Figure6:SlingandRopeusedforTesting
Analysis Two results from each test were obtained from the analysis; cable natural frequencies and modal dampingvalues.Foreachofthetests,twomethodswereapplied.First,analysiswasdoneusingMatlab, applyingthefollowinggeneralsteps: 1) TruncatethedatatocaptureonlythefreeͲvibrationsegment 2) BandͲpassfilterthedatatoanalyzeonemodeatatime 3) ExtractthepositivepeakvaluesfromthefilteredfreeͲvibrationsegment 4) Computethelogdecrementrepetitivelyforsubsequentpeaks 5) Computetheaveragelogdecrementvalueforthedatasegment Whilethelogdecrementisausefulandsimpletechnique,itisderiveddirectlyfromtheresponseofan underdamped single degree of freedom system [Chopra, 2000]; this does not necessarily apply well in practicetoarealmultiͲdegreeoffreedomsystem. Forexample,ifastructureisexcitedinsuchawayastoonlyexcitethefirstmode,thenfreeͲvibrationof thatmodemaybeobtainedwithoutinterferenceoftheothermodes.However,asinthecasewiththe cable, if they are excited in such a way as to excite more than one mode, then the response will be a combinationofthosemodes.Eachmodehasadifferentmodaldampingvalue,differentmaximummodal amplitudeanddifferentfrequencyatwhichitwillvibrate. Consequently it may not be possible to obtain a pure freeͲvibration signal for each mode. Therefore a morerobustmethodshouldbeusedtocaptureallofthemodaldampingvalues.Forthis,theStochastic Subspace Identification algorithm was applied via the ARTeMIS Extractor software [SVS, 2009]. Three variations of the method are available in the software, and using the automated mode identification feature,frequencyanddampingestimateswereobtaineduptothe5Hzbandwidth.
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TheSSImethodestimatesdampingvaluesbyobtainingthecomplexeigenvaluesforeachmode.Detailsof thederivationareavailablefrom[OverscheeandDeMoor,1996].Priorlaboratoryandanalyticalstudies by the author demonstrated a high degree of accuracy of the method in predicting the damping value [Turek,2007].
Results Resultsfromtestingofcables8,6and4oftheBackspanofTowerE1,andoftheMainspanofTowerE1, arepresentedinthispaper.AnexampleoftheresultsforCable4,Backspanisshown.Figure7showsthe rawtimehistorydataand thespectrum(using theWelchalgorithm[Welch,1967])for each ofthe two tests. Inbothtests, twomodes are identified below5Hz.Itisseeninthespectra thatthemodalpeaks haveshiftedtotheright,bothincreasinginfrequency. 8
3
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Test 1 Data Run 13
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2.5 3 Frequency (Hz)
3.5
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Figure7:Cable4BSrawdataandspectraldensities
3.5
4
4.5
5
Figure8showsthedatafilteredtothefirstmodeforTest1.Table1summarizestheresultsofallofthe tests.Thetablepresentsthefrequencyforeachoftheidentifiedmodesbelow5Hz,thedampingfrom boththeSSIandLDresults.FortheSSIresults,theaverageoftheresultsfromthethreevariationsonthe method(PC,UPCandCVA;seeRef)iscomputedandpresented. As discussed in the previoussection, the LD method does not necessarily apply well to multiͲdegree of freedom systems. Some of the filtered timeͲhistory records do not show consistent decay, and the LD methodcannotbereasonablyapplied.Forconsistency,theLDmethodisappliedtoallmodes,butaseries ofcategoriesareestablishedtoqualifytheuseofLDonthedata.Thecategoriesaredenotedas‘good’, highlightedingreen,‘moderate’,highlightedinyellowand‘poor’,highlightedinred.Thegoodcategory results displayed an ideal freeͲdecay allowing for successful application of LD. The moderate category
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results, shown in the table, showed a general decay of the signal with some inconsistencies that deterioratetheLDresults.Thepoorcategoryresults,inthetable,didnotshowaconsistentdecayinthe signal. ComparisonofthegoodcategoryresultsshowsthereisagreementbetweenLDandSSIwithanaverage difference of about 5%; this confirms the use of the SSI method for this analysis. For the moderate categorytheaveragedifferenceisabout40%,whileforthepoorcategorythedifferenceisabout100%.It isworthnotingthatofatotalof39modesevaluatedfrombothtests,8weredenotedasgood,17were denotedasmoderate,and14denotedaspoor.Ofthemoderateresults,6hadcloseagreementwithan averagedifferenceof0.5%.
Figure8:Cable4BS,FilteredtoMode1forLDanalysis
MonitoringSystemandFutureStudies Apermanentvibrationmonitoringsystemhasbeeninstalledonthebridge,withaccelerometersmounted on selected locationsoncables, towers and deck.An example of a sensor mounted onCable 7, middle fan,backspanisshowninFigure9. Inaddition,externalcabledampershavebeeninstalledoneachofthetwolongestcables(7and8).The dampers were not in operation during the testing described in this paper. A future study will be undertaken to evaluate the performance of those external dampers; data will be collected during wind eventsandcomparedtopreviouslycollectedambientvibrationdatafromcable7(notreferencedinthis paper).
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Table1:Testresults Frequency [Hz] 0.98
TEST1 SSIDamping [%] 0.52
LDDamping [%] 0.90
1.93
0.62
2.91
0.35
3.86
0.42
0.58
4.86
0.33
0.41
1.05
0.41
0.51
1.27
0.22
0.27
2.08
0.54
0.74
2.49
0.56
0.39
3.12
0.42
0.57
3.80
0.16
0.16
4.16
0.38
0.47
1.46
0.12
0.48
1.66
0.33
0.61
2.89
0.39
0.66
3.31
0.26
0.48
4.33
0.44
0.46
1.52
0.24
1.49
1.65
0.40
0.86
3.01
0.29
0.42
3.31
0.22
0.54
4.50
0.58
2.19
4.94
0.28
0.35
CABLE 4BS
2.06
0.29
0.22
2.42
0.45
0.45
4.14
0.44
0.46
3.10
0.33
0.34
CABLE 4MS
2.04
0.43
0.42
2.36
0.78
0.66
4.09
0.20
0.33
4.70
0.32
0.38
CABLE 8BS
CABLE 8MS
CABLE 6BS
CABLE 6MS
Frequency [Hz] 1.28
TEST2 SSIDamping [%] 0.14
LDDamping [%] 0.36
0.88
2.56
0.20
0.20
0.69
3.84
0.36
0.33
0.22
CABLE 3.26 0.37 0.30 4.23 0.62 0.77 2BS CABLE 3.26 0.27 0.53 4.35 0.54 0.71 2MS Note: Green/bold denotes ‘good’ category results; Yellow/italics denotes ‘moderate’ catergory results; Red/plaindenotes‘poor’categoryresults
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Figure9:AccelerometerMountedonCable–PermanentMonitoringSystem
SummaryandConclusions This paper presented the results of a freeͲvibration damping study on stay cables from the Pitt River Bridge, in British Columbia Canada. A total of 8 cables were tested and analyzed on two separate occasions (referred to as Test 1 and Test 2), with the primary purpose of comparing behavior of the 6 shortestcables,whichhadneoprenedampinginsertsinstalledbetweenthetwosetsoftests.Inaddition, duringTest1,thecableswerealoadofabout10Ͳ15%lowerthanthefinalload.Thelongestcableswill have a permanent monitoring system and external dampers installed, both of which will be a part of futurestudies. Fromtheresultsofthestudy,inallcasesthemeasurednaturalfrequenciesincreasedbetweenTest1and Test2.ThisisexpectedduetotheincreaseinloadfromTest1toTest2.Fortheresultsofthedamping analysis,thetwoshortestcablesbothexhibitedanincreaseindampingbetweenTest1andTest2.For the third longest cable,the first mode damping value increased, while the higher modes either did not changeordecreasedinvalue.Thisbehaviorillustratesthecomplexitiesofmodalinteractionanditseffect onthecabledamping. For the longest cables, the conditions of the test differed from that of the other cables. The neoprene inserts were installed but included a gap of approximately 10mm, to allow for displacement of the external dampers. The dampers themselves were installed but not in their final configuration, and consequently were not in operation during the testing. The damping analysis was performed on those cablesdespitethisfact,andgenerallytheresultsshowedadecreaseindampingvalueafterTest2.This behaviorcanpotentiallybeexplainedbythechangeinloadbetweentests;afterthefinaltensioningof the cable certain damping sources such as movement between individual strands and movement between the strands and the plastic pipe have been diminished. Despite these results, conclusions regardingthelongestcabledampingvaluescannotbemadeuntiltheexternaldampersareinoperation andmoredataisobtained. Additionaluncertaintyinthetestresultsmaycomefromthesteelexitpipesatthedeckendofthecables. Foreachofthefirsttests,thesteelexitpipeswerenotboltedinplaceandinsteadwerefreelyrestingon thecables.Thismayhavetheeffectofincreasingthedamping,dependingonhowhighupthepipeswere
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placedonthecables.Duringthesecondtests,theexitpipeswereboltedtotheendconnectionforeach ofthecables.
Acknowledgements TheauthorswouldliketoacknowledgeVinceStolenyandLeonKondratiewofofDSIfortheirassistancein performing the tests. They would also like to acknowledge Devin Sauer, of the University of British Columbiaforhisworkontheanalysisofthedata.
References
Chopra, A. K., Dynamics of Structures: Theory and Applications to Earthquake Engineering (2nd Edition) PrenticeHall,2000 ARTeMISExtractor,Copyright1998Ͳ2009StructuralVibrationSolutionsDenmark Overschee,P.van&B.DeMoor:Subspaceidentificationforlinearsystems–Theory,Implementation, Applications.KluweracademicPublishers,1996. Turek, M., A Method for Implementation of Damage Detection Algorithms for Civil SHM Systems, PhD Thesis,DepartmentofCivilEngineering,UniversityofBritishColumba,2007 Welch,P.D,"TheUseofFastFourierTransformfortheEstimationofPowerSpectra:AMethodBasedon TimeAveragingOverShort,ModifiedPeriodograms,"IEEETrans.AudioElectroacoustics,Vol.AUͲ15(June 1967),pp.70Ͳ7
BookID 214574_ChapID 32_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Identification of Stiffness and Damping in Nonlinear Systems Mikaya LD Lumoria, Johan Schoukensb, John Lataireb a
Department of Engineering (Electrical Engineering Program), University of San Diego, 5998 Alcala Park, San Diego, California 92110-2492, USA b
Department of Fundamental Electricity (ELEC), Vrije Universiteit Brussel, Pleinlaan 2, B1050 Brussels, Belgium
ABSTRACT Resonance nonlinearities in vibrating mechanical structures are either due to stiffness, damping, or a combination of both. A method is presented to detect the nonlinear distortions, determine and quantify the distortion levels. This is achieved by configuring a nonlinear device as a second-order, single-input-single-output (SISO) closed-loop feedback system such that static nonlinearities are confined to the feedback path, and the dynamic linear part is modeled as the forward gain. The closed-loop system is then subjected to random phase multisine excitations. This makes it possible to model the linear part by its frequency response function, thus facilitating the characterization of the nonlinear part. There is a good agreement between the estimated and experimental data. The results indicate distortion nonlinearities due to stiffness and damping with distinguishable levels. 1. Introduction Nonlinear (NL) vibrating mechanical systems inherently exhibit single- or multiple-resonant modes, accompanied by nonlinear distortions due to nonlinear stiffness, nonlinear damping, or a combination of both. This paper presents a method whose goal is to detect and establish distinguishable levels of the nonlinearities. The configuration of the nonlinear dynamic device under test (DUT) is shown in Fig. 1. Identification methods, based on the Best Linear Approximation (BLA) and random phase multisine excitations have been developed for these systems [1], [2].
u(t)
+
G0
−
y(t)
NL Fig. 1. Static nonlinear (NL) feedback model The theory and mathematical models for the DUT are presented in Section 2. Experimental validation of the nonlinear model is accomplished physically by a resonant electric circuit shown in Fig. 3; it is modeled on the electronic dynamic system S2 of the DUT in Fig. 2, with switching elements for the feedback loop. In Section 3 the identification algorithm, based on the BLA, is formulated in a linear-in-the-parameters framework, using input and output regressors. The measurements procedures, the measured data, and the estimated data are discussed and compared in Section 4. Finally, comments and conclusions are given in Section 5. 2. System model theory 2.1 Linear system model The physical device to be modeled and investigated is shown schematically in Fig. 2. It is a dynamic, second-order, single-input-single-output (SISO) mechanical system, S1, and its electrical counterpart, a dynamic, second-order electrical SISO system, S2. Both the linear and nonlinear mathematical models are presented subsequently.
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_32, © The Society for Experimental Mechanics, Inc. 2011
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i(t)
S1
k
u(t)
m d
+
y(t)
−
S2 R
v(t)
C L
Fig. 2. Schematics of mechanical (S1) and electrical (S2) systems The respective mathematical linear models of the dynamic systems S1 and S2 in Fig. 2 are:
my(t ) + dy (t ) + ky (t ) = u (t ) di(t ) 1 t L + Ri (t ) + ³ i (τ )d (τ ) = v(t ) dt C −∞
(1a) (1b)
where m, d, k, u(t), and y(t) denote the mass, dashpot friction constant (or damping coefficient), spring constant (or stiffness coefficient), prime mover (or forcing function/excitation), and the response function (or displacement function) of S1, respectively; L, R, C, v(t), and i(t) denote the inductance, resistance, capacitance and voltage source (or forcing function/excitation), and the response function (current) of S2 respectively. 2.2. Nonlinear system model The mathematical model of the approximate nonlinear dynamic mechanical system S1 is of the form
my(t ) + ¦ (d n y n (t ) + k n y n (t )) = u (t )
(2)
n
where dn and kn are the damping and stiffness coefficients, respectively. Equation (2) can be n1
n2
extended easily to include the mixed nonlinear terms y y , etc which have been excluded for rd simplicity. It is also assumed that the highest, significant nonlinearity is a 3 -degree of the DUT, i.e. n =1, 2, 3. Hence, equation (2) reduces to the form:
my(t ) + [ d1 y (t ) + d 2 y 2 (t ) + d 3 y 3 (t )] + [ k1 y (t ) + k 2 y 2 (t ) + k3 y 3 (t )] = u (t )
(3)
As a further approximation the second-degree nonlinearities are removed from (3), resulting in the following mathematical model used in the identification algorithm for the DUT shown in Fig. 3):
[my(t ) + d1 y (t ) + k1 y (t )] = [u (t ) − ( d 3 y 3 (t ) + k3 y 3 (t ))] u
+
y
G0
−
k3 • 3 d3 •3
vout v˙ out
+
vin
s
−
L +
− +
−
C R
y
•3
y˙
•3
y3 y˙ 3
k3 d3
SNL
Fig. 3. Schematics of (i) Top – Closed-loop system and (ii) - DUT (Silver Box II)
(4)
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343 Fig. 3 depicts closed-loop (top) and electronic (bottom) equivalents of equation (4), where:
s=
d 1 , and G0 = . In Fig. 3 (bottom) the input vin is into a difference amplifier, and 2 dt ms + d1s + k1
the output vout is proportional to the integral of the current, consistent with Fig. 2. 3. Identification 3.1 Random phase multisine excitation A random phase multisine signal is used to excite the nonlinear DUT. This type of excitation provides a straightforward determination of the frequency response function (FRF) of the linear part of the system. The FRF is then used for characterizing the nonlinear part. Besides, a random phase multisine provides a good detection tool of nonlinearity at the unexcited frequencies, known as ‘detection lines’ [1], [2]. A random phase multisine is a periodic signal that comprises a sum of harmonically related sine waves, defined as:
u (t ) =
1 N
l=N
¦U l e
j ( 2πf max
l t +φ k ) N
(5)
l =− N
In equation (5) l ≠ 0 and the factor 1/ N normalizes u(t) such that as N ĺ (asymptotic) the multisine is maintained at both a constant power level and a constant root mean square (RMS) value. The amplitude levels Ul and the fundamental frequency, f0 = (fmax/N), are chosen by the user, consistent with the desired power spectrum. If the amplitudes vary uniformly, good practical results are possible only for N ≥ 20 excited lines [2], [4], The phases φl are selected from an independent random phase distribution process (e.g. uniformly distributed over [0, 2π]) such that the expected j value E{e φl} = 0. Furthermore, some of the spectral lines are randomly unexcited in the excitation spectrum, viz: ∃l : U l = 0 , thus rendering it easy to observe the behavior of the nonlinear response at the output (detection lines) at those frequencies [1]. In order to guarantee that u(t) is a real signal, the following constraints are imposed:
U l = U −l ® ¯φl = −φ−l Multisine excitations have the ability to engender excitations that indicate whether a nonlinear system has even-mode or odd-mode nonlinearities. The SISO nonlinear system (DUT) is excited by an odd random phase multisine signal whose amplitude levels are defined as:
U for l = 2n + 1 l ≤ N , n ∈ 1 ° U l = ®0 elsewhere °∃l : U = 0 l ¯
(6)
In equation (6) the frequency lines of the signal are grouped into blocks, each of length 4. One randomly selected line is unexcited ( ∃l : U l = 0 ) in each block [5], thus providing detection lines for detecting odd nonlinearities [1], [2] at the output. Consider a multisine input signal Ul that has odd frequency components (f0, 3 f0, 5 f0, etc.) corresponding to odd spectral components (l = 2n + 1, N, n ∈ N), depicted in Fig. 4.
8O
QRQOLQHDU V\VWHP …
O
…
O
Fig. 4. Response of a nonlinear system to an odd multisine excitation
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344 Clearly, the even nonlinearities engender output spectral components |Yl| at even frequencies. This is simply because any even combination of odd frequencies adds up to an even frequency. Therefore, the even spectral lines at the output can be used as detection lines of even-mode nonlinearities. Moreover, odd combinations of any odd frequencies will result in odd frequencies. Hence, randomly unexcited odd frequency lines will facilitate the detection of odd nonlinearities. 3.2 The Best Linear Approximation of a nonlinear system [2], [3] The Best Linear Approximation (BLA), GBLA, is defined as the model G belonging to the set of linear models *such that the BLA is given by the argument that minimizes the expected value E{..}: 2 GBLA = arg min E{|y(t) – G{u(t)}| } (7) G ∈* where u(t) and y(t) are input and output of the nonlinear system, respectively. Note that the expected value is taken over random phases of the input. The input is assumed to be noiseless unlike the output, which has normally distributed measurement noise. Generally, however, the Best Linear Approximation (GBLA) of a nonlinear system depends on the characteristics of the stochastic input u(t), namely [2], [6], [7]: (1) the amplitude distribution, (2) the power spectrum, and (3) the higher order moments. Consider a SISO nonlinear system S that is excited by a single input u resulting in a single output y. Let V be a class of Gaussian excitation signals with a user-defined spectrum. If y converges in a mean square sense for u ∈ V, assumed to be a uniformly bounded Voltera series, then S can be modeled as a sum of a noise source ys and a linear system GBLA(jω), known as the Best Linear Approximation (BLA). This is computed without imposing causality via the Fourier transform, based on the Wiener-Hoppf equation, and derived from the classic result of equation (7) [8] as:
GBLA ( jω ) =
S yu ( jω ) Suu ( jω )
(8)
where Suu(jω) is the auto-power spectrum of the input, and Syu(jω) is the cross-power spectrum between the output and the input. It has been shown in [9] that if a Gaussian noise and a random multisine both have the same power spectrum, then their BLAs are identical asymptotically. Hence, the pdf of a random phase multisine is Gaussian-distributed asymptotically A nonlinear SISO system (DUT) is depicted in Fig. 5(a) with its BLA equivalence in Fig. 5(b). In the latter, ys is a nonlinear noise source for the output portion of y that cannot be captured by the linear model GBLA(jω).
ys u
y
⇔
u
yBLA
GBLA
+
y
Fig. 5. (a) SISO nonlinear system (DUT) and (b) BLA SISO system equivalence The BLA in Fig. 5(b) is determined at the radian frequency ωl as follows: Y(jωl) = GBLA(jωl)U(jωl) + Ys(jωl) (9) The nonlinear noise source Ys(jωl) depends on each particular realization of u, exhibiting a stochastic behavior from one realization to another, with its expected value E{Ys(jωl)} = 0. Therefore, it is possible to determine GBLA(jωl) by averaging the system response over several realizations of the input u.
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345 3.3 Identification algorithm With reference to equation (4), Fig. 3 and Fig. 5, the goal of the estimation algorithm is to determine the Cost Function, which is defined later, immediately before equation (12). The SISO nonlinear system (DUT) is excited by an odd random phase multisine signal whose amplitude spectra are defined by equation (6). The nonlinear output contributions Y(NL) are predominantly at the unexcited lines, and they are assumed to be negligible at the excited lines compared to the linear contributions Y(L). Hence, the total response at each spectral line l is given by the spectra Yl = Y( L )l + Y( NL ) l where
Git Y([L3)]1 Git Y([L3)]1 · § Y1 · §¨ GitU1 ¸ ¨ ¸ Git Y([L3)]2 GitY([L3)]2 ¸ ¨ Y2 ¸ ¨ GitU 2 ¸ §¨ 1 ·¸ ¨ . ¸ ¨ . . . ¸ ¨ d3 ¸ ¨ ¸ =¨ . . . . ¨ ¸¨ ¸ ¨ ¸ k [ 3] [ 3] ¨ Y ¨ N −1 ¸ GitU N −1 Git Y( L ) N −1 Git Y( L ) N −1 ¸ © 3 ¹ ¸ ¨ Y ¸ ¨¨ Git Y([L3)]N Git Y([L3)]N ¸¹ © N ¹ © GitU N
(10a)
where Git is an estimate of G0 evaluated iteratively in Section 4.3 from GBLA. The matrix vector form of equation (10a) is:
Y = Y( L ) + Y( NL ) = G it U + Kș
(10b)
In equation (10b) Y is a matrix vector of the measured (known) output at both the excited and the unexcited (detection) lines.
§ Git Y([L3)]1 GitY([L3)]1 · § Y1 · § GitU 1 · ¨ ¸ ¨ ¸ ¨ ¸ [ 3] [ 3] G Y G Y ¨ Y G U it ( L ) 2 it ( L ) 2 ¸ ¨ 2 ¸ ¨ it 2 ¸ ¨ ¸ ¨ . ¸ ¨ . ¸ . . §d · ¨ ¸ , and ș = ¨¨ 3 ¸¸ where k3 and Y=¨ , G U = , K = ¨ ¸ ¸ it . . ¨ ¸ © k3 ¹ ¨ . ¸ ¨ . ¸ [ 3 ] [ 3 ] ¨ Git Y( L ) N −1 Git Y( L ) N −1 ¸ ¨ YN −1 ¸ ¨ GitU N −1 ¸ ¨¨ ¸¸ ¨Y ¸ ¨GU ¸ [ 3] [ 3] G Y G Y © N ¹ © it N ¹ it ( L ) N it ( L ) N © ¹ d3 are the coefficients of the nonlinear stiffness and the nonlinear damping, respectively. At the unexcited lines the linear system response is zero, i.e.
Y( L ) = G it U = 0 , thus Y = Y( NL ) in
equation (10b). Therefore, the actual measured nonlinear response at the detection lines is:
Y( NL ) = Kș
(11)
In equation (11) the parameters ș are estimated by the estimates (arg) of the Cost Function,
șˆ that minimize the argument
2
V (ș) = Kș − Y( NL ) , viz:
(
2
șˆ = arg min Kș − Y( NL ) ș
)
(12)
The estimates of the parameters are given by the Least Square Estimate (LSE) as (Matlab function):
§ dˆ · șˆ = K + Y( NL ) = ¨¨ 3 ¸¸ ˆ © k3 ¹
(13)
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346 where
K + is the pseudo-inverse of K . Hence the estimated output at the detection lines is YS = Kșˆ
(14)
The estimated results YS are then compared with the actual known results Y(NL). This is implemented in Section 4. 4. DUT – Experimental measurements and their estimates 4.1. A comparison of estimated data with actual, simulated data A modified Duffing oscillator is used to generate simulated data for estimation by the proposed LS algorithm. The Duffing oscillator is a second-order nonlinear dynamic system with a static nonlinearity due to stiffness. It is excited by a time-harmonic signal, and is mathematically modeled by the following nonlinear second-order differential equation:
d 2 y (t ) dy (t ) +a + by (t ) + cy 3 (t ) = A cos(ωt ) 2 dt dt
(15)
where a, b, c are parameters of the system, and A an arbitrary amplitude of the excitation signal. Rewriting the mathematical model of the DUT in equation (4) differently:
my(t ) + d1 y (t ) + d 3 y 3 (t ) + k1 y (t ) + k 3 y 3 (t ) = u (t )
(16)
With reference to equation (16), it is clear that the mathematical model of the DUT is a generalization of the Duffing oscillator, used in the estimation algorithm. A good agreement between the actual simulated data and their estimates validate the LS estimation algorithm. 4.2 Experimental set-up and procedure for the measurements Fig. 6 shows the schematic of the experimental set-up comprising (i) an actuator together with the DUT whose electronic components are illustrated in Fig. 3, and (ii) a BLA with its NL noise source as an equivalent of the DUT. The DUT is excited by an odd random-phase multisine signal u(t) whose amplitude spectra are defined by equation (6). The measured input signal is uLP(t) = u0,LP(t) + nu(t), where nu(t), is the measurements noise. At the output the measurement noise is ny(t). The following settings were made in advance of the measurements comprising several experiments: • RMS values of the input signal u(t): 50 mV • Fundamental frequency of the input signal: f0 = 10 Hz • Number of phase realizations (M) per an experiment: 100 • Sampling frequency: fs = 19.540 KHz • Measurements frequency bandwidth: 10 Hz – 3000 Hz. ny (t) u(t)
nu (t)
u0,LP (t)
y0 (t)
+
y(t)
+
uLP (t)
u(t)
nu (t)
u0,LP (t)
GBLA (jω)
ny (t) + ys (t) yBLA (t)
+
y(t)
+
uLP (t)
Fig. 6. Schematics of the measurement set-up (top) and the BLA (bottom) During the experimental procedure, the input signal was applied to the DUT through an actuator comprising an anti-alias filter. Several experiments over a frequency band of 3000 Hz were performed at a sample frequency of 19.54 KHz over 112 excited lines used for computations. Each
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347 experiment was performed over 100 phase realizations at a frequency resolution of 10 Hz. The measured data were then processed and estimated as presented in the next section. 4.3 Experimental measurement results and the LS-estimated simulations With reference to Fig. 1and Fig. 3, an iterated frequency response function is computed from an initial estimate. The iteration procedure is as follows: • Initialization: GBLA = Y/ U, where Y = fft(y) and U = fft(u) in Fig. 1 and Fig. 3 •
* = 8 −ᤡ1/< ᤢᤣ<ᤤ where i = 0, 1, 2, 3,…., i_max
(17)
L
L
Using equation (17) for i_max = 5, (FRF)it = _Git_ = _Gi_max_ was computed at the unexcited lines via interpolation. Note that at i = 0, NL0 (nonlinearities) = 0 in the feedback path, and we get GBLA = Y/ U. This gives the FRF without iteration, as a first approximation. Fig. 7 shows the experimental response data for both the output signal and the nonlinearities to an RMS input of 50 mV, together with estimates of the distortion nonlinearities (determined by use of the GBLA) and the identification estimates of the damping and stiffness parameters, d3 and k3 respectively, consistent with equation (10). Assuming negligible disturbing noise at the input and output (SNR = 100 dB), the ratio of output signal to distortion for nonlinear damping (d3 = 5) and nonlinear stiffness (k3 = 50) were estimated to be 32.8258 dB and 50.0402 dB respectively. 30 0 Phase (rad)
|G| (dB)
20 10 0
−1 −2 −3
1000 2000 Frequency (Hz)
3000
0
1000 2000 Frequency (Hz)
40
40
20
20
20
0 −20 −40
Output (dB)
40
Output (dB)
Output (dB)
−10 0
0 −20 −40
0
1000 2000 Frequency (Hz)
3000
0 −20 −40 −60
−60
−60
3000
0
1000 2000 Frequency (Hz)
3000
0
1000 2000 Frequency (Hz)
3000
Fig. 7. Measured and estimated results of the output signal and nonlinearity distortions. Top Row – Magnitude and phase: (i) (FRF)it: black line, x; (ii) GBLA: gray line, o Bottom Row – Output response left to right: (1) Left : (i) Output signal: black circles; (ii) NL damping distortion: black x; (iii) NL stiffness distortion: black o (2) Middle & right figs: (i) Output signal: black circles; (ii) Net NL distortion: black o = measured; black x = estimated using GBLA (middle fig.) and (FRF)it (right fig.); (iii) Black stars, *: = difference between measured and estimated net NL distortions Details of the results are included in the caption of Fig. 7. It is clear from the measured and estimated NL distortions (bottom row figures) that there is a good agreement between the estimated nonlinearities and the measured nonlinearities. The following observations are pertinent:
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348 •
• •
Although 100 phase realizations were measured per experiment, the results at the top row, and at the middle and right of the bottom row are from 1 realization only. However, the figure at the left of the bottom row has 100 realizations per experiment, which explains the smoothness of the graphs compared to the other graphs at the bottom row. The magnitude and phase of the FRF (top row) were computed as a first approximation using GBLA (gray line, o), which in turn is an initial estimate used in equation (17) to determine (FRF)it iteratively. These two FRFs were used to estimate the distortions in the nonlinear DUT. The following details pertain to the 3 figures at the bottom row of Fig. 7: (1) Left figure (100 phase realizations): Respective estimates of the measured distortions are about 30 dB for nonlinear damping (middle graph, black x) below the output signal (black circles), and about 50 dB below the output signal for nonlinear stiffness, at a level of noise disturbance (not shown). These results agree with the theoretical estimates discussed earlier. It is clear that the nonlinear damping is the main cause of the distortions in the DUT. (2) Middle figure (1 phase realization): Estimates of the net distortions based on GBLA (black x) have good agreement with measured net distortions (black o). The error between the estimated data and the measured data is shown as black stars for the middle and right graphs. (3) Right figure (1 phase realization): Estimates of the net distortions based on (FRF)it have an improved agreement (black x) with the measured net distortions (black o), compared to the middle figure. This is evident from the error levels (black stars) which are lower than those for the middle figure.
5. Conclusion By judiciously restricting the nonlinearities of the DUT to the feedback path and applying the BLA, a simple, elegant approach has been implemented successfully to study NL distortions. A good agreement exists between the estimated data and the experimental measurements of the nonlinear DUT. The results have distinguishably quantified levels of the NL distortions due to NL damping and NL stiffness. Thus, a vital insight into NL resonant circuits has been gained. Besides, the identification/estimation method can potentially be adapted for application to many nonlinear vibrating systems, including those with multiple resonances. This is an attractive goal for future research. References [1] T. D'haene, R. Pintelon, J. Schoukens, E. Van Gheem, Variance analysis of frequency response function measurements using periodic excitations, IEEE Transactions on Instrumentation and Measurement 54 (4) (2005) 1452–1456. [2] R. Pintelon, J. Schoukens, System Identification: A Frequency Domain Approach, IEEE Press, New Jersey (2001). [3] J. Paduart, Identification of Nonlinear Systems Using Polynomial Nonlinear State Space Models, PhD dissertation, Vrije Universiteit Brussel – ELEC Department, Belgium (2008). [4] J. Schoukens, T. Dobrowiecki, R. Pintelon, Identification of linear systems in the presence of nonlinear distortions: a frequency domain approach, IEEE Transactions on Automatic Control, 43 (2) (1998) 176–190. [5] R. Pintelon, G. Vandersteen, L. De Locht, Y. Rolain, J. Schoukens, Experimental characterization of operational amplifiers: a system identification approach, IEEE Transactions on Instrumentation and Measurement, 53 (3) (2004) 854–876. [6] M. Enqvist, Linear Models of Nonlinear Systems, PhD dissertation, Linkoping Studies in Science and Technology, Dissertation no.985, (2005). [7] M. Enqvist, L. Ljung, Linear approximation of nonlinear FIR systems for separate input processes, Automatica, 41 (3) (2005) 459–473. [8] P. Eykhoff, System Identification: Parameter and State Estimation, Wiley, New York (1974). [9] J. Schoukens, J. Lataire , R. Pintelon, G. Vandersteen, T. Dobrowiecki, Robustness Issues of the Best Linear Approximation of a Nonlinear System, IEEE Transactions on Instrumentation and Measurement 58 (5) (2009) 1737–1745.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Truncation Effects on the Dynamic Damage Locating Vector (DDLV) Approach Matthew Maddalo1 and Dionisio Bernal2 1
Graduate Student, IGERT Fellow, Northeastern University, 360 Huntington Ave., Boston, MA 02115 2 Professor, Dept. of Civil and Environmental Engineering, Center for Digital Signal Processing, Northeastern University, 360 Huntington Ave., Boston, MA 02115
ABSTRACT Localization of damage in a system is an issue that has been widely studied in structural health monitoring. The Dynamic Damage Locating Vector (DDLV) method localizes damage from information in the null space of the change in transfer matrices (ΔG). A precisely known ΔG and an accurate model of the reference state represent ideal conditions in the DDLV approach. Since ideal conditions are not realized in practice, the question of robustness arises. This paper examines the performance of the DDLV approach under error in ΔG coming from truncation of the modes to a limited bandwidth. The numerical results are obtained using a rectangular thin plate model.
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_33, © The Society for Experimental Mechanics, Inc. 2011
349
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350 NOMENCLATURE φ – eigenvectors of the system matrix Ac λ – eigenvalues of the system matrix Ac ΔG – change in the transfer matrix G(s) from undamaged to damage states Ac,Bc,Cc,Dc – state space realization matrices G(s) – linear map between input and output for any value s in Laplace domain (transfer matrix) Gd – transfer matrix in damaged condition Gu – transfer matrix in undamaged condition Kd – damaged system stiffness matrix Ku – undamaged system stiffness matrix L – vector in null space of ΔG (DDLV) M – system mass matrix u – system input x – state vector y – system output INTRODUCTION In recent years, a great deal of research in the area of structural health monitoring has been aimed toward the use of vibration data to detect, locate, and quantify damage. With an aging infrastructure, non-destructive methods for evaluating the condition of a structure are in demand. One method for damage localization that has shown promise is the Damage Locating Vector (DLV) method proposed in Bernal [1]. This method, presented in 2002, locates potentially damaged elements using the difference in flexibility matrices between the undamaged and damaged conditions. Its effectiveness has been shown through numerical examples and physical experiments [3,4]. It is important to note that the quantity used to implement the DLV method, the change in flexibility matrices, only contains information related to the static response of a system at sensor locations. Damage that does not produce changes in this response will be undetectable. For this reason, there may be particular damage or sensor configurations which make damage localization impossible with static flexibility. In a typical system identification setting, however, information regarding the dynamic behavior of a system is also obtained. A method that makes use of this information should result in more robust and effective damage location. In 2007, the DLV approach was extended from the static flexibility case to the dynamic case, resulting in a method that has been designated as the Dynamic Damage Locating Vector (DDLV) method [2]. From an operational perspective, the implementation of the DLV and DDLV methods is similar. The fundamental difference resides in the fact that the flexibility matrices of the DLV method are replaced by the more general transfer matrices and that, as a result, the damage locating vectors are complex. A feature of the DDLV approach is that one can generate damage locating vectors (in principle) at as many s values as desired, and this allows generation of multiple results that can be aggregated to promote robustness. The fact that the DDLV approach can isolate damaged elements, even when static flexibility remains unchanged, is also worth noting explicitly. In this paper, the theory of the DDLV method is briefly explained. Also, an example is presented in which the method’s robustness is investigated with respect to transfer matrix accuracy. DYNAMIC DAMAGE LOCATING VECTOR METHOD The fundamental ideas of the Dynamic Damage Locating Vector (DDLV) approach to damage localization are presented in this section, while further derivation may be found in Bernal [2]. Consider an undamaged system for which the input-output response is contained in a transfer matrix, G(s). In such a case, the matrix G(s) linearly maps the inputs (u) to the outputs (y) for any value of s in the Laplace domain. If a state space realization of the following form can be computed for the system, (1) (2)
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351 Then, taking a Laplace transform of qs.1 and 2 and combining, one gets that the transfer matrix, G(s), is given by (3) where we’ve assumed that the output is displacement so there is no direct transmission term. For the purpose of modal truncation studies the pole-residue form of the transfer matrix is convenient. To pass from Eq.3 to the poleresidue form one begins with a spectral decomposition of Ac, namely (4) where λ is a diagonal matrix containing the eigenvalues and, φ, contains the eigenvectors. Substituting into Equation 1 and premultiplying by yields (5) Setting Y=
one finds that Equations 1 and 2 become (6) (7)
Or, with Г=
Bc and ψ=Cc (8) (9)
One finds, therefore, that G(s) is also given by (10) In Eq.10 the matrix that is being inverted is diagonal so it can be written as
(11) where j is the j-th column of and bj is the j-th row of The form in Eq.11 is convenient since the modes to be included in an analysis are dictated by the values of j included in the sum. If the transfer matrix is known for both the undamaged, Gu, and damaged, Gd, conditions of the system, the matrix, ΔG, may be computed as follows. (12) If ΔG is non-zero and rank deficient, then there must be at least one non-zero vector, L, in its null space such that the following equation is true. (13) Conceptually, the null space of ΔG contains the load vectors for which the (Laplace domain) output response of the damaged and undamaged system is identical. It has been proven that when treated as loads on the system, the vectors L that satisfy the above equation lead to stress fields that are zero over damaged elements. Therefore, by applying these load vectors, called Damage Locating Vectors (DLVs), elements experiencing stress may be eliminated from the possibly damaged set. Depending on the number of sensors, damage may successfully be located to within a particular element or set of elements. Note that for a value of s=0, the transfer matrix, G(s), is simply the static flexibility matrix. In this special case, the DLV represents a set of static loads. For any other value of s, the vectors in the null space represent Laplace
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352 transforms of dynamic loads and are called Dynamic Damage Locating Vectors (DDLVs). Their effect for the purposes of damage location is the same, as they induce stress fields of zero over damaged elements of the system. A point of emphasis concerning the DLV and DDLV methods is that they do not directly indicate damaged elements. Rather, they provide a criterion for establishing that particular elements are undamaged. In this paper, the effects of transfer matrix inaccuracies (associated with truncation) on DDLV effectiveness are investigated. A situation is modeled in which only a portion of the modes of a system, a plate in this case, are identified. Several damage cases will be evaluated to determine if the method remains accurate under such conditions. NUMERICAL EXAMPLE A model of a 6x6 square plate was used to test the robustness of the DDLV method with respect to modal truncation. Each square unit of the plate was divided into two triangular elements, for a total of 72 elements. The plate was modeled to have supports at each corner, and ten displacement sensors were placed on the left and right sides of the plate. The element designation and the sensor placement are illustrated in Figure 1.
Fig. 1. Plate Element Designation and Assumed Sensor Placement In this experiment, 25 damage cases were considered. In each case, one of the 72 elements in the plate was assigned a reduced stiffness (30% less than that of the other elements). Figure 2 illustrates the various damage locations that were considered in these simulation runs.
Fig. 2. Damaged Element Locations for DDLV Runs
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353 For each case, an undamaged stiffness matrix, Ku, and a damaged stiffness matrix, Kd, were obtained using a Matlab Finite Element Toolbox. Using this information, a state space representation (Ac,Bc,Cc) was generated for both system conditions. The transfer matrices for the damaged and undamaged cases were then generated using all 270 modes of the structure. For each damage case, the DDLV method was used at several s-values along the imaginary line (near the undamaged eigenvalues). For each value of s, a Dynamic Damage Locating Vector was found and applied to the undamaged system. Multiplication of the load times the dynamic flexibility matrix 2 -1 (Ms +Ku) (no damping was considered), produced the displacements of the plate in the Laplace domain. The displacements were then multiplied by Ku to convert them to equivalent static loads, so that the stress field could be computed using convention software. The elements where ranked on the basis of the generalized subspace angle between the nodal displacement vector and the null space of the stiffness matrix for each individual triangular element, Ke. This null space has three basis vectors that represent a basis for rigid body motion. When the stresses are zero the subspace angle between the null space of Ke and the individual displacement vector for a damaged element is zero. The process of computing the subspace angle for each element was repeated in each damage case, using several s-values and summing normalized subspace angle results. As expected from the theory, the DDLV method successfully located the damage in all 25 cases when there is no truncation. A typical result is shown in Figure 3 below, where damage is simulated in Element 60. 0
10
-2
Subspace angle
10
-4
10
-6
10
-8
10
-10
10
0
10
20
30 40 50 Element Number
60
70
80
Fig. 3. DDLV Results for Damage in Element 60 After confirming the adequacy of the computational set up, the robustness of the method was tested as the transfer matrix was truncated. A series of runs was completed in which only 25 pairs of modes were used to generate the transfer matrices. This represents roughly 18% of the total modes of the system. The modes used in the analysis were always consecutive, but the series of modes used was varied. When the DDLV method was used to select as damaged only the element with the lowest subspace angle, a high level of success (greater than 80%) was seen if modes of lower frequency were included in the transfer matrix. The results from this series of tests are shown in Figure 4.
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354
Fig.4. Successful Damage Location % Vs. Modes Included: 50 Modes with 1 Selected Element If the method is allowed a little more flexibility, the success rate can be further improved. In the previous graph, the method is considered to fail if the damaged element is not identified as the most likely element (whichever has the smallest subspace angle). If the success criterion is relaxed to identifying the damaged element within the four most likely elements, the success rate over several mode intervals is improved. This case tests the ability of the method to successfully locate the damage when narrowing the possibly damaged set to about 5% of the total elements. The results (shown in gray), compared with the previous runs (black), are presented in Figure 5.
Fig. 5. Successful Damage Location % with the Selection of 1 and 4 Elements: 50 Modes Statistically, the likelihood of locating the damaged element with four randomly selected choices is less than 6%. With only one randomly selected choice, the percentage drops to less than 2%. From the graph, it is clear that the DDLV method provided significant damage location results, even when only 18% of the modes were included in the transfer matrix. The fact that the success rate is 100% when the modes are in the 81-130 and 101-150 and only 15% in the 161-210 shows that bandwidth selection (when the investigator has an option of where to direct the energy) can play an important role in the results.
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355 This experiment was repeated, this time using only 15 pairs of modes in the transfer matrices. Even though this was equivalent to only 11% of the total number of modes, the DDLV still returned moderately successful results within the lower frequency modes. Figure 6 summarizes the results when only one damaged element was selected (black) and when four possibly damaged elements were selected (gray).
Fig. 6. Successful Damage Location % with the Selection of 1 and 4 Elements: 30 Modes Even with a significant portion of the modes excluded, the DDLV method still regularly located the damaged element within four selections in many of the bandwidths tried. In fact, in cases where mode numbers are less than 170, the success rate ranges from 68% to 92% in this region. CONCLUSIONS The exercise presented in this paper suggests that the Dynamic Damage Locating Vector method may successfully isolate damaged elements even when a significant portion of modes are excluded from the transfer matrix. With only 18% of the system’s modes included, damage was still isolated at various locations on the plate element with a high degree of accuracy. When only 11% of the system’s modes were included, the success rate suffered, but the method still provided meaningful results over certain ranges of modes. It appears that optimization of the bandwidth for the identification may prove possible using numerical models. Further research may revolve around removing some of the simplifying assumptions that are made in this paper. For example, the DDLV method’s effectiveness may be tested when the transfer matrices are obtained from noisy input-output data, rather than directly from mass, damping and stiffness matrices. ACKNOWLEGMENT The material presented in this paper is based upon work supported by the National Science Foundation under Grant No. DGE – 0654176. The first author is grateful to the IGERT program established under this grant and centered on the topic of Intelligent Diagnostics.
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356 REFERENCES [1] Bernal, D. “Load Vectors for Damage Localization.” Journal of Engineering Mechanics, Vol. 128, No. 1, 7-14. (2002) [2] Bernal D. “Damage Localization from the Null Space of Changes in the Transfer Matrix.” AIAA Journal, Vol. 45, No. 2, 374-381. (2007) [3] Gao, Y., Spencer, B.F., and Bernal, D. “Experimental Verification of the Flexibility-Based Damage Locating Vector Method,” Journal of Engineering Mechanics, Vol. 133, No. 10, 1043-1049. (2007) [4] Tran, V.A., Duan, W.H., and Quek, S.T. “Structural Damage Assessment using Damage Locating Vector with Limited Sensors,” Proceedings of SPIE, The International Society for Optical Engineering, Vol. 6932, 693226-1-693226-11. (2008)
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Residual Analysis for Damage Detection: Effects of Variation on the Noise Model
Christopher Wright1 and Dionisio Bernal2
1. Graduate student IGERT program, Northeastern University, Department of Civil and Environmental Engineering. Address: 360 Huntington Ave., Stearns 11, Boston, MA 02115 2. Northeastern University, Department of Civil and Environmental Engineering, Center for Digital Signal Processing.
ABSTRACT Deviations from whiteness in the innovations of a Kalman filter indicate that the filter is not optimal for the given data. Lack of optimality can come from changes in the system properties but also from discrepancies between the noise statistics used to formulate the filter and the actual values. This paper examines the importance that changes in the noise part of the filter have on its ability to function as a damage detector. Analysis and numerical results show that the sensitivity is such that a realistic assessment of performance demands that potential fluctuations in the noise statistics be accounted for.
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_34, © The Society for Experimental Mechanics, Inc. 2011
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NOMENCLATURE State Space Model Time Step Index State Vector Output Vector Process Noise Vector Measurement Noise Vector Number of Output Channels Number of Input Channels Total Number of Time Steps Covariance of the Process Noise Covariance of the Measurement Noise Estimate for the State Estimate for the Output Innovation Sequence Kalman Gain State Error Covariance of the State Error Gain Matrix Covariance of the Innovations Mean of the Innovations Normalized Innovations Covariance Function for the Innovations Number of Lags Test Statistic Threshold Value for
INTRODUCTION The goal of damage detection in civil engineering is to identify conditions that pose a threat to the safety and reliability of a structure. This goal is accomplished when one can determine with confidence that the structure is healthy, or that some undesirable change has taken place. There are two general approaches to the damage detection problem, parameter estimation techniques and residual analysis [1]. In the parameter estimation technique mathematical models for the system are formulated for each data set and changes in the model parameters are tracked to determine whether significant changes have taken place. In the residual based approach, a single reference model is used and changes in the system are judged through discrepancies between predictions obtained with the model and the measurements. The key components of a residual analysis scheme are: 1) formulation of a reference model for the healthy system, 2) generation of a residual or innovation sequence using new data and the reference model, 3) calculation of a metric or test statistic from the residual sequence, and 4) criteria for deciding whether the metric represents a damaged or healthy system at a given confidence level. Information about the system can be extracted from the residual sequence by looking at either amplitude or frequency content. The main drawbacks of amplitude analysis are that the input must be known (measureable) and that the distribution of the test statistic must be determined empirically. The classical
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frequency content detector, introduced by Mehra and Peschon [2], is a whiteness test on the innovations of a Kalman filter [3]. Advantages of the whiteness test are that it can be applied to systems excited by stochastic inputs and that the theoretical distribution of the test statistic under the null hypothesis (i.e., that the system hasn’t changed) is available. Changes to the system are detected by observing a deviation from whiteness in the innovations. In the formulation of a Kalman filter the statistics of the noise in the system, namely the covariance of the process and measurement noise play a key role. The issue at hand is that changes in the noise statistics and system parameters (caused by damage) both have the ability to effect a deviation from whiteness in the innovations. Over a lengthy health monitoring period (steps 1-2 in the previously outlined scheme), variability in the noise covariance is a likely scenario. In the literature one finds several authors who have successfully used the Kalman filter approach to detect damage in both simulated and experimental capacities, among them Frank [4], Zhou [5], Liu [6], and Bodeux [7]. The issue of damage detection under changing noise conditions, however, has been touched on indirectly at best, with no definitive results or characterization of the problem. The purpose of this paper is to evaluate the sensitivity of a Kalman filter to changes in the system parameters when the statistics of the noise have changed since the reference model was created. The beginning of the paper reviews the relevant theory following the 4 steps to set up a damage detector using a Kalman filter. A numerical examination is presented and the paper is finished with a summary and concluding remarks. THE REFERENCE MODEL The Kalman Filter A linear time invariant system with degrees of freedom and disturbances can be described in discrete time as [8]
sensors (output channels) subjected to (1a) (1b)
where, is the state vector, is the dimensional process noise vector, is the dimensional output vector , is the dimensional measurement noise vector, is the time step index ( are the system and state to output matrices. The process and and ), and ), and white. and measurement noise sequences are assumed Gaussian, zero mean ( If, in addition, one assumes the noise process is stationary then the covariance matrices are independent of time and one has (2a) (2b) where
is the expected value operator. A constant gain observer estimate of the state,
, is given by [9]
(3) The state error is and its steady state covariance is the trace of the state error covariance is the Kalman gain, which is given by
. The gain
(4) where
is the solution to the algebraic Riccati equation
that minimizes
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(5) The innovations are defined as the difference between the measurements and the estimate of the output, namely (6) A Data-Driven Kalman Filter In the original paper by Mehra and Peschon [2], the reference Kalman filter was established under the premise that an accurate model for the system is available and that the covariance matrices of the process and the measurement noise are known. In practice, it is common that neither of these conditions are satisfied. For damage detection purposes, however, only the innovation sequences are of interest and the basis of the state is immaterial. For these conditions it is well-known that one can formulate the Kalman filter directly from the data [10], [11]. In the numerical examinations presented subsequently, the reference Kalman filter is formulated from measurements. Note that some approximation is introduced because the length of the sequences used is finite. THE METRIC (TEST STATISTIC) The statistic in the whiteness test can take several forms but here we use the sum of the covariance of the innovations for a preselected number of lags. One begins by obtaining a unit variance normalized innovation sequence. To do this, the empirical covariance matrix of the innovations, (10) is computed, where
is the length of the sequence and
innovations are obtained as computed as
is the mean (assumed to be 0). The normalized
. The covariance function for the normalized innovations is then
(11) for , where is the number of lags, under the null hypothesis each represents an identically distributed random variable with variance [2]. Here we opted to normalize the covariance function so that each entry has unit variance. Namely, we take . In this way the test statistic (12) under the null hypothesis (i.e., that there are no changes to the system) has a distribution with p degrees of freedom (DOF). Figure 1 shows the CDF for 20 DOF and depicts the threshold above which, under the null hypothesis, there is only 5% chance of finding q.
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Fig. 1. Theoretical
CDF with 20 DOF showing the threshold, , above which there is a 5% chance of finding under the null hypothesis.
Type I and Type II Errors, Power of Detection If and one asserts, incorrectly, that the structure is damaged, a Type I error has been incurred. The opposite situation, in which and the structure is judged to be healthy, when it is in fact damaged, constitutes a Type II error. Note that the null hypothesis (no change) requires not only that the system be undamaged, but also that the noise statistics be unchanged from those used to formulate the filter. Since the noise statistics do change one realizes that the actual distribution of the metric in the healthy state is not going to be the theoretical , but will have to reflect the correlations that arise from changes in and . An additional contribution to the departure for the theoretical comes from the fact that the model of the reference state is approximate because it is based on finite duration sequences. The probability of Type II errors cannot be assessed without knowledge of the probability distributions of the damaged states. In practice one may be able (in some cases) to estimate these probabilities in a simulation context but here we take the position that this is impractical and that the best one can hope for is to have the distribution of the healthy state. What can be done in practice in a data-driven fashion is to select a threshold based on an acceptable level of Type I error. In practice it is common to use one minus the chance of a Type II error and call it the Power of the Test; in this paper we call this index the Power of Detection (POD) to emphasize the damage detection application. To develop an appreciation for the importance of the fluctuations in and we compute the POD for several damage scenarios for various parameterizations of the fluctuations in and . In doing so we impose that the maximum permissible chance for a Type I error is 10%. NUMERICAL EXAMINATION We perform the computations using the 5-DOF mass and spring system depicted in Figure 2. Damping is 2% in all the modes and the sensor arrangement is a single output sensor placed on mass #3. The system is excited by stochastic disturbances applied at all the coordinates. The following list describes the noise scenarios, with #1 being the control case. 1) Q: constant, R: constant. This is the control where and at the interrogation stage are the same as when the model was formulated. 2) Q: variable scale, R: proportional. varies from the model by a random multiplier between 0.25 and 4. is consistent with a noise that has 5% of the standard deviation of the output. 3) Q: variable pattern, R: proportional. Each entry in is allowed to vary independently between 0.25 and 4 times the value from the model. as in #2.
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4) Q: variable scale, R: constant. as in #2, as in #1. 5) Q: variable pattern, R: constant. as in #3, as in #1. Damage is considered by reducing the stiffness of the springs (one at a time) from 0 to 30%. For the statistical analysis one case consists of 100 simulations of a given noise scenario, one damage location, and one damage extent. The POD’s for each case were computed by fitting a generalized extreme value (GEV) density function to the data.
Fig. 2. Five DOF mass and spring system used for numerical simulations with in some set of consistent units.
,
, and
The probability density function (PDF) of the control case is depicted in Figure 3. As can be seen, the discrepancy with the theory is significant and stems from the approximate nature of the Kalman filter extracted from the finite duration data of the reference state.
Fig. 3. PDFs from theory and noise model #1.
Figure 4a shows the density functions for all the noise scenarios when the system is undamaged and Figure 4b has the distributions for noise model #3 as a function of the damage severity in the case of damage on spring #6. As can be seen, the distribution functions for the healthy system vary widely with the noise model. The relative similarity of the curves for models #1 and #2 is expected as the Kalman filter is invariant to equal scaling of and . Note that keeping proportional to the output in noise model #2 implements this condition. In models #3, #4, and #5, where either the pattern of is changing or and are not scaled by the same amount, the greatly increased variance of the healthy condition makes detection of low levels of damage a difficult task. As can be seen from Figure 4b, 5% damage is essentially impossible to detect in the selected spring.
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Fig. 4. a) Distribution of
for healthy system under different noise models b) Distributions for increasing damage levels in spring #6 under noise model #3.
In Figure 5, the POD is calculated for a range of damage severities ranging from 0 to 30% for each spring under noise model #1, where and are invariant. As one expects, there is a damage threshold below which damage cannot be detected at a given POD confidence level (the limit coming from the fact that the model is approximate and the data sequences finite). Damage Severity vs. POD, Noise Model #1
1 0.9
Spring 1 Spring 2 Spring 3 Spring 4 Spring 5 Spring 6 Spring 7
0.8 0.7
POD
0.6 0.5 0.4 0.3 0.2 0.1 0 0
0.05
0.1
0.15 0.2 Damage Severity
0.25
0.3
0.35
Fig. 5. POD calculated for a range of damage severities for the case of noise model #1. Note that at 0% damage, the POD is just the permissible level of Type I error, in this case 10%.
The results in Figure 5 show that 5% damage can be detected with 80% POD or better in all springs except for #2 and #7, where the minimum detectable damage is around 10% and nearly 20% respectively for the same POD. This result shows that the sensitivity of the whiteness test can vary notably as the location of the damage changes from one spring to the next (for a given sensor placement) and that a fusion of several strategies (with different sensitivities) may be an attractive solution strategy. For contrast, Figure 6 shows POD versus damage severity for noise model #4, which proved to be the most detrimental to the damage detection task (although not by much as similar results were achieved with noise models #3 and #5). Minimum detectable damage thresholds are greatly increased and we note that damage up to 30% on spring #2 and #7 (as well as #4) is nearly impossible to detect.
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1 0.9 0.8 0.7
POD
0.6
Spring 1 Spring 2 Spring 3 Spring 4 Spring 5 Spring 6 Spring 7
0.5 0.4 0.3 0.2 0.1 0 0
0.05
0.1
0.15
0.2
0.25
0.3
Damage Severity
Fig. 6. POD calculated for a range of damage severities under the most detrimental noise model, #4.
CONCLUDING REMARKS This study shows that changes in the noise model have a notable effect in the detection capability of a whiteness test on the innovations of a Kalman filter. When and are invariant, damage as low as 5% can be detected with a POD of 80% or better in 5 out of the 7 springs in the example considered. In the case of noise fluctuation as in model #4, however, the minimum detectable damage at 80% POD increases by more than 300% in the same springs. Although these results are for an academic example they suggest that the issue of fluctuation in noise statistics is potentially important in determining the capacity of the Kalman filter as a detector. It is worth noting that scaling and by the same amount (noise model #2) had little effect on the distribution of the test statistic because the Kalman filter gain is invariant under these changes. Whether this “benign” scenario is a good approximation of some practical conditions is unclear at this time. Although not central to this study, it is worth restating that the location of the damaged can play a significant role in the ability for a whiteness test to detect damage. In the example considered springs #2 and #7 were identified as being particularly problematic and even at high levels of damage (25%), the detector could not perform with any reasonable level of confidence. While one anticipates that the noise encountered in practice will differ from case to case, studies to determine realistic fluctuation for “typical situations” would be useful to guide future simulation studies. Since the sensitivity of any technique is likely to vary with the damage scenario the fusion of methods that are complementary may be a fruitful strategy to improve the performance of damage detection. The results of this study suggest that a Kalman filter detector may be significantly less powerful than one would surmise if the fluctuations in the noise model are not considered. ACKNOWLEDGEMENT The first author’s time was supported by a grant from the National Science Foundation under the Integrative Graduate Education and Research Traineeship (IGERT) Intelligent Diagnostics Fellowship (Award Number NSF DGE-0654176), this support is gratefully acknowledged.
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REFERENCES [1] Patton, R. J., Frank P. M., and Clark R. N. Issues of Fault Diagnosis for Dynamic Systems. Springer. 2000. [2] Mehra, R. K., and Peschon, J. “An innovations approach to fault detection in dynamic systems.” Automatica. 7 637–640. 1971. [3] Kalman R. E. “A new approach to linear filtering and prediction problems.” Journal of Basic Engineering, Vol.82, 34-45. 1960. [4] Frank, Paul M. "Fault Diagnosis in Dynamic Systems Using Analytical and Knowledge-based Redundency--A Survey of Some New Results." Automatica 26.3 (1990): 459-74. [5] Zhou, Li, Shinya Wu, and Jann Yang. "Experimental Study of an Adaptive Extended Kalman Filter for Structural Damage Identification." Journal of Infrastructure Systems 42 (2008). [6] Liu, X., P.J Escamilla-Ambrosio, and N.A.J. Lieven. "Extended Kalman Filtering for the Detection of Damage in Linear Mechanical Structures." Journal of Sound and Vibration 325 (2009): 1023-046. [7] Bodeux, J. B., and J. C. Golinval. "Application of ARMAV models to the Identification and Damage Detection of Mechanical and Civil Engineering Structures." Smart Materials and Testing 10 (2001): 47989. [8] Brogan, W. Modern Control Theory. Upper Saddle River, NJ: Prentice Hall. 3rd ed. 1974. [9] Kaylath, T., Sayed, A. H., and Hassibi, B. Linear Estimation. Prentice-Hall. Information and System Science Series. Upper Saddle, River, N.J. 2000. [10] Di Ruscio, D. “Combined Deterministic and Stochastic System Identification and Realization: DSR— A Subspace Approach Based on Observations; Modeling, Identification, and Control.” 17(3), 193-230. 1996. [11] Van Overschee, P., and Moor, B. L. “Subspace identification for linear systems: Theory, Implementation, Applications.” Kluwer Academic, Boston. 1996.
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BookID 214574_ChapID 35_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Minimizing Distortions from Time Domain Zooming
David Abramo, Research Assistant, Northeastern University, 360 Huntington Ave. 11 Stearns, Boston, MA 02115 Dionisio Bernal, Associate Professor, Northeastern University, Center for Digital Signal Processing 360 Huntington Ave. 400 Snell Engineering, Boston, MA 02115 ABSTRACT The highest frequency to be identified imposes a lower bound on the sampling rate, and, given a sampling rate, the lowest frequency that can be identified is limited by noise and finite precision. When the bandwidth to be identified is large these two requirements imply that identification of all modes of interest in a single analysis is not feasible and some type of zooming becomes necessary. The traditional approach to realize time domain zooming is decimation. This paper reviews another alternative, presented within the framework of the Eigensystem Realization Algorithm (ERA), where zooming is realized without filtering by appropriate shifting of the two data matrices used in the algorithm. The method is known as ERA/S, where the S indicates shifting and skipping, and the skipping is due to the arrangement of the Markov Parameters (MP) in the data matrices. Since ERA/S does not filter the data, the modes outside the (shifted) Nyquist band appear aliased. The aliased modes, however, can be easily identified and removed. NOMENCLATURE t : Sampling Time of Input and Output data Imax : Maximum frequency of the system to be identified Ad, Bd, Cd, Dd : Discrete state space matrices of the system Ac, Bc, Cc, Dc : Continuous state space matrices of the system H : Data matrix associated with Eigensystem Realization Algorithm Pα : Observability matrix Qβ : Controllability matrix <ℓ : Markov Parameters Įȕ: Data matrix dimension parameters : Rectangular matrix associated with SVD of H Rs , Ss : First s columns of R and S associated with SVD of H q: Shifting/Skipping parameter : Data matrix associated with ERA/S IG Identified system modes I,' Expected location of aliased modes I1\T Nyquist frequency ȟ Identified damping ȟ,' Calculated value of aliased damping INTRODUCTION Due to the nature of sampled data, the useful bandwidth is limited to 1/ t. In applications, this bandwidth is centered at the origin on the frequency line so the maximum frequency that can be seen un-aliased is 1/2 t. It follows that if the highest frequency to be identified from sampled data is Imax the sampling rate has to be selected such that
W
I PD[
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_35, © The Society for Experimental Mechanics, Inc. 2011
(1)
367
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368 If Imax is large, a high sampling rate is needed; however, a high sampling creates difficulties in the identification of the low frequency modes. The reason being is that the eigenvalues of the discrete time model approach unity as the time step becomes very small compared to the period of the mode. Therefore, noise combined with finite precision make it impossible to differentiate them [1]. The common approach for such situations is to divide the full bandwidth into a series of smaller segments by changing the sampling rate. The change in sampling rate is obtained by low pass filtering and down-sampling the data, i.e., by decimation. The decimated realizations for the various sampling rates are typically taken to continuous time using the zero-order-hold (ZOH), from which they then can be combined to form a realization of the whole system. Since the real input does not satisfy the ZOH, the discrete to continuous (d2C) transform is approximate and error increases as the sampling rate to modal frequency decreases. One way to reduce the d2C error is to keep the sampling rate high relative to all the modes of interest but, as noted previously, this can lead to trouble due to noise and finite precision when the bandwidth is large. One way to get around the need for decimation, originally presented by Bernal and Tigli [2] and designated as ERA/S, is to realize the time domain zooming by appropriate time shifting of the data matrices used in the ERA. Specifically, in ERA/S, where the S stands for skipping and shifting, the Hankel matrix H1 (used in the standard algorithm) is replaced by Hq, where q is some integer greater than or equal to one. It is shown that this is equivalent to having increased the time step by a factor of q. This approach does not introduce any filtering and, as a consequence, the modes that are outside the shifted Nyquist limit appear aliased in the identification. As shown in the original paper and reviewed here, however, the aliased modes can be easily identified and removed. The objective of this paper is to provide further numerical evidence on the merit of ERA/S by comparing its performance with decimation results. EIGENSYSTEM REALIZATION ALGORITHM (ERA) A brief summary of the ERA is presented next. The observability and controllability matrices of a discrete time system are defined as
3
&G
4
&G $G
%G
&G $G
7 7
(2)
$G %G
$G %G
(3)
and from these definitions one can define +k as
+N
3 $GN 4
(4)
where
<
&G $G %G
(5)
are the discrete time Markov Parameters, where Ɛ 1 and for Ɛ = 0, <0 = Dd. In Eq. 4 and are arbitrary. The Markov Parameters, MP, can be found for a set of input and output data using well known procedures. Once the matrix in Eq. 4 is formed from the data, the ERA extracts a triplet {Ad ,Bd,Cd} as follows: taking k = 0 in Eq. 4 one has
+ Performing a singular value decomposition of +0 gives
34
(6)
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+ where
5 67
(7)
(8)
is a rectangular matrix that can be written as V
where the non-zero entries is a diagonal matrix with s entries. Letting 5s and 6s be matrices corresponding to the first s columns of 5 and 6, +0 is also
5V
+
V
6V7
(9)
One gathers, therefore, (from Eqs. 6 and 9) that one can take
3
5V
V
4
V
From Eqs. 2 and 3 it follows that the first m rows of 5V and the first r columns of
V
V
(10)
6 V7
gives a realization of the discrete time state matrix Cd
6 V7 gives a realization of the discrete time state matrix Bd, where m is the number of
outputs and r is the number of inputs. Next, taking k = 1 in Eq. 4,
3 $G 4
+
(11)
Substituting Eq. 10 into Eq. 11 and recognizing the orthogonal properties of 5s and 6s, one gets
$G
V
5V7 +6 V
V
(12)
which completes the realization. EIGENSYSTEM REALIZATION ALGORITHM WITH SHIFTING AND SKIPPING (ERA/S) Shifting Note that in the formulation of ERA, H1 is calculated to find a realization of Ad (Eq. 12). Consider, instead of H1, the matrix Hq, where q is an integer value greater than or equal to one. From Eq. 4, Hq is
3 $G T 4
+T
(13)
T
Similar to Eq. 12, a realization for $G can be found as
$G T
V
5V7 + T 6 V
V
(14)
The relationship between the discrete and the continuous time system matrix is known to be
$G Therefore, raising both sides by the power q one gets
H $F
W
(15)
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$G T H $F T
W
(16)
From where it follows that
OQ $G T
$F
(17)
T W
Solving for Ac in Eq. q 15 and comparing to Eq. 17, one concludes that the shifting parameter q creates an effective time step of q t . Since no filtering is done, however, modes outside the shifted Nyquist limit will appear aliased. Shifting and Skipping The idea is to encompass both skipping and shifting by simply selecting a shifting parameter q and a skipping parameter p, where q is selected to shift the Nyquist limit and p is selected to reduce the size of the data matrices [3] and, in theory, they do not have to be equal. Numerical results show, however, that selecting p = q is good practice because in this way the same MP’s are included in the H0 and Hq matrices, which improves consistency given the inevitable approximations that come from noise and finite precision. Therefore, with this requirement, Eq. 4 takes the form
where T
(T
3 $G T 4
< T
<TT
< T
<
T
< < <
T T
(18)
T
results in data matrix H1 in the ERA).
Discriminating the Aliased Modes The location of aliased modes can be obtained, as shown in [2], from
I ,'
PRG I G I 1\T IRU1=
I ,'
I 1\T PRG I G I 1\T IRU1=
(19)
where NZ is the Nyquist zone, which depends on the where the identified damped system modes, Id, are in relation to the Nyquist limit, I1\T. That is, for each Id, if 0
,'
where D
7G
Q
I G and 7,'
D D
and ȟ is the previously identified damping, Qis the ratio between periods, Q
(20)
7,' 7 G , where
I ,,' .
EXAMPLE A 4-degree of freedom system shown in Figure 1 has been created to demonstrate the ERA/S. The parameters of the model have been selected such that the highest frequency is 187.5 Hz and the lowest 1.12 Hz, thus st rd creating a wide bandwidth system. The input is on the 1 and 3 masses while there are velocity sensors at all locations. The sampling rate is 1000 Hz. The excitation consists of white noise sampled at 1000 Hz passed
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371 th
through an 8 order Chebyshev Type-I low pass filter with cutoff frequency at 125 Hz. The output is computed under the premise that the input satisfies the ZOH. Both the input and output signals are contaminated by colored noise. The damping distribution is not classical but examination shows that the deviation of the eigenvectors from purely real (as one expects because the eigenvalues are well separated) is minimal. The damping ratios ( ) are
L
computed conventionally by taking the complex eigenvalues as
, which gives damping ratios of
[0.0234 0.0053 0.0031 0.0284].
N
N P F N
F
N P
N P
F
F
P F
Figure 1: 4-DOF system used to illustrate ERA/S (k = 75 N/m, m1= m3 = m 4 = 0.5 kg, m2= 0.015 kg, and c = 0.5 N-sec/m) ERA/S vs. DECIMATION The ERA/S is carried out with q=1, q=25, and q=250. The original ERA is the same as ERA/S with q =1 and the results are in Table 1 which shows the identified frequencies, damping ratio’s, and weighted modal phase collinearity (mpcw) indices [4] for the selected order. The damping ratio of unity in mode 1 is an artifact of the computational algorithm reflecting that the identified damping is larger than critical. Since the damping ratios are small we take this result to be spurious. The higher modes of 187.62 Hz, 82.93 Hz, and 48.01 Hz and their damping ratios are, however, accurately identified. Table 1: Results of ERA/S with q=1 and an order of 7 selected Mode
Frequency
Damping Ratio
Mpcw
1 2 3 4 5 6 7
0.91 48.01 48.01 82.93 82.93 187.62 187.62
1 0.0056 0.0056 0.0032 0.0032 0.028 0.028
100 99.97 99.97 99.99 99.99 99.99 99.99
The q=25 case results in a Nyquist frequency of 20Hz and no system modes are identified. Only previous modes that have already been identified appear aliased at expected locations calculated from Eqs. 19 and 20. The q=250 case results in a Nyquist frequency of 2Hz and the results are shown in Table 2. Based on the examination of the damping ratios, modes 1, 2, 3, 6, 7, and 8 are ruled out leaving an identified mode at 1.12 Hz with a damping ratio of 0.024, which are virtually exact for the first mode of the real system. The results from decimation are presented in Table 3, of which, we went directly to a Nyquist frequency of 2Hz. It
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372 is apparent that modes 3, 4, 5, and 6 are not system modes, but modes 1 and 2 provide an accurate identification of the first system mode and its damping. Note that the decimation results, as far as frequencies and damping ratios go, are accurate. As noted previously, the distortion one anticipates from decimation is on the input to state matrix and this can be best appreciated by looking at transfer functions. We illustrate, therefore, the magnitude of rd the transfer functions from the input on the 3 mass to all outputs. Figure 2 compares the results from the decimated identification to the exact solution and Figure 3 compares the exact solution to the estimation obtained using ERA/S. As can be seen, the distortion introduced by the decimation process is significant. Table 2: Results of ERA/S with q=250, Nyquist of 2 Hz with an order of 8 selected Mode
Frequency
Damping Ratio
Mpcw
1 2 3 4 5 6 7 8
0.258 1.11 1.11 1.12 1.12 2.82 2.82 3.32
1 0.235 0.235 0.024 0.024 0.967 0.967 0.811
100 99.98 99.98 99.99 99.99 52.65 52.65 100
Table 3: Results of ERA considering data decimated by 250, Nyquist of 2 Hz with an order of 6 selected Mode
Frequency
Damping Ratio
Mpcw
1 2 3 4 5 6
1.12 1.12 -1.39 -1.39 -1.64 -1.64
0.027 0.027 0.268 0.268 0.023 0.023
99.99 99.99 99.32 99.32 92.32 92.32
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rd
Figure 2: Magnitude of the transfer matrix at each coordinate due to input on the 3 mass for the exact system shown in Figure 1 compared to the identified system from a realization using decimated data.
rd
Figure 3 : Magnitude of the transfer matrix at each coordinate due to input on the 3 mass for the exact system shown in Figure 1 compared to the identified system from a realization using ERA/S.
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374 CONCLUSION This paper reviews the ERA/S, a modification of ERA that can be used to treat systems with large bandwidth without the need to decimate the data. The advantage of ERA/S compared to decimation is the fact that the d2C is performed much more accurately because, in effect, the high sampling rate is preserved throughout. The procedure is relatively easy to incorporate in codes that already implement the ERA. A disadvantage over the standard decimation approach is the fact that there is a need to keep track of the modes that will appear aliased and the fact that the number of MP’s that have to be computed at the outset is large. This is because they are obtained at the high sampling rate and must cover sufficient duration to contain a significant fraction of the period of the slowest mode in the system. ACKNOLEDGMENT The first author is currently supported by the National Science Foundation under IGERTGrant No. DGE – 0654176, this support is gratefully acknowledged. REFERENCES [1] Soderstrom, T., & Stoica, P. System Identification. Englewood Cliffs, NJ: Prentice-Hall International, 1988. [2] Bernal, D., & Tigli, O. The ERA/S Algorithm. Proc. XXV International Modal Analysis Conference. Orlando, 2007. [3] Juang, J.-N. Applied System Identification. Upper Saddle River, NJ: Prentice Hall PTR, 1994. [4] Pappa, R. S., Elliott, K. B., & Schenk, A. Consistent-mode indicator for the eigensystem realization algorithm. Journal of Guidance, Control, and Dynamics , 16 (5), 852-858, 1993.
BookID 214574_ChapID 36_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII
February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc. 3URFHVVDQG0HDVXUHPHQW1RLVH(VWLPDWLRQIRU.DOPDQ)LOWHULQJ Yalcin Bulut1, D. Vines-Cavanaugh2, Dionisio Bernal3 Department of Civil and Environmental Engineering, 427 Snell Engineering Center Northeastern University, Boston 02215, MA, USA
[email protected],
[email protected],
[email protected] 1
PhD Candidate,
2
PhD Student,
3
Associate Professor
ABSTRACT The Kalman filter gain can be extracted from output signals but the covariance of the state error cannot be evaluated without knowledge of the covariance of the process and measurement noise Q and R. Among the methods that have been developed to estimate Q and R from measurements the two that have received most attention are based on linear relations between these matrices and: 1) the covariance function of the innovations from any stable filter or 2) the covariance function of the output measurements. This paper reviews the two approaches and offers some observations regarding how the initial estimate of the gain in the innovations approach may affect accuracy. Keywords: Kalman Filter, Process Noise, Measurement Noise NOMENCLATURE
A
Discrete system matrix
B
Input matrix of dynamic system
C
Measurement matrix of state space model
E
Expected value operator
K
Steady state Kalman gain
K0 lˆ
Initial steady state Kalman gain
lj
Covariance function of the innovation process
Ob
Observability matrix
P0
Initial state error covariance matrix
P
State error covariance matrix
x0
Initial state vector
xk
State vector at time k
j
Estimate of the covariance function of the innovation process
xˆ
k
State estimate after the measurement at time k is taken into consideration
xˆ
k
State estimate before the measurement at time k is taken into consideration
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_36, © The Society for Experimental Mechanics, Inc. 2011
375
BookID 214574_ChapID 36_Proof# 1 - 23/04/2011
376
yk
Discrete measurement vector
wk
Discrete process noise
vk Rˆ
Discrete measurement noise
R Qˆ
Covariance matrix of the measurement noise Estimate of the covariance matrix of the process noise
Q
Covariance matrix of the process noise
6
State covariance matrix
/j
Covariance function of the output
†
Pseudo inverse matrix operator
Estimate of the covariance matrix of the measurement noise
INTRODUCTION The basic idea in estimation theory is to obtain approximations of the true response by using information from a model and from any available measurements. The mathematical structure used to perform estimation is known as an observer. The optimal observer for linear systems subjected to broad band disturbances is the Kalman Filter (KF). In the classical presentation of the filter the gain, K, is computed given the model parameters and the covariance of the process and the measurement noise, Q and R, respectively. Since Q and R are seldom known a priori work to determine how to estimate these matrices from the measured data began soon after introduction of the filter. An important result, apparently first published by Son and Anderson [1], is the fact that the information needed to determine K is encoded in the output signals and thus, given sufficiently long data sequences K can be estimated from the measurements without the need to know or estimate the Q, R matrices. Perhaps the most widely quoted strategies to carry out the estimation of K are due to Mehra [2] who built on the work by Heffes [3] and the subsequent paper by Carew and Bellanger [4], both techniques being iterative in nature. A noteworthy contribution from this early work is the contributions by Neethling and Young [5], who suggested some computational adjustments that could be used to improve accuracy. The problem of computing K can be approached, of course, as one of estimating Q and R because once these matrices are known K follows from well known relations. If all that is of interest is K, therefore, one can take the direct approach, i.e., from the data to K, or the indirect one that estimates Q and R first. Nevertheless, if in addition to estimating the state one is interested in the covariance of the state error then the computation of Q and R becomes necessary since there is no direct approach (at least not known to the writers) to go from the data to the state error covariance P. The two approaches that have received most attention in the Q, R estimation problem are based on: 1) correlations of the innovation sequence and 2) correlations of the output. In the innovations approach one begins by “guessing” a filter gain (usually be estimating some Q and R from whatever a priori knowledge there may be) and then the approach obtains the optimal
BookID 214574_ChapID 36_Proof# 1 - 23/04/2011
377 gain, the K, from analysis of the resulting innovations. In the limiting case where the output sequences are very long (infinite in principle) the results converge to the exact solution independently of the initial “guess”. In the real situation, however, where one must operates with finite length sequences (finite precision and so on) the results for Q and R do depend on the initial estimation and a question that arises is whether the answer is sensitive or not. . In this regard, Mehra [6] suggested that the estimation of Q and R could be repeated by starting with the gain obtained in the first attempt but this expectation was challenged by Carew and Bellanger [4], who noted that the exact gain is used as the initial guess the approximations in the approach are such that the correctness will not be confirmed. Some observations on the issue of the initial gain are presented later in this paper. The output correlation approach can be viewed as a special case of the innovations where the initial gain is taken as zero. In practice, however, the output correlation alternative has been discussed as a separate approach (probably) because the mathematical arrangements that are possible in this special case cannot be easily extended to the case where K 0. It is opportune to note that some recent work on the estimation of Q and R has taken place in the structural health monitoring community but the work appears to have been
done in isolation of the classical works noted
previously, [7]. Part of the motivation for this paper is to offer a concise review of the classical correlation based approaches that may prove useful in the structural engineering community. The paper is organized as follows: the next section provides a brief summary of the KF particularized to a time invariant linear system with stationary disturbances (which is a condition we have implicitly assumed throughout the previous discussion). The following two sections review the formulations to estimate Q and R using the innovations and output correlations and the paper concludes with a numerical example. THE KALMAN FILTER
Consider a time invariant linear system with unmeasured disturbances w(t) and available measurements y(t) that are linearly related to the state vector x(t). The system has the following description in sampled time
xk 1 yk
Axk Bwk
(1)
Cxk vk
(2)
A \ nxn , B \ nxr and C \ mxn are the transition, input to state and state to output mx1 nx1 matrices, yk \ is the measurement vector and xk \ is the state. The sequence nx1 mx1 wk \ is known as the process noise and vk \ is the measurement noise. In the where
treatment here, it is assumed that these are uncorrelated Gaussian stationary white noise sequences with zero mean and covariance of Q and R, namely
E ( wk )
0 E ( wk w j T )
QG kj
(3a,b)
E ( vk )
0 E (vk v j T )
RG kj
(4a,b)
E ( vk w j T )
0
(5)
BookID 214574_ChapID 36_Proof# 1 - 23/04/2011
378 where
G kj denotes the Kronecker delta function and ( () denotes expectation. For the system in
eqs.1 and 2 the KF estimate of the state can be computed from
xˆk1 k
Axˆk
(6)
k
k
xˆ K ( yk Cxˆ )
xˆ
(7)
where xˆk is the estimate after the information from the measurement at time k is taken into
consideration and xˆk is the estimate before. The (steady state) Kalman gain K can be expressed in a number of alternative ways, a popular one is [8]
PC T (CPC T R)1
K
(8)
where P, the steady state covariance of the state error, is the solution of the Riccati equation
P
A( P PC T (CPC T R) 1 CP) AT BQBT
(9)
The KF provides an estimate of the state for which P is minimal. The filter is initialized as follows
xˆ0
P0
E[ x0 ]
(10)
E[( x0 xˆ0 )( x0 xˆ0 )T ]
(11)
INNOVATIONS CORRELATION APPROACH FOR Q AND R ESTIMATION We begin with the expression for the covariance function of the innovation process for any stable observer with gain K0. As shown by Mehra [2] this function is
lj
CPC T R
j
C [ A( I K 0C )] j 1 A [ PC T K 0 (CPC T R )]
lj
(12)
0 j ! 0
(13)
where the covariance of the state error in the steady state, as shown by Heffes [3] is the solution of the Riccati equation
P
A( I K 0C ) P ( I K 0C )T AT AK 0 RK 0T AT BQBT
(14)
Estimation of PCT Listing explicitly the covariance function for lags one to A one has
l1
CA( PC T K 0l0 )
(15)
l2
C [ A( I K 0C )] A( PC T K 0l0 )
(16)
l3 C [ A( I K 0C )]2 A( PC T K 0 l0 ) ..................... lA C [ A( I K 0C )]A 1 A( PC T K 0l0 )
(17)
Z ( PC T K 0l0 )
(19)
(18)
from where one can write
L
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379 In which,
L
ªl1 º «l » « 2» «l3 » « » «...» «¬lA »¼
Z
ªCA º « » «C [ A( I K0C )] A » «C [ A( I K0C )]2 A » « » «...... » «C [ A( I K C )]A 1 A» 0 ¬ ¼
(20a,b)
As can be seen, matrix Z is the observability block of an observer whose gain is K0 post-multiplied by the transition matrix A. On the assumption that the closed loop is stable and observable, one concludes that Z attains full column rank when
A is no larger than the order of the system, n.
Accepting that the matrix is full rank one finds that the unique least square solution to eq.19 is
PC T where
K 0 l0 Z † L
(21)
Z † is the pseudo-inverse of Z . On the premise that the model is known without error T
(which is a strong assumption), error in solving eq.21 for PC results from the fact that the innovations covariance is approximate due to the finite duration of the signals. Estimation of R Having obtained an estimate for PCT, the covariance of the output noise R can be estimated from eq.12 as
Rˆ
ˆ ˆT ) lˆ0 C ( PC
(22)
where the hats are added to emphasize that the quantities are estimates. Estimation of Q Derivation of an expression to estimate Q is significantly more involved than the one for R. The process begins by replacing, in eq.14, the covariance R by its expression in terms of the autocorrelation at zero lag. After some algebra one gets
APAT M BQB T
(23)
A [ K 0CP PC T K 0T K 0l0 K 0T ] AT
(24)
P where
M
Now consider a recursive solution for P in eq.23. In a first substitution one has
A( APAT M BQBT ) AT M BQBT
(25)
A2 P( A2 )T AMAT ABQBT AT M BQBT
(26)
P or
P
BookID 214574_ChapID 36_Proof# 1 - 23/04/2011
380 and after q substitutions one gets
P
q 1
q 1
j 0
j 0
Aq P( Aq )T ¦ A j M ( A j )T ¦ A j BQBT ( A j )T
(27)
Before solving eq.27 for Q it is first necessary to extract the set of equations for which only knowledge of PCT (estimated by eq.21) is necessary. These equations are attained by post-multiplying both sides of eq.27 by CT and pre-multiplying by CA-q. This gives q 1
q 1
j 0
j 0
CP ( Aq )T C T CA q ¦ A j M ( A j )T C T CA q ¦ A j BQBT ( A j )T C T
(28)
Given that P is symmetric, the CP product to the right of the equal sign can be expressed as
( PC T )T ,
CA q PC T
and one has q 1
q 1
¦ CA j q BQBT ( A j )T C T
CA q PC T ( PC T )T ( Aq )T C T ¦ CA j q M ( A j )T C T
j 0
(29)
j 0
For convenience we transpose both sides of eq.29 and get q 1
¦ CA BQB j
T
q 1
( PC T )T ( A q )T C T CAq PC T ¦ CA j M ( A j q )T C T
( A j q )T C T
j 0
(30)
j 0
where we’ve accounted for the fact that M is symmetrical. To shorten eq.30 we define
CA j B
Fj T
Gj
B (A
(31)
j q T
) C
T
(32)
and q 1
sq
( PC T )T ( A q )T C T CAq PC T ¦ CA j M (A j q )T C T
(33)
j 0
So eq.30 becomes q 1
¦ F QG j
j
sq
(34)
j 0
Applying the vec operator to both sides of eq.34 one has q 1
¦ (G
T j
Fj ) vec(Q) vec( sq )
(35)
j 0
where
denotes the Kronecker product. Eq.35 can be evaluated for as many q values as one desires, although it is evident that all the equations obtained are not necessarily independent. Selecting q from one to p gives
H vec (Q ) where
S
(36)
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381
q 1
hq
¦ (G
T j
Fj )
j 0
H
ª h1 º «h » « 2» S « .. » « » ¬« hp ¼»
ª vec ( s1 ) º « vec( s ) » 2 » « « .. » « » ¬« vec( s p ) ¼»
(37a,b,c)
Eq.36 is the expression used in the innovations approach to obtain an estimate of Q. From its inspection one finds that H has dimensions m2 p x r2, where we recall that m and r represent the numbers of outputs and independent disturbances, respectively. Structure in Q The matrix Q is symmetrical and can often be assumed diagonal. The constraints of symmetry and/or a diagonal nature of Q can be reflected in a linear transformation of the form
vec(Q) T vec(Qr )
(38)
where vec(Qr) is the vector of unknown entries in Q after all the constraints are imposed. In the most general case, where only symmetry is imposed, the dimension of T is ଶ ൈ ሺ ͳሻȀʹ. However, since structural engineering problems are such that the largest possible value of r is n/2, T has a maximum dimension of ଶ ൈ ሺ ͳሻȀͶ. In the most general case in structures, therefore, the matrix H, after symmetry is imposed, is ଶ ൈ ሺ ͳሻȀͶ. A necessary (albeit not sufficient) condition for a unique solution for Q, therefore, is ξ ͳȀʹ. Enforcing Positive Semi-Definitiveness By definition, the true matrix Q is positive semi-definite (i.e., all its eigenvalues are 0), however, due to approximations, the least square solution may not satisfy this requirement. In the general case one can satisfy positive semi-definitiveness by recasting the problem as an optimization with constraints. Namely, minimize the norm of ሺ ܪൈ ܿ݁ݒሺܳሻ െ ܵሻ subject to the constraint that all eigenvalues of Q 0. This problem is particularly simple for our case, where Q is diagonal. In this scenario the entries in vec(Q) are the eigenvalues and all that is required is to ensure that they are not negative. Also, here is a suitable application for the always converging non-negative least squares solution [9]. This algorithm is used for the numerical example provided at the end of this paper. OUTPUT CORRELATION APPROACH FOR Q AND R ESTIMATION As in the previous case, it is assumed that the state is stationary and that the process and measurement noise are white and uncorrelated with each other, namely
E ( xk 1 xkT1 )
Ȉ
(39)
E ( xk vkT )
0
(40)
E ( xk wkT )
0
(41)
BookID 214574_ChapID 36_Proof# 1 - 23/04/2011
382
E ( wk vkT )
0
(42)
Estimation of R The covariance function of the output is
/i
E ( yk i ykT )
(43)
Substituting eqs.2 and enforcing the assumptions in 40-42 one gets
C ( E xk i xkT )C T
ȁi
(44)
From eq.1 one can show that
xk i xkT
Ai xk xkT Ai 1 Bwk xkT Ai 2 Bwk 1 xkT } Bwk i 1 xkT
(45)
Taking expectations on both sides of eq.45 gives
E ( xk i xkT )
Ai E ( xk xkT )
(46)
which when substituted into eq.44 gives
ȁ0
ȁi
C ȈC T R
C Ai ȈC T
for i
0
for i z 0
(47) (48)
Defining
E ( xk 1 ykT )
G
(49)
and substituting eqs.1 and 2, then imposing the assumptions in eqs.40-42 one gets
G
AȈC T
(50)
Writing out the covariance function in eq.48 for i=1,2…p one has
/1 ½ °/ ° ° 2° ® ¾ ° . ° °¯/ p °¿
ª C º « CA » « » A6C T « . » « p 1 » ¬CA ¼
(51)
or, substituting Eq.50 into Eq.51 and solving for G one has
G
/1 ½ °/ ° † ° 2° Obp ® ¾ ° . ° °¯ / p ¿°
(52)
BookID 214574_ChapID 36_Proof# 1 - 23/04/2011
383 From eq.4 47 and eq.50 one gets
R
ȁ 0 CA1G
(5 53)
Estimation n of Q Substiituting eq.1 in nto eq.39 and enforcing the e operating as ssumptions gives
Ȉ
AȈAT BQBT
(5 54)
Applying the t vec opera ator to eqs.54 and eq.50 on ne obtains
vec(6)
I ( A
A )1 ( B
B)vec(Q)
(5 55)
C
A vec(6)
(5 56)
and
vec(G ) Combining g eqs.55 and 56 gives
vec(G ) V vec (Q )
(5 57)
where
V
C
A I ( A
A )1 ( B
B)
(5 58)
Structure in Q The ob in the ca also. bservations made m ase of the inn novations app proach apply here h Enforcing Positive Sem mi-Definitivene ess The ob in the ca also. bservations made m ase of the inn novations app proach apply here h NUMERIC CAL EXAMINA ATION We co onsider the 5--DOF spring--mass structu ure depicted in n fig.1. The fiirst un-dampe ed frequency is 2.66Hz an nd the 5th is s 16.30Hz. Damping D is cclassical with 2% in each h mode and the stochasttic disturbancce, having a Q = 1, is app plied at the 1sst coordinate. Velocity mea asurements are a taken at th he 3rd coordiinate and the exact respon Noiise in the outp nse is computed at 50Hz sampling. s put is such th hat R = 5x10-44.
Fig.1 5-DOF F spring-masss structure (m m1-m5 = 0.05; k1,k3,k5,k7 =100; k2,k4,k6=1 120)
BookID 214574_ChapID 36_Proof# 1 - 23/04/2011
384 Two hundred simulations are performed on each of four different cases to investigate the affect of duration and number of lags on the estimation of the steady state error covariance matrix P. The cases are defined as follows: Case I: duration 200sec lags=40; Case II: duration 200sec. lags=10; Case III: duration 20sec. lags=40; and Case IV: duration 20sec. lags=10. Estimates of Q and R are obtained from the presented methods and substituted into eq.9 for the calculation of P. On the basis that eq.9 requires Q and R to be positive definite, all negative estimations of R are replaced by zero. Results are presented for the diagonal elements of the covariance of the state error as histograms in figs. 2 and 3; m and s represent the mean and standard deviation of the estimate and t represents the true value. We note the expected value of the maximum displacement response in the first degree of freedom is 0.07 and %10 error represents 4.9x10-5 variance. For conciseness only cases I and II are presented in figs. 2 and 3. These cases represent the most accurate estimations for both approaches. An interesting observation is that case I is most accurate for the innovations approach (many lags) while II (few lags) is most accurate for the output approach. One notes that both approaches have means in relatively good agreement with the true values but comparison of standard deviations show that the innovations approach is the more consistent estimator. As expected, the results for the shorter duration signals deteriorated notably (not shown for brevity).
P 11 Case I
0.2
2
2.5
P 33
0.2
m=2.4 s=0.15 t=2.4
0.1
0
P22 0.2 m=0.718 s=0.164 t=0.756
0.1
3
0
0
1
0
0.2
m=0.509 s=0.167 t=0.551
0.1
2
P44
0
0.5
0
0.4
m=1.62 s=0.25 t=1.67
0.1
1
P55
1
m=0.845 s=0.345 t=0.929
0.2
2
0
3
0
1
2
Fig.2 Histograms of the first five diagonal elements of the state error covariance obtained using estimates of Q and R from the innovations approach.
Case II
P11 0.4
0.2
m=2.4 s=0.2 t=2.4
0.2 0
P22
0
0.4
4
0
0
1
2
0
P44 0.2
m=0.549 s=0.238 t=0.551
m=0.752 s=0.213 0.2 t=0.756
0.1
2
P33
0
0.5
0
0.4
m=1.66 s=0.276 t=1.67
0.1
1
P55
1
2
m=0.946 s=0.514 t=0.929
0.2
3
0
0
1
2
Fig.3 Histograms of the first five diagonal elements of the state error covariance obtained using estimates of Q and R from the output approach. Influence of the Initial Gain Intuitively one expects that the closer the initial guess is to the true Kalman gain the more accurate the results will be. This is generally true but exceptions are possible and tend to occur if some poles of the closed loop observer are small enough to become negligible for powers less than the system order because in this case some of the singular values in Z (eq.20b) can become very small and can lead to numerical trouble. To exemplify the previous observations, we considered the same system illustrated
BookID 214574_ChapID 36_Proof# 1 - 23/04/2011
385 in fig.1, except that output measurements are assumed available not only at coordinate 3 but also at 5. The covariance matrices are Q=1, R=5*10-7I. Fig.4 shows histograms of the estimated Q based on 200 simulations for the case where the initial gains are: a) zero b) the exact KF and c) a filter based on Q = 1 R = 3I. The duration of the signals is 40 secs and 10 lags are used. As can be seen, the poorest
RelativeFrequency
result is obtained when the initial gain is the exact Kalman gain.
0.25
0.35
0.20
(a) 0.15
Mean=0.988 mean=0.988 std=0.098
0.30
std=0.098 (a)
0.25 0.20
0.25
Mean=0.997 mean=0.997
Mean=1.455 mean=1.455 std=0.866 std=0.866
0.20
(b) (b)
0.15
0.15
0.10
std=0.076 std=0.076
(c) (c)
0.10
0.10 0.05
0 0.5
0.05
0.05
1
1.5
0 -5
0
Q
5
0 0.5
1
1.5
Fig.4 Histograms of Q with measurement noise R = 5e-7 x I: a) gain = 0, b) gain = exact KF and c) gain = KF for Q = 1, R = 3 x I CONCLUDING COMMENTS The classical innovations and output covariance techniques to estimate Q and R from output measurements were reviewed. Both methods lead to the solution of a linear system of equations that is rather poorly conditioned so accuracy demands the use of long duration signals to minimize error in the correlation function estimates. In the numerical example signals with duration on the order of 40 times the period of the slowest mode proved inadequate. When the duration is 400 times the fundamental period the mean of 200 simulations proved in good agreement with the covariance of the state error but even then the coefficients of variation are not small. Specifically, in the innovations approach, which proved most accurate, the coefficient of variation of the 5th entry in the covariance of the state error is 0.4. Numerical results suggest that accuracy is promoted by having an initial gain that is close to the Kalman gain but care must be exercised to avoid initial gains that exacerbate the poor conditioning of the coefficient matrix that must be pseudo-inverted. While it is true that a second cycle where the “guess” for K is taken as the value from the first cycle is not guaranteed to provide improvement, this strategy, as suggested by Mehra [6] is expected to improve accuracy when the initial gain is far from the mark.
REFERENCES [1] Son, L.H. and B.D.O. Anderson, “Design of Kalman Filters Using Signal Model Output Statistics,” Proc.IEE, Vol.120, No.2, February 1973, pp.312-318
BookID 214574_ChapID 36_Proof# 1 - 23/04/2011
386 [2] Mehra, R. K. “On the identification of variance and adaptive Kalman filtering”, IEEE Transactions on Automatic Control, Vol. 15, No.2, 1970, pp. 175-184. [3] H. Heffes, “The effect of erroneous models on the Kalman filter response,” IEEE Transactions on Automatic Control, AC-11, July, 1966, pp. 541-543. [4] Carew, B. and Belanger, P.R., “Identification of Optimum Filter Steady-State Gain for Systems with Unknown Noise Covariances”, IEEE Transactions on Automatic Control, Vol.18, No.6, 1974, pp. 582–587. [5] C. Neethling and P. Young. “Comments on identification of optimum filter steady-state gain for systems with unknown noise covariances”, IEEE Transactions on Automatic Control, Vol.19, No.5, 1974, pp. 623-625. [6] Mehra, R.K. “Approaches to adaptive filtering,” IEEE Transactions on Automatic Control, Vol. AC-17, Oct., 1972, pp. 693-698. [7] Yuen, K.V., Hoi, K.I., and Mok, K.M., “Selection of Noise Parameters for Kalman Filter”, Journal of Earthquake Engineering and Engineering Vibration (Springer Verlag), Vol.6, No.1, 2007, pp. 49-56. [8] Dan Simon, “Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches”, WileyInterscience, 2006 [9] Lawson, C.L., & Hanson, R.J. "Solving Least Squares Problems", Prentice-Hall, 1974, Chapter 23, p.161.
BookID 214574_ChapID 37_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Model Calibration for Fatigue Crack Growth Analysis under Uncertainty
Shankar Sankararaman, You Ling, Chris Shantz and Sankaran Mahadevan Department of Civil and Environmental Engineering Box 1831-B, Vanderbilt University, Nashville, TN – 37235 United States of America
NOMENCLATURE a N ș ǻKth ıf ǻK Y C, m, n r ij
Crack Size No. of Loading Cycles Equivalent Initial Flaw Size Threshold Stress Intensity Factor Fatigue Limit Stress Intensity Factor Geometry Factor Parameters of Modified Paris’ Law Retardation coefficient
ABSTRACT This paper presents a Bayesian methodology for model calibration applied to fatigue crack growth analysis of structures with complicated geometry and subjected to multi-axial variable amplitude loading conditions. The crack growth analysis uses the concept of equivalent initial flaw size to replace small crack growth calculations and makes direct use of a long crack growth model. The equivalent initial flaw size is calculated from material and geometrical properties of the specimen. A surrogate model, trained by a few finite element runs, is used to calculate the stress intensity factor used in crack growth calculations. This eliminates repeated use of an expensive finite element model in each cycle and leads to rapid computation, thereby making the methodology efficient and inexpensive. Three different kinds of models – finite element models, surrogate models and crack growth models - are connected in this framework. Various sources of uncertainty – natural variability, data uncertainty and modeling errors - are considered in this procedure. The various component models, their model parameters and the modeling errors are integrated using a Bayesian approach. Using inspection data, the parameters of the crack growth model and the modeling error are updated using Bayes theorem. The proposed method is illustrated using an application problem, surface cracking in a cylindrical structure.
1. INTRODUCTION Mechanical components in engineering systems are often subjected to cyclic loads leading to fatigue, crack initiation and progressive crack growth. It is essential to predict the performance of such components to facilitate risk assessment and management, inspection and maintenance scheduling and operational decision-making. Several studies in the past have used fracture mechanics-based models for crack growth analysis, and thereby to predict the performance of the component. These models such as Paris law, NASGRO equation, etc. are calibrated from experimental testing of coupons.
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_37, © The Society for Experimental Mechanics, Inc. 2011
387
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388 The direct application of such fracture-mechanics based models to complicated structures and multi-axial variable amplitude loading conditions is difficult. Reliable models are available only to predict the growth of the crack only in the crack region. The behavior of short cracks is not clear and hence is difficult to model. In this paper, an equivalent initial flaw size [1] is used to replace the short crack growth calculations and make direct use of a long crack growth model for analysis. The next step in crack growth modeling is to choose a particular crack growth model. This paper uses a modified Paris law for purposes of illustration, and the methods developed can be applicable to any chosen crack growth law. The modified Paris’ law calculates the incremental crack growth in every loading cycle as a function of the current crack size and the stress intensity factor. While the stress intensity factor can be expressed as a function of the crack size in closed form for structures with simple geometry and crack configurations, it needs to be calculated using a finite element analysis for complicated structures subjected to multi-axial variable amplitude loading conditions. Hence, the finite element model is replaced by an efficient Gaussian process surrogate model for cycle-by-cycle integration to calculate the crack size as a function of number of load cycles. Crack growth is a stochastic process and the various sources of uncertainty – natural variability, data uncertainty and model uncertainty – need to be considered in crack growth analysis. Natural variability includes variability in loading, material properties, geometry and boundary conditions. The geometry of the specimen and boundary conditions are considered deterministic in this paper. The uncertainty in experimental data collected for inference needs to be accounted for. The crack growth analysis procedure uses several models in sequence and each model has its own error. For example, an error term (treated as a random variable) is added to the crack growth law to represent the fitting error since experimental data were used to estimate the coefficients. Further, the model coefficients are also treated as random variables. The discretization error in finite element analysis is deterministic and is calculated using the Richardson extrapolation method [10]. This finite element analysis is replaced by a Gaussian process model which introduces additional uncertainty in the overall prediction. These different sources of uncertainty do not combine linearly and the interaction between them is unknown. Hence, a systematic procedure is required to account for the various sources of uncertainty. The goal of this paper is to calibrate the parameters of the crack growth model and also infer the error in the crack growth model. In the Bayesian inference technique, the various quantities of interest are assigned prior probability distributions which are then updated to calculate the posterior probability distributions. Such techniques have been employed by various researchers in the past [2, 3], but they have been used to calibrate either one quantity [4] or the distribution parameters of one model [5]. These efforts studied simple structures under uniaxial constant amplitude loading conditions. In this paper, several different models are combined and the overall crack growth parameters are calibrated using the experimental evidence, while accounting for the different sources of uncertainty in a systematic procedure. The following section discusses the crack growth modeling procedure used in this paper. Section 3 discusses the several sources of uncertainty and proposes methods to handle them. Section 4 outlines the Bayesian inference technique used for calibrating the crack growth model. Section 5 illustrates the proposed methods using an example, surface cracking in a cylindrical crack.
2. CRACK GROWTH MODELING The rigorous approach to fatigue life prediction would be to perform crack growth analysis starting from the actual initial flaw, accounting for voids and non-metallic inclusions. The behavior of short crack growth is not completely understood and the concept of an equivalent initial flaw size was proposed to bypass small crack growth analysis and make direct use of a long crack growth law for fatigue life prediction. An equivalent initial flaw size ș is calculated from material properties (ǻKth, the threshold stress intensity factor and ıf, the fatigue limit) and geometric properties (Y) as explained in Liu and Mahadevan [1]. ଵ
௱
ߠ ൌ గ ሺ ఙ ሻଶ ሺͳሻ
Consider any long crack growth law used to describe the relationship between da/dN and ǻK, where N represents the number of cycles, a represents the crack size and ǻK represents the stress intensity factor. This paper uses modified Paris’ law for illustration purposes and includes the effects of Wheeler’s retardation model as:
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݀ܽ ߂ܭ௧ ൌ ߮ ܥሺ߂ܭሻ ሺͳ െ ሻ ݀ܰ ߂ܭ r
In Eq. 2, ij refers to the retardation parameter which can calculated as explained by Sheu et al. [6]. Eq. 2 calculates the incremental crack growth in every loading cycle as a function of the stress intensity factor. In each cycle, the stress intensity factor can be expressed as a function of the crack size (a) and loading (L). Hence, the crack growth law in Eq. 2can be rewritten as
݀ܽ ൌ ݃ሺܽǡ ܮሻሺ͵ሻ ݀ܰ By integrating the expression in Eq. 2 starting from the equivalent initial flaw size ș, the number of cycles (N) to reach a particular crack size aN can be calculated as shown in Eq. 4. ಿ
ܰ ൌ න ݀ܰ ൌ න
ఏ
ͳ ߂ܭ ߮ ܥሺ߂ܭሻ ሺͳ െ ߂ܭ௧ ሻ
݀ܽ ሺͶሻ
For structures with complicated geometry and loading conditions, the integral in Eq. 4 is to be evaluated cycle by cycle, calculating the stress intensity factor in each cycle of the crack growth analysis. Further, if the loading is multi-axial (for example, simultaneous tension, torsion and bending), then the stress intensity factors corresponding to three modes need to be taken into account. This can be accomplished using an equivalent stress intensity factor. If KI, KII, KIII represent the mode-I, mode-II and mode-III stress intensity factors respectively, then the equivalent stress intensity factor Keqv can be calculated using a characteristic plane approach proposed by Liu and Mahadevan [7]. Thus, the computation of ǻK requires the use of a finite element analysis represented by Ȍ.
߂ܭ௩ ൌ ߖሺܽǡ ܮሻሺͷሻ Repeated evaluation of the finite element analysis in Eq. 5 renders the aforementioned cycle by cycle integration extremely expensive, perhaps impossible in some cases. Hence, it is necessary to substitute the finite element evaluation by an inexpensive surrogate model. A Gaussian process (GP) surrogate modeling technique has been employed in this paper. A few runs of the finite element analysis are used to train this surrogate model and then, this model is used to predict the stress intensity factor for other crack sizes and loading cases (for which finite element analysis has not been carried out). The basic idea of the GP model is that the response values Y (Keqv in this case), are modeled as a group of multivariate normal random variables, with a defined mean and covariance function. Suppose that there are m training points, x1, x2, x3 … xn of a d-dimensional input variable (the input variables being the crack size and loading conditions here), yielding the resultant observed random vector Y(x1), Y(x2), Y(x3)… Y(xn). R is the m x m matrix of correlations among the training points. Under the assumption that the parameters governing both the T trend function (f (xi) at each training point) and the covariance (Ȝ) are known, the conditional expected value of the process at an untested location x* is calculated as in Eq. 6 and Eq. 7 respectively.
ܻ כൌ ܧሾܻሺ כ ݔሻȁܻሿ ൌ ݂ ் ሺ כ ݔሻߚ ் ݎሺ כ ݔሻܴ ିଵ ሺܻ െ ߚܨሻሺሻ ߪଶ כൌ ܸܽݎሾܻሺ כ ݔሻȁܻሿ ൌ ߣሺͳ െ ି ܴ ் ݎଵ ݎሻሺሻ T
In Eq. 6 and Eq. 7, F is a matrix with rows f (xi), r is the vector of correlations between x* and each of the training points, ȕ represents the coefficients of the regression trend. McFarland [3] discusses the implementation of this method in complete detail. The greedy point algorithm developed by McFarland [3] has been implemented for the selection of training points. The Gaussian process surrogate model can be used to calculate the equivalent stress intensity factor in the cycle-by-cycle integration of the crack growth law. The following section discusses the various sources of uncertainty in the crack growth analysis procedure.
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390 3. SOURCES OF UNCERTAINTY The various sources of uncertainty can be classified into three different types – physical variability, data uncertainty and model uncertainty: I. Physical Variability a. Loading b. Equivalent initial flaw size c. Material Properties (Fatigue Limit, Threshold Stress Intensity Factor) II. Data Uncertainty a. Material Properties (Fatigue Limit, Threshold Stress Intensity Factor) b. Output Measurement Error (measured crack size after number of loading cycles) III. Model Uncertainty/Errors a. Crack growth law uncertainty b. Uncertainty in calculation of Stress Intensity factor i. Discretization errors in finite element analysis ii. Uncertainty in surrogate model Each of these different sources of uncertainty is briefly discussed below. A block loading history is illustrated in this paper. In this paper, the block length is assumed to be a uniform distribution (U(0,500)) and the maximum amplitude and minimum amplitude for that block are assumed to follow normal distributions (N(8,2) and N(24,2) respectively, in KNm). A sample loading history is shown in Fig. 1.
Fig. 1. Sample Loading History The equivalent initial flaw size derived in Eq. 1 depends on ǻKth, the equivalent mode-I threshold stress intensity factor, ǻıf, the fatigue limit of the specimen and the geometry factor Y. This formula is used to derive the prior distribution of EIFS which is later updated during model calibration in Section 4. Experimental data are available in literature to characterize the distribution of material properties such as threshold stress intensity factor (ǻKth) and fatigue limit (ǻıf). McDonald et al. [9] proposed a method to account for data uncertainty, in which in the quantity of interest can be resampled and represented using a probability distribution, whose parameters are in turn represented by probability distributions. Then the family of distributions can be integrated using the principle of total probability [8] to calculate a single probability distribution. In this paper, a modified Paris law has been used for the illustration of crack growth analysis; however, the methodology can be implemented using any kind of crack growth model. The uncertainty in crack growth model can be subdivided into two different types: crack growth model error and uncertainty in model coefficients. In each cycle, a normally distributed random variable İcg is added to the modified Paris law in Eq. 2. Prior probability distributions are assumed for model coefficients and model error and these are updated after collecting evidence. The calculation of stress intensity factor ǻK is done in two stages. First, a few finite element analysis runs are required to train the GP model. Second, the GP model is used to predict the stress intensity factor. Finite element solutions are subject to discretization errors, whereas the prediction of any low-fidelity model such as the GP model also has error. The discretization errors in finite element solutions are calculated using Richardson
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391 extrapolation [10] while the output of the Gaussian process surrogate model is a normal distribution [3]. Hence, the value of stress intensity factor is sampled from this distribution in every loading cycle. 4. MODEL CALIBRATION This section explains the calibration technique used to infer the probability distributions of the model parameters and modeling errors using experimental data. First, a set of variables to be calibrated is chosen. In this paper, for the purpose of illustration, the equivalent initial flaw size (ș), model parameters (C) and crack growth model error (İcg) are chosen for calibration. Let ȍ denote the vector of these quantities. Assume that there is a set of m experimental data points (Ai, Ni), i.e. the crack size A after N loading cycles. Using this information, it is possible to calculate the probability distributions of ȍ using Bayes theorem. In this paper, experimental data have been simulated by assuming true distributions for ȍ. Bayesian updating is a three step procedure 1) Choice of prior distribution 2) Construction of likelihood function 3) Calculation of posterior distribution. Prior probability distributions are assumed for each of the parameters. The prior distribution of EIFS is calculated using the expression proposed by Liu and Mahadevan [1]. A non-informative prior is used for the standard deviation of crack growth model error (İcg). A zero mean is assumed for this quantity because it represents the fitting error and only the variance is updated. The prior distribution of C is obtained from Liu and Mahadevan [1]. The likelihood of ȍ is calculated as being proportional to the probability of observing the given data conditioned on ȍ. The likelihood is a function of ȍ and for every given ȍ a Monte Carlo analysis is required for this calculation, so as to account for the various sources of uncertainty. I.
Construct the Gaussian process surrogate model as explained in Section 2. Include the sources of uncertainty as explained in Section 3.
For a given ȍ II. Generate a loading history (N cycles) as explained in Section 3. III. Use the deterministic prognosis methodology in Section 2 to calculate the final crack size at the end of Ni (for i = 1 to m) cycles. IV. Repeat steps II and III and calculate the probability distribution of crack size at the end of Ni (for i = 1 to m) cycles. Let this distribution be denoted by f(a). Then the likelihood of ȍ can be calculated as being proportional to f(Ai | Ni). Combining all m experimental data points, the effective likelihood can be calculated as:
ܮሺߗሻ ൌ ς ݂ሺܣ ȁܰ ሻሺ݅ ൌ ͳ݉ݐሻሺͻሻ Finally, the likelihood is multiplied with the prior and normalized to calculate the posterior distribution [3]. Then this joint distribution can be marginalized to calculate the individual distributions [8]. This calibration methodology is illustrated using an example in the following section. 5. NUMERICAL EXAMPLE A two-radius hollow cylinder with an elliptical crack in fillet radius region is chosen for illustrating the proposed methodology. This problem consists of modeling an initial semi-circular surface crack configuration and allowing the crack shape to develop over time into a semi-elliptical surface crack. This is shown in Fig. 2.
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Fig. 2. Surface Crack in a Cylindrical Structure The finite element software ANSYS is used to build and analyze the finite element model. A sub modeling technique has been used near the region of the crack for the accurate calculation of stress intensity factor. The material and geometrical properties of the specimen are listed in Tables 1 and 2 respectively. Table 1 Material properties
Table 2 Geometrical Properties
Aluminium 7075- T6
Cylinder Properties
Modulus of Elasticity
72 GPa
Length
152.4 mm
Poisson Ratio
0.32
Inside Radius
8.76 mm
Yield Stress
450 MPa
Outside Radius
14.43 mm
Ultimate Stress
510 MPa Outside Radius
17.78 mm
Variables such as the geometry of the structure, Young’s modulus, Poisson ratio, boundary conditions, friction coefficient between crack faces, etc are treated to be deterministic in this paper. This finite element model is run for 10 different crack sizes and 6 different loading cases and these results are used to train the Gaussian process surrogate model. Experimental evidence of 20 data points is simulated and used for calibration as explained in Section 4. The posterior probability distribution and the true distribution of EIFS are shown in Fig. 3.
Fig. 3. Posterior Distribution of EIFS
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393 In Fig. 3, the prior distribution (mean = 0.5 and standard deviation = 0.1) is not shown because its variance is much larger than the posterior distribution. It is observed that with more measurements the posterior probability distribution approaches the true distribution. The results of calibration of variables are summarized in Table 3. Table 3. Results of Calibration Quantity EIFS (ș) (mm) C (m/cycle) İcg (mm)
True Distribution Mean Std. Dev. 0.375 0.025 6.0 E -13 1.0 E -13 0 0.001
Prior Distribution Mean Std. Dev. 0.5 0.1 6.5 E -13 4.0 E -13 0 0.01
Posterior distribution Mean Std. Dev. 0.37 0.024 6.1 E-13 1.3 E -13 0 0.005
From Table 3, it is seen that while the distribution of EIFS is almost the same as the true distribution, the distributions for the other parameters do not agree with the true distribution. During calibration, the equivalent initial flaw size is used only once to start the crack growth propagation procedure, whereas the model parameter C and the error İcg are used in every loading cycle, thereby adding uncertainty with every additional loading cycle. Hence, the uncertainty in their estimates is high as the number of loading cycles increase. This uncertainty may be decreased by frequent inspections and collecting more data for calibration. 6. SUMMARY This paper proposed a method to calibrate fatigue crack growth analysis for structures with complicated geometry and multi-axial loading. The concept of equivalent initial flaw size was used to replace small crack growth analysis and use a long crack growth model, specifically Paris law, for crack propagation. Expensive finite element analysis was replaced by an inexpensive surrogate, i.e. the Gaussian process model, to evaluate the stress intensity factor in each cycle for use in crack growth law. Several sources of uncertainty – physical variability, data uncertainty and modeling errors - were included in the calibration procedure. Physical variability included loading conditions and material properties such as threshold stress intensity factor and fatigue limit. The uncertainty in data used to characterize these parameters was accounted for. Three different kinds of modeling errors – discretization errors, surrogate modeling error and crack growth model error – were considered in this paper. A probabilistic methodology was proposed to incorporate these sources of uncertainty into the calibration methodology. A Monte Carlo based sampling approach was used to calculate the distribution of crack size as a function of number of loading cycles. This distribution was used to calculate the likelihood, and in Bayesian updating. Experimental data is simulated using assumed probability distributions and the results of calibration are compared with these assumptions. This research work modeled coplanar cracks only. The effect of non-coplanar cracks will be considered in future work. 7. ACKNOWLEDGEMENT The research reported in this paper was partly supported by funds from NASA Ames Research Center (Project Manager: Dr. K. Goebel) through subcontract to Clarkson University (Principal Investigator: Dr. Y. Liu). The support is gratefully acknowledged. 8. REFERENCES [1] Liu, Y., and Mahadevan S. Probabilistic fatigue life prediction using an equivalent initial flaw size distribution. 2008. Int J Fatigue, doi:10.1016/j.ijfatigue.2008.06.005 [2] Urbina, A. Uncertainty Quantification and Decision Making in Hierarchical Development of Computational Models. 2009. Doctoral Thesis Dissertation. Vanderbilt University, Nashville, TN. [3] McFarland J. 2008. Uncertainty analysis for computer simulations through validation and calibration. Ph D. Dissertation, Vanderbilt University, Nashville, TN. United States of America. [4] Makeev., A, Nikishkov., Y, and Armanios., E. 2007. A concept for quantifying equivalent initial flaw size distribution in fracture mechanics based life prediction models, Int J Fatigue (2006), Vol. 29, No. 1, Jan. 2007.
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394 [5] Cross, R., Makeev, A., and Armanios, E. 2007. Simultaneous uncertainty quantification of fracture mechanics based life prediction model parameters, Int J Fatigue, Vol. 29, No. 8, Aug. 2007. [6] Sheu B.C, Song P.S, and Hwang S. 1995. Shaping exponent in wheeler model under a single overload. Eng Fract Mech 1995;51(1): 135–43. [7] Liu, Y. and S. Mahadevan. 2005. Multiaxial high-cycle fatigue criterion and life prediction for metals. International Journal of Fatigue, 2005. 27(7): p. 790-80. [8] Haldar, A., and Mahadevan, S, Probability, Reliability and Statistical Methods in Engineering Design, Wiley, New York, 2000. [9] McDonald, M., Zaman, K., and Mahadevan, S. Representation and First-Order Approximations for Propagation of Aleatory and Distribution Parameter Uncertainty. In the Proceedings of 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 4 - 7 May 2009, Palm Springs, California. [10] Rebba, R., Mahadevan, S., and Huang, S. Validation and error estimation of computational models, Reliability Engineering & System Safety, Volume 91, Issues 10-11, DOI: 10.1016/j.ress.2005.11.035.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
BAYESIAN FINITE ELEMENT MODEL UPDATING USING STATIC AND DYNAMIC DATA Boris A. Zárate, Post-Doctoral Fellow, Department of Civil and Environmental Engineering, University of South Carolina, 300 Main, Columbia, SC 29208 Juan M. Caicedo, Assistant Professor, Department of Civil and Environmental Engineering, University of South Carolina, 300 Main, Columbia, SC 29208 Glen Wieger, Doctoral Student, Department of Civil and Environmental Engineering, University of South Carolina, 300 Main, Columbia, SC 29208 Johannio Marulanda1, Assistant Professor, Universidad del Valle, Calle 13 # 100-00, Edf. 350, Cali, Colombia
Nomenclature ܝሷ ܖ ܙሷ ܖ ܠ ܜ ܘ ܞ ۳ ۷ ܔܕ ۺ ۿ ܃
Acceleration response n-th mode shape Acceleration response of the n-th mode in generalized coordinates x-axis coordinate Time instant Applied force Velocity of the mobile sensor Elasticity modulus Moment of inertia Mass per unit length Length Matrix with modal coordinates Matrix with the response in generalized coordinates Matrix with the response in geometric coordinates
Abstract Finite element models of current structures often behave differently than the structure itself. Model updating techniques are used to enhance the capabilities of the numerical model such that it behaves like the real structure. Experimental data is used in model updating techniques to identify the parameters of the numerical model. In civil infrastructure these model updating techniques use either static or dynamic measurements, separately. This paper studies how a Bayesian updating framework behaves when both static and dynamic data are used to updated the model. Displacements at specific structure locations are obtained for static tests using a computer vision method. High density mode shapes and natural frequencies are obtained using a moving accelerometer structure. The static data and the modal characteristics are combined in a Bayesian modal updating technique that accounts for the incompleteness and uncertainty of the data as well as the possible
1
Doctoral Candidate at the University of South Carolina.
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_38, © The Society for Experimental Mechanics, Inc. 2011
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nonuniqueness of the solution. Results show how the posterior probability density function changes when different type of information is included for updating. Introduction Numerical models of existing structures are commonly used to evaluate the changes in existing structures or determine the reliability of a structural system. However, modeling an existing civil structure is not an easy task. The analyst is presented with many options to make, such as selecting the correct numerical representation and the correct parameters for the model. Model updating techniques use experimental data to assist the analyst in determining the most appropriated numerical model of an existing structure. Several model updating methodologies that involve a probabilistic approach have been proposed in the literature. Among these methodologies Bayesian approaches have become popular in recent years. Beck and Katafygiotis [1,2] proposed a Bayesian approach to compute the updated probability distribution of the uncertain parameters based on dynamic measurements. The predictive probability distribution function of the model parameters is found using a weighted average of the probability distributions of the optimal parameters in an asymptotic approximation because the probability distribution is highly peaked at the optimal parameters. Many model updating studies have been performed using dynamic data only. This paper focuses on studying how the shape of the posterior probability density function changes when information from different sources is applied to the model updating process. Three different types of data sources are investigated: static displacements, natural frequencies and mode shapes. A numerical model of a simple supported beam is used to study the behavior of the posterior probability density function. Bayesian analysis Let, Ȃ, represent a chosen model (e.g. finite element model from a real structure), which is a function of some structural parameters 4 (i.e. elasticity modulus, moment of inertia of a concrete cracked section, etc), and D represents the experimental information obtained from the real structure (i.e. natural frequencies, vibration mode shapes, etc). The Bayes’ theorem can be written in model updating context, as
(1)
P(4 D, M ) v P( D 4, M ) P(4 M )
where, P(4 D, M ) corresponds to the probability density function (PDF) of the parameters 4 for the chosen model M after being updated with the observation D. P(4 D, M ) is also called the posterior PDF. P(4 M ) is the PDF of the parameters 4 for the chosen model M before updating, or prior PDF, and P( D 4, M ) is the likelihood of occurrence of the measurement D given the vector of parameters 4 and the model M. If it is assumed that no prior information is known, the prior PDF would be a uniform distribution. Using a Gaussian distribution of the difference between the numerical model and the measured data for the likelihood, the posterior probability can be calculated with the equation 2 2 id fe id fe id fe § 1 n m I j ,i I j .i (4) 1 p u j u j ( 4) ¨ 1 n w j w j (4 ) f p (4; D) exp¨ ¦ ¦¦ ¦ 2 j1i1 2 j1 V uj V jw V Ij .i ¨ 2 j1 © 0
2
· ¸ ¸¸ 4l 4 4 u ¹ otherwise
(2)
where, n is the number of identified modes of vibration, m is the number of modal coordinates, p is the number of displacement measurements obtained, Z idj is the j-th identified natural frequency, Z jfe (4) is the j-th natural id fe frequency of the finite element model, I j,i is the i-th modal coordinate of the j-th identified mode shape, I j ,i (4) is the i-th modal coordinate of the j-th mode shape of the finite element, u idj is the j-th measured displacement, u jfe (4) is the j-th displacement at the finite element model, V jw is the standard deviation of the error in the j-th I natural frequency identified, V j,i is the standard deviation of the error in the i-th modal coordinate that corresponds to the j-th identified mode shape, V ju is the standard deviation of the error in the j-th displacement, 4 l and 4 u are the lower and upper bounds of the parameter 4 . The function f p (4; D) is a posterior
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Lasers
Structure
Structure’s representation
Figure 1. Reduced Structure with Lasers proportional probability density function (PPDF) of the structural parameters 4 given a set of experimental modal parameters D. Equation (2) can also be used when only one type of experimental data is used (i.e. static displacements), by dropping the other terms from the equation. Displacements measurements Measuring displacements on full scale civil structures is a complex task due to the need of a reference point. The methodology used on the numerical simulations of this paper is designed to reduce the civil structure down to a size such that a computer vision technique can be used to track displacements (Figure 1) [3-5]. A large structure could be measured with a single camera, reducing cost or synchronization errors that could occur on a structure of significant size. In addition, multiple points can be tracked without the increased costs that would be associated with other techniques. Lasers are placed at the point of interest on the structure and pointed to a displacement recording station (DRS). The DRS consists of two inclined receptive surfaces made of two way mirrors (Figure 2a). A camera to observe the movement of the lasers marks is placed behind the mirrors. Two laser marks, one from each receptive surface, can be seen from the back of the receptive surfaces as shown on Figure 2b. The image marks are automatically identified (Figure 2c) and used to calculate the global location of the laser located at the structure [5]. The error of the displacement measurements has shown to have a standard deviation of 0.31 mm when the distance between laser marks at the DRS is 2m. A Gaussian random number created with these characteristics is included on the simulations of the test structure described above. Detailed information about this methodology can be found in [3-5] High Density Mode Shapes Using Mobile Sensors The standard modal identification approach is to process data from vibration sensors located at fixed strategic Lasers
Camera
Beam Receptive surface a. Lasers and DRS
b. Image with laser marks
Figure 2.. Displacement Recording Station
c. Identified Laser Marks
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ݑሷ (ݔ, )ݐ
ݔ(, )ݐ ݔ
Figure 3. One-dimensional dynamic system. locations in the structure (stationary sensors). Due to physical and economic limitations, only a few degrees of freedom of the total degrees of freedom of the structure are used for modal identification. Two alternatives to increase the spatial resolution of the identified mode shapes are the use of high density sensor networks and the application of mode shape expansion methods. The first approach is limited by physical and economic constraints and the sensitivity of the algorithms to sensor distribution among other challenges [6]. In the second case, mode shape expansion methods can introduce errors into the modal identification process due to: i) discrepancy between the location of the sensors and the location of DOF in numerical models, ii) measurement errors, and iii) modeling errors [7, 8]. A technique to increase the spatial resolution of the identified mode shapes that has using of mobile sensors was introduced in [9]. Mobile sensing allows the identification of a fine grid of discrete points representing the operational mode shape using the vibration of the structure, avoiding the need for modal expansion or the use of dense sensor networks. Furthermore, the methodology only requires the use of one or few single sensors to calculate highly spatial mode shapes. The acceleration response of a one-dimensional system in the x െ axis subjected to a dynamic excitation (Figure 3) is described by the equation λ
uሷ (x, t) = Ԅn (x)qሷ n (t)
(3)
n=1
where Ԅn (x) is the n െ th natural vibration mode and qሷ n (t) is the response of the n െ th mode in generalized coordinates. If the response of the system is measured with a mobile sensor travelling at a constant speed v, the response can be written as a function of time only λ
uሷ (t) = Ԅn (vt)qሷ n (t)
(4)
n=1
If sufficient information about the system is known and properly organized, it is possible to identify the mode shapes of vibration using the equation [9] Ȱ = Qെ1 U
(5)
where the matrix Ȱ contains the modes of vibration, the matrix Q contains the response of the structure in generalized coordinates, and U is a matrix containing the response in geometric coordinates. The matrix Q does not change with time, can be computed ahead of time and it is dependent on the type of excitation only (i.e. Q for impulse loads is different than for ambient vibration). However, the target natural frequencies and damping ratios must be previously identified to create the matrix Q. The matrix U is formed using cross-correlation functions for the case of ambient vibration, in a similar fashion as with the Natural Excitation Technique [10]. Numerical simulations
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u(x,t) 10m
20m
30m
40m
50m
x
L, E, I, ml Figure 4. Simply supported beam for the numerical simulations. A uniform simply supported beam is considered to numerically evaluate the effect of the different type of measurements on the posterior PDF (Figure 4). The properties of the beam are specified below in Table 1. 60 beam elements with 1 meter length each were used in the finite element model. The moments of inertia of all sixty elements were allowed to vary for updating the finite element model. The boundary conditions were set to allow only in-plane rotation at the ends. The interior nodes were allowed to have 2 degrees of freedom (vertical displacement and in-plane rotation), for a total of 120 degrees of freedom. T able 1. Properties of the beam Properties Length Moment of inertia Modulus of elasticity Mass per unit length Damping ratio (all modes)
L [m] 4 I [m ] E [GPa] ml [kN/g·m] ] [%]
60 6.75 25 150 5
Four tests were simulated on the beam having data sources: i) static displacements, ii) natural frequencies, iii) mode shapes, and iv) all previous data. Test 1: Static displacements. Five equidistant points at 10, 20, 30, 40, and 50 meters were selected for displacement measurements. A static test with a point load with magnitude of 3.6x106N was added at 20 meters from the left end for this simulated test. Test 2: Modal analysis with stationary sensors. Five stationary acceleration sensors were simulated at the same locations that the displacement sensors for the stationary modal updating. All sensors use an acquisition frequency of 600 Hz. A random force with normal distribution, zero mean and amplitude of 1 kN is applied at all the displacement degrees of freedom of the beam. One hundred tests of ten minutes each were performed using the first ten modes in the simulation. The Natural Excitation Technique [10] and the Eigensystem Realization Algorithm [11] were used to identify frequencies of the structure. It is assumed that only one mode (3rd) is identified.
Figure 5: Finite Model of Beam
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c.
6
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ȣ = 6.75 m
4
d. ȣ = 6.75 m4
Figure 6: Proportional PDF – Case 1: static data. Test 3: Modal analysis with mobile sensors. The third test used a mobile sensor moving from one end of the beam to the other at a constant velocity of 1.67 cm/s. One hundred tests of sixty minutes each were performed using the third mode in the simulation. One stationary accelerometer at the center of the beam was used as reference sensor. Both sensors use an acquisition frequency of 60 Hz. Assuming that the natural frequency and damping ratio are known, mean values and standard deviations of the third mode shape were identified using the cross-correlation function between the acceleration of the reference sensor and the acceleration recorded by the mobile sensor. Originally 54000 points were identified for the mode shape and then were averaged to 59 points separated 1 meter each. Test 4: All sources. The last test assumes that all the data sources considered on the previous cases are available. Results The posterior proportional probability density function (PPDF) of moment of inertia for the 60 elements was calculated for each test presented above. A posterior PPDF was also calculated for the case when the u information of the three tests is available for updating. The values of V jw , V j.I i and V j for equation (2) were obtained by multiplying the standard deviations of the respective measurements by a factor of two. 10
1
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ȣ = 6.75 m4
Figure 7: Proportional PDF – Case 2: 3 n atural frequency.
8
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8
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5
d. ȣ = 6.75 m
4
e. ȣ = 6.75 m4
Figure 8: Proportional PDF – Case 3: Third mode with mobile sensor. To simplify the visualization of the results, the results shown focus on the parameters for elements 10 and 30 from left to right on the beam. Figure 6 shows the posterior PPDF when the other 58 values are fixed at ȣ = 2.00 m4 4 4 4 6.75 m and 7.5 m . As expected, the results show a higher probability at ȣ =6.75 m (Figure 6.b). However at ȣ = 6.75 m4 the values of high probability for ȣ10 and ȣ30 is close to 2m4 (Figure 6.c). The parameters of ȣ10 and ȣ30 would have been incorrectly identified if the other parameters were not considered as part of the updating 4 4 process and were selected to have a value of 7.5 m . There is no probability when ȣ = 2.00 m (Figure 6.a). Figure 7 shows the results when only the third natural frequency of the structure is used for updating. In this case the area of high probability is very wide and a large range of values could be used as a solution of the problem. A 4 4 similar behavior to the case 1 is obtained when the values of the other parameters are at ȣ = 2.00 m and 7.5 m . Infinite solutions are also observer on Figure 7.c. The results of case 3 are shown on Figure 8. In this case the third mode of vibration is used for the updating process as calculated for the mobile sensor. These results are closer to case 1 (static displacements were considered) except that a small probability is observed at ȣ = 2 m4. Figure 9 contains the case when all the measures are available for the updating process. The probability is zero 4 4 for the values of ȣ = 2.00 m and 7.5 m . In addition, an area of high probability is encountered around the 4 theoretical value of ȣ = 6.75 m clearly demonstrating the benefits of including different type of measurements when updating a model. -4
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Figure 9: Proportional PDF – Case 4: All data sources.
f.
ȣ = 6.75 m4
8
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Conclusions The numerical model of a simple supported beam was updated using 4 data sets: i) static displacements, ii) natural frequencies from stationary acceleration sensors, iii) mode shapes from a mobile sensor and iv) all previous data sources. A function proportional to the posterior probability function was calculated based on each of these cases and all the measurements using a Bayesian framework. The beam was simulated with 60 independent sections. For visualization purposes the results focused on the identified moment of inertia of elements 10 and 20. Results demonstrate that having all data sources provide a posterior PPDF concentrated around the theoretical solution. The posterior PPDF of cases with only one data source are broadly distributed and in the case of only one natural frequency, infinite solutions were found. Acknowledgements This material is based upon work supported by the National Science Foundation under Grant No.CMMI-0846258 (Dr. Mahendra P. Singh program director). The last author would like to acknowledge the support of the Universidad del Valle in Colombia, South America. References [1]. Beck, Jim L., and Lambros S. Katafygiotis. 1998. Updating Models and Their Uncertainties. I: Bayesian Statistical Framework. Journal of Engineering Mechanics 124 (4):455-461. [2] Katafygiotis, Lambros S., and Jim L. Beck. 1998. Updating Models and Their Uncertainties. II: Model Identifiability. Journal of Engineering Mechanics 124 (4):463-467. [3] Caicedo, J. M. 2005. Displacement measurements in civil structures using digital cameras and lasers. In XXIII International Modal Analysis Conference. Orlando, Florida: SEM. [4] Wieger, G. R., and J. M. Caicedo. 2008. Displacement Records of Civil Structures. In ASCE Inaugural International Conference of the Engineering Mechanics Institute. Minneapolis, Minnesota. [5] Wieger, G. R. 2009. Development and Verification of a Computer Vision Technique to Measure the Response of civil Structures, Department of Civil and Environmental Engineering, University of South Carolina, Columbia. [6] Lynch, Jerome P., and Kenneth J. Loh. 2006. A Summary Review of Wireless Sensors and Sensor Networks for Structural Health Monitoring. The Shock and Vibration Digest 38 (2):91-128. [7] Balmes, E. 2000. Review and evaluation of shape expansion methods. In IMAC XVIII : a conference on structural dynamics. San Antonio, Texas. [8] Pascual, R., R. Schalchli, and M. Razeto. 2005. Improvement of damage-assessment results using highspatial density measurements. Mechanical Systems and Signal Processing 19 (1):123-138. [9] Marulanda A., Johannio, and J. M. Caicedo. 2008. Modal Identification Using a Smart Mobile Sensor. In ASCE Inaugural International Conference of the Engineering Mechanics Institute. Minneapolis, Minnesota. [10] James, George H., Thomas G. Carne, James P. Lauffer, and Arlo R. Nord. 1992. Modal testing using natural excitation. In 10th International Modal Analysis Conference. San Diego, CA. [11] Juang, Jer-Nan, and Richard S. Pappa. 1985. An Eigensystem Realization Algorithm for Modal Parameter Identification and Model Reduction. Journal of Guidance 8 (5):620-627.
BookID 214574_ChapID 39_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
)UHTXHQF\'RPDLQ7HVW$QDO\VLV&RUUHODWLRQLQWKH3UHVHQFHRI 8QFHUWDLQW\
SonnyNimityongskul DanielC.Kammer DepartmentofEngineeringPhysics,UniversityofWisconsinͲMadison 1500EngineeringDr.,Madison,WI53706 SethLacy AirForceResearchLaboratory 3550AberdeenAveSE,KirtlandAFBNM87117 VitBabuska SandiaNationalLaboratory1 POBox5800,MS0847,Albuquerque,NM87185 $EVWUDFW The aerospace community traditionally relies on modal-based test-analysis correlation metrics, such as natural frequency matching and cross-orthogonality, but these techniques are known to breakdown in the presence of high modal density. This paper shows how the frequency domain orthogonality metric can parallel validation criteria that are currently used with modal analysis, while avoiding the complications and pitfalls associated with modal analysis beyond the low-frequency range. This work also examines a modification of the correlation criteria in the presence of uncertainty contained in an ensemble of test data. A 6-DOF spring-mass system and a General Purpose Spacecraft example are included to illustrate the use of the correlation criteria. 1RPHQFODWXUH M : Mass matrix K : Stiffness matrix C : Damping matrix Z : Impedance matrix h : Frequency response I : Mode shape matrix W : Weighting matrix DOF : Degree of freedom
FDO : Frequency domain orthogonality FBC : Frequency band correlation FRF : Frequency response function a : input locations s : Sensor DOF n : Sensor DOF without inputs ı : Standard deviation Ȧ : Frequency
1
SandiaisamultiprogramlaboratoryoperatedbySandiaCorporation,aLockheedMartinCompany,fortheUnitedStates DepartmentofEnergyundercontractDEͲAC04Ͳ94AL85000.
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_39, © The Society for Experimental Mechanics, Inc. 2011
403
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,QWURGXFWLRQ The development of high-precision spacecraft requires models that are valid beyond the low-frequency, modally sparse, range. For regions with well spaced modes the aerospace community has successfully used modalbased correlation and error metrics to determine the accuracy of a model. The metrics most often used for testanalysis correlation are orthogonality and cross-orthogonality of the mode shapes and error in the natural frequencies. Government agencies, such as NASA and the United States Air Force, require the use of these metrics with specific criteria for determining modeling accuracy. For instance, the Air Force requires the test/analytical mode shape cross-orthogonality values for the diagonal terms to be greater than 0.95 and the offdiagonal terms must be less than 0.1 for a valid model [1, 2], as well as less than 3% error in the paired natural frequencies. However, these modal-based correlation techniques break down in the presence of high modal density. Beyond the low-frequency range, the modal density can increase dramatically, making it difficult to distinguish and accurately extract the target modes from the frequency response. Additionally, at higher frequency the modal parameters become increasingly sensitive to modeling errors and uncertainty in the structure. The combination of the test errors and model uncertainty with a high modal density makes the application of modal-based correlation metrics impractical. To avoid the use of modal based metrics, test-analysis correlation has also been developed directly in terms of the frequency response functions (FRF). The use of FRFs for correlation, as opposed to mode shapes, is appealing because experimental FRFs are more easily generated, while the extraction of mode shapes can be difficult and introduce errors into the correlation. There exists a number of shape-based metrics that compare the test and analytical FRFs both in frequency and in space [3]. A drawback to these metrics is that they only give information regarding the accuracy of the FEM at the measured degrees of freedom (DOF). These shape-based metrics contain no check on the accuracy of FEM at the unmeasured DOF. In contrast, the inclusion of a reduced model representation within a metric, such as what is used in modal orthogonality, provides additional information about the FEM in the form of a consistency check between the unmeasured and measured degrees of freedom. This work uses the frequency domain orthogonality (FDO) metric [4], which compares a reduced analytical impedance matrix to the test frequency response in an equation of motion based metric. The paper shows how the FDO metric can parallel validation criteria that are currently used with modal analysis, while avoiding the complications and errors involved with the extraction of test modes. This paper also examines correlation criteria in the presence of uncertainty in an ensemble of FRFs. One may encounter an ensemble due to variation among nominally identical structures, or from multiple tests run on a single structure. At higher frequencies, nominally identical structures may produce a wide spread in the frequency response due to small variations in the material properties and geometry. The FRFs from a single structure can vary due to slight changes in test configurations or time dependent changes in material properties. This leads to the problem of how to correlate a model with respect to an ensemble of test data with an inherent level of uncertainty. Using the FDO metric, the correlation criteria are modified based on the spread of the FRF data within the ensemble. )UHTXHQF\'RPDLQ2UWKRJRQDOLW\ Frequency domain orthogonality metrics are used to examine the consistency of the test frequency response with respect to the analytical impedance matrix. These metrics are based on the fact that the displacement frequency response matrix, also known as the receptance matrix, is the inverse of the impedance matrix [4]
h(Z )
Z Z
1
Z
2
M i ZC K
1
(1)
This property can be used to create an orthogonality-like metric between the FEM dynamic stiffness matrix and the test frequency response, Ʃ.
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FDO(Z )
Z Z hˆ Z # >I @
(2)
Frequency domain orthogonality can be evaluated at each point in the frequency range of interest. This allows the metric to highlight the regions in the frequency band where the model has good consistency with the test data and where it is deficient. The correlation of each input-output pair can be viewed as a vector in frequency, and therefore possible validation measures may use various statistics associated with the metrics, such as the mean, standard deviation, and maximum and minimum values over the frequency band of interest. Various forms of the FDO metric have been used to define the error vectors for use in model updating schemes [5-8] &RPSDULQJ0RGDO2UWKRJRQDOLW\DQG)UHTXHQF\'RPDLQ2UWKRJRQDOLW\ A comparison can be made between the widely used modal orthogonality metric and the frequency domain orthogonality metric. It can be shown that FDO is simply an alternate form of modal orthogonality [4], which examines all of the modes simultaneously and automatically accounts for errors in the natural frequencies. However, a major advantage to the FDO metric is that it does not require the extraction of modes from the test FRFs. A connection between the two metrics can be seen by examining how the modes are related to the individual entries of each metric. The individual entries of the modal orthogonality matrix corresponding to the ijth mode pair can be expressed as the summation over the DOF
Oij
m
¦I
T i
M k I jk
(3)
k 1
in which m is the number of DOF, Mk is the kth column of the mass matrix and Ijk is the kth element of mode shape Ij. The summation used in Eq.(3) shows that the orthogonality of each mode pair is examined independently with respect to the entire mass matrix. The individual entries of the FDO matrix correspond to the ijth input-output pair. Assuming mass normalized modes and using the modal representation for the frequency response [9], the ijth entry of the frequency domain orthogonality can be expressed as
FDOij Z
Z i Z h j Z
Z M 2
i
1ZCi K i
¦ Z r 1
n
¦ r 1
Z M 2
i
I r I rj
n
2 r
Z 2 1] r ZZ r 2
(4)
1ZCi K i I r I rj
Z r2 Z 2 2 1] r ZZ r
where n is the number of mode shapes, Zi is the ith row of the impedance matrix and hj is the jth column of the receptance FRF. Using the assumptions of no error in the test and analysis natural frequencies and modal damping, C
MI 2]Z r I T M , the FDO metric can be simplified to
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FDOij Z
n
¦
Z M 2
i
¦
> Z M 2 r
¦
i
2
@
K i I r Z r2 Z 2 2 1] r ZZ r M iI r I rj
Z r2 Z 2 2 1] r ZZ r
r 1
n
Z Z 2 1] r ZZ r 2 r
r 1
n
1Z M iI 2]Z r I T M K i I r I rj
> Z
@
Z 2 2 1] r ZZ r M iI r I rj
Z r2 Z 2 2 1] r ZZ r
r 1
n
¦I
2 r
(5)
rj
M iI r
r 1
The simplified form given in Eq.(5) shows that the FDO of each input-output pair is simply the orthogonality of all the modes with respect to a single row of the mass matrix. A comparison of Eq.(5) and Eq.(3) shows that the two metrics have a very similar structure, while computing two alternate forms of mode-based orthogonality. The individual terms of the modal orthogonality matrix examine the orthogonality of a mode pair with respect to the entire mass matrix, while the frequency domain method looks at the orthogonality of all the modes contained in the frequency response with respect to a single row of the mass matrix. This comparison suggests that it is reasonable to use similar correlation criteria for the FDO as is used in modal orthogonality. It should be noted that the simplified form of the FDO metric shown in Eq.(5) will not be computed in practice, but it was included for the purpose of drawing a comparison to modal orthogonality. 8VHRI)'20HWULFLQ7HVW7$0RU)(07$0&RUUHODWLRQ A major obstacle for the FDO metric is the size mismatch between the test data and the analytical model. Experimental frequency response data will only contain rows corresponding to sensor locations and columns corresponding to the input locations. In order to use the FDO metric, the impedance matrix must be either reduced to the sensor locations or the test data must be expanded to the full DOF size. This work will focus on the use of a test-analysis model (TAM) reduced representation of the impedance matrix. Frequency domain model reduction techniques, such as dynamic reduction [9] and principal direction reduction [10], can be used to generate the TAM. The FDO metric can be used to determine the accuracy of the TAM with respect to the full FEM, or the accuracy of the TAM with respect to the test data. Frequency domain orthogonality for the reduced model is calculated by taking the product of the TAM impedance matrix and the frequency response data at each frequency point. The TAM impedance matrix can be row partitioned to its input locations (a-set) and the sensor DOF without inputs (n-set), producing
ª Z TAM a (Z )º ˆ ª Ia º hsa Z # « » «Z » ¬0 na ¼ ¬ TAM n (Z )¼
(6)
where Ʃsa is the experimental or FEM frequency response with rows corresponding to sensor locations (s-set) and columns corresponding to input locations. The FDO metric provides an orthogonality-like measure between the TAM impedance matrix and the test or FEM frequency response for each point in frequency. For perfect correlation, the diagonal entries of the identity block will be equal 1.0, while the off-diagonal entries will be equal to zero. The correlation criteria for FDO are designed to mimic what is used with modal orthogonality and cross-
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orthogonality. However, while the terms in modal orthogonality are real and bounded from 0 to 1, the terms in FDO are complex and unbounded. Equation (7) will be used to calculate a positive real-valued FDO error.
FDOerror Z
I Z TAMa Z hˆsa Z
(7)
Akin to what is used in modal orthogonality, possible validation criteria for FDO error may require the individual terms to be less than 0.1. The FDO metric is sensitive to errors in the natural frequencies between the test data and analytical model. For the case of identical test and analysis modes, the FDO metric will not necessarily be an identity matrix due to the differences in the natural frequencies. This leads to the notion that the FDO metric should be allowed to roam in frequency in order to account for small differences in the natural frequencies. By allowing the FDO metric to roam about a central frequency, the FDO metric is comparable to both the current Mil Handbook validation requirements for natural frequency error and modal orthogonality. To account for model uncertainty and uncertainty in the nominally identical structures, the FDO metric will be allowed to roam by an amount įȦ in a small frequency band to determine the best correlation between the test FRF and the model. The use of the allowable variation in frequency will be referred to as frequency band correlation (FBC) [11]. The frequency band used in the FDO metric can either be a constant width or it may vary as some percentage of the center frequency. Allowing the frequencies to vary will help neutralize the effect of frequency shifts between the test FRFs and the analytical model. Frequency band correlation can be used in a fashion akin to the Mil Handbook validation criterion which requires that there is less than 3% error in the test and FEM modal frequencies. In practice, the allowable frequency error can be set and then the metric can be examined to see if the model meets the correlation criteria. The FDO metric will select the change in frequency that minimizes the Frobenius norm of the FDO error.
Error (Z , GZ )
ªI º min « » Z Z GZ hˆ Z >rGZ @ 0 ¬ ¼ Fro
(8)
Since the impedance matrix varies quadratically with respect to frequency, the roaming FDO metric does so as well. Therefore making the frequency correlation window wider will only improve the result up to a certain point. If the quadratic error function does not have a zero crossing then no allowable frequency window will provide perfect correlation. This means that not all of the error in the FDO metric can be accounted for by frequency errors. &RUUHODWLRQZLWK5HVSHFWWRDQ(QVHPEOH An additional goal of this work involves developing correlation criteria for an ensemble of test data containing an inherent level of uncertainty. The variation in the ensemble may be due to small differences among nominally identical structures, or from multiple tests run on a single structure over time. A single model cannot be expected to correlate perfectly with an entire ensemble of varying test data, and therefore the model correlation criteria should take into account the spread in the test data. The variation contained in the ensemble can be quantified using the FDO metric. The goal is to calculate the variation in the FRFs relative to an experimental impedance matrix derived from the FRF data. Since test FRFs are only taken at a limited number of sensor and input locations, only the corresponding entries of the experimental impedance matrix can be computed using a pseudoinverse of the test FRF. The entire ensemble can be evaluated against the test impedance matrix that is generated from the median FRF. The median test FRF is used to generate the experimental impedance matrix in an attempt to minimize the spread in the FDO
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error from all of the remaining test cases. The final result shows the variation in the FDO error for the entire family of FRFs exclusively in terms of the test data. 6HOHFWLRQRI0HGLDQ)5) The selection of the median FRF is the first step in characterizing the variation in the ensemble of test FRFs. A common sense approach may suggest averaging over the ensemble, and then generating an averaged impedance matrix. However, the problem with averaging over the ensemble is that peaks and zeros tend to be smoothed out, and often times the average FRF may look nothing like any single member of the ensemble. Alternatively, this work aims to pick a FRF from the ensemble which most closely resembles the entire family of FRFs. This is accomplished by selecting the FRF which resembles the mean FRF away from the peaks and zeros. Near the peaks and zeros it is known that the mean FRF values are skewed due to frequency shifts. The averaging process lowers the magnitudes of the peaks and raises the magnitudes of the zeros. Away from the peaks and zeros, the FRF ensemble tends to have less variation in magnitude, and therefore the mean value is likely to be close to one or more of the test FRFs. At higher frequencies the selection of frequency points may not be as critical because the FRF peaks and zeros may become smoothed out due to high modal density. The median FRF will be determined by finding the FRF that is the nearest to the mean values at a set of selected frequency points away from sharp peaks or zeros. The mean of the ensemble of FRFs, h , is calculated for each input-output pair at the selected frequency points. The relative error between the mean and test cases is defined as
E
§ hij Z k hij Z k · ¨ ¸ ¦¦¦ ¸ ¨ hij Z k j 1 i 1 k 1© ¹ na
ns
nf
2
(9)
The median FRF is chosen by selecting the FRF that minimizes the error between mean FRF and itself for each input-output location at the selected frequency points. The error between the mean and test data is normalized by the mean value in order to treat all frequency points equally regardless of magnitude. *HQHUDWLQJ7HVW,PSHGDQFH0DWUL[ The median FRF is used to create the test impedance matrix at each frequency based on their inverse relationship. The entire square test impedance matrix cannot be created since only the input columns of the test FRF exist. A pseudoinverse of the median FRF can create a test impedance matrix corresponding to the input rows and output columns. The problem with taking the standard Moore-Penrose pseudoinverse of the FRF is that test impedance matrix will not vary smoothly in frequency like the FEM impedance matrix. This may make the test impedance matrix appear to be more sensitive to errors in the FDO metric, as compared with the model impedance matrix. It is desirable to have a test impedance matrix that behaves similarly to the model impedance matrix, so that they both posses a similar sensitivity to errors. This allows the accuracy of the model to be compared with the true spread in the test data; otherwise the variation across the ensemble may be exaggerated. Weighting matrices can be used in the pseudoinverse calculation to create a test impedance matrix that mimics the smooth behavior of the model impedance matrix. This can be done by using a least squares optimization that aims to minimize the error between the test and TAM impedance matrices at the rows corresponding to the input locations. The residual vector, İ, used in the optimization is defined as
H Z vec Z x Z Z TAMa Z
(10)
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where
Z x Z
h Z W Z h Z * sa
1
sa
hsa* Z W Z
(11)
The objective function, J(Ȧ), is optimized at each frequency to find the weighting matrix, W(Ȧ), which provides the smallest errors between the test and model impedance matrices.
J Z H Z H Z
(12)
This technique requires that the model impedance matrix closely represent the test data. To ensure a close match, the model impedance matrix is first updated to match the median FRF. Frequency domain model updating can be performed using a gradient based least squares algorithm known as the Response Function Method [6-8]. (YDOXDWLRQRIWKH9DULDWLRQLQ)'20HWULFRYHUDQ(QVHPEOHRI7HVW'DWD The test impedance matrix is used to examine the spread in an ensemble of test data relative to the FDO metric. The error in the FDO metric in Eq.(7) is calculated over the entire ensemble, and used to determine the variance in the test data relative to the ensemble at each frequency. Bounds on the accuracy of the ensemble with respect to the median FRF are generated by calculating the standard deviation of the FDO metric over the ensemble at each frequency. The comparison of the test FRFs with the test impedance matrix in the FDO metric will be referred to as test-median correlation, while the comparison of the test FRFs with respect to the FEM will be referred to as test-analysis correlation. With a set of bounds on the error in the FDO metric due to the variation in the ensemble of FRFs, the correlation criteria can be relaxed from what was used with a single test FRF. The suggested validation criteria for the single FRF are augmented to factor in larger errors due to variations in the ensemble of test data. The correlation criteria are relaxed by allowing for greater errors in the FDO metric for frequency bands where the standard deviation of the test FDO error is greater than the maximum allowable error of 0.1. The standard deviation is calculated based on test-median correlation over the ensemble with no FBC. In regions where the 1ı bound exceeds the 0.1 threshold, 1ı will be used as the upper limit for allowable FDO error, while in regions where the 1ı bounds are less than 0.1, the passing criteria will remain unchanged. In frequency regions with large variation (1ı > 0.1), the metric examines various levels of FBC to determine how greatly the frequency uncertainty affects the error in the FDO metric. The end goal is to determine what level of FBC is required for the model to be within the augmented correlation criteria for a sufficient portion of the test cases. Note that since FDO varies quadratically with frequency, there may not be a frequency band that accomplishes this goal. Correlation in the presence of an ensemble is determined by the level of FBC needed to get a sufficient amount of the test cases inside the 1ı error bounds at the frequency points with large variation and inside the standard passing criteria at all the remaining frequency points. To illustrate the need for a hybrid correlation criteria, a pair of examples cases, one with very small variation and one with larger variation in the FRFs, is examined. For an ensemble with small variation, the 1ı bounds may be very small and therefore it is unrealistic to expect the model to fall within those bounds at each frequency. Here the dispersion of the data is tight so the errors from the FDO metric may be well inside of the 0.1 bounds, but the model may contain errors greater than the variation of the data. However these errors would be deemed to be of an acceptable size. For this case, getting inside of the 1ı bounds is too restrictive for what is required of the model. Figure 1 shows an ensemble of FRFs with very small variation and the subsequent FDO error between the model and the ensemble.
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Figure 1: Receptance FRF (top) and FDO error (bottom) for an ensemble with low variation Figure 1 shows that the FDO results between the model and the ensemble pass the 0.1 criteria for each test case, but none of the cases are remotely close to passing the 1ı criterion. This shows that the model possesses some errors not due to the spread of the test data, but they are at an acceptable level. This case shows that the use of the 1ı bound alone would be too restrictive and would exclude sufficiently accurate models. A second example shows larger variation among the test FRFs. The variation in the ensemble is reflected in the spread of the FDO errors. No matter how well the model matches any particular FRF it has no chance of meeting the 0.1 correlation criteria for the entire ensemble. There is a need to relax the correlation criteria to account for the uncertainty in the ensemble. Figure 2 displays the receptance FRFs and the resulting FDO error for the exact experimental model.
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Figure 2: Receptance FRF (top) and FDO error (bottom) for an ensemble with larger variation It can be seen in Figure 2 that the FDO error has no chance of meeting the 0.1 criterion for the exact test model. This emphasizes the need to relax the correlation criteria to include the 1ı criterion. For regions with high modal density, the use of the 0.1 criteria may become less relevant and the correlation may be dominated by the 1ı criterion. Neither the 1ı nor the 0.1 correlation criteria stand alone as a viable metric for use with an ensemble of test data. However these two simple examples demonstrate the need for a hybrid criterion for model correlation with an ensemble. $SSOLFDWLRQV '2)6SULQJ0DVV6\VWHP The following example, the 6-DOF spring mass system shown in Figure 3 [12], is used to illustrate the main ideas. While the 6-DOF model is a very simple example, it contains 3 closely spaced modes near 7 Hz, which could complicate the model correlation process. The model has 6 masses and 10 springs, all of which are used as design variables in the updating process. The model is assumed to have 1% modal damping. For this example there are outputs at each mass and an input at mass 2. Table 1 lists the nominal model and “experimental” values for the design variables.
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Design Variable K1 K2 K3 K4 K5 K6 K7 K8 K9 K10 M1 M2 M3 M4 M5 M6
Figure 3: 6-DOF spring mass model
Nominal Value 3000 (N/m) 1500 (N/m) 1000 (N/m) 2000 (N/m) 1100 (N/m) 1400 (N/m) 1250 (N/m) 5000 (N/m) 3000 (N/m) 1000 (N/m) 1.0 (kg) 1.5 (kg) 1.2 (kg) 2 (kg) 2.5 (kg) 1.1 (kg)
Experimental Value 3600 (N/m) 1725 (N/m) 1200 (N/m) 2200 (N/m) 1320 (N/m) 1330 (N/m) 1500 (N/m) 5250 (N/m) 3600 (N/m) 850 (N/m) 1.0 (kg) 1.4 (kg) 1.2 (kg) 2.2 (kg) 2.5 (kg) 0.9 (kg)
Table 1: Mass and stiffness parameters
An ensemble of test FRFs was generated analytically by allowing all 16 experimental design variables to vary normally with the standard deviation set to 5% of the experimental value. Fifty test cases were generated, and the median FRF was chosen using the technique outlined in Section 3.1. The frequency points at 1, 9, 11, and 22 Hz were used to determine the median FRF. Figure 4 shows the ensemble of FRFs for the output at mass 6 along with the median FRF and the FRF generated with the exact experimental mass and stiffness values. Note that the exact experimental FRF was not included in the ensemble, and it is only shown for visual comparison. -1
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Figure 4: Receptance FRF for 6-DOF spring mass model (output 6, input 2) It can be seen in Figure 4 that the median FRF closely resembles the exact experimental FRF, and its peaks appear to fall near the center of peak distributions for the ensemble. It is interesting to note that if the standard deviation on the design variable is reduced from 5% to 1% for one of the test cases, then the algorithm for determining the median FRF will generally select that particular FRF as the median. This agrees with intuition, because the FRF with a smaller design variable variance would likely have far less variation in the FRF as compared to the other test cases. This implies that the median finding technique is satisfactory for picking the median FRF for this example problem. The above property did not hold when frequencies near the resonant
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peaks are used in the median finding algorithm. In general, the algorithm did not select the reduced variance FRF as the median when frequencies close to the peaks are used. This can be attributed to the fact that the mean values of the FRF near the peaks are substantially lower due to the averaging process. The nominal 6-DOF model was updated to match the median test FRF using a least squares cost function based on the FDO metric with the Response Function Method [6-8]. The FRFs at input 2 and output 6 for the nominal model, test median, and updated model are shown in Figure 5. -1
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Figure 5: FRF for nominal model, updated model, and median test case In Figure 5 the FRFs for the updated model and the median FRF are indistinguishable. With the updated model tuned to the test median, it can then be used to shape the test impedance matrix using the method outlined in Section 3.2. Figures 6 shows the real part from entries of the FEM and two test impedance matrices that were calculated using the standard Moore-Penrose inverse (pinv) of the FRF and the weighted pseudoinverse described in the optimization. 4
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Figure 6 demonstrates that the optimization in Section 3.2 forces the pseudoinverse to behave more like the actual model. The variation in the test impedance matrix generated from the Moore-Penrose inverse does not vary smoothly in frequency like the weighted test impedance matrix. This makes the unweighted test impedance matrix more sensitive to errors, as compared with the weighted impedance matrix in the FDO metric. Figure 7 shows the weighted test impedance matrix has a very similar sensitivity to errors in the FDO metric as compared to the analytical model. The unweighted impedance matrix is more sensitive in the neighborhood of 15 and 20 Hz, where the impedance matrix generated from the unweighted pseudoinverse is unsmooth. The FDO results in Figure 7 were obtained using 0% FBC. Frequency Domain Orthogonality for Test and pinv Median FRF 6 5 4 3 2 1 0
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Figure 7: FDO metric versus the ensemble for the unweighted test Z, weighted test Z, and the Z from the exact experimental model The error in the FDO metric for the test-median correlation is used to determine the variance in the ensemble of test data and to augment the FDO correlation criteria. Bounds for the test-median FDO are created for a 0%
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frequency correlation window. The correlation criteria are augmented in frequency regions where the 1ı bound on the test-median FDO metric is greater than 0.1. The updated model is examined with respect to the ensemble for various frequency windows. As the frequency window is widened the error in the FDO metric is reduced. Figure 8 shows the FDO metric for various levels of FBC with the augmented correlation criteria based on the 1ı bound from the test-median FDO. FDO Error with 2% FBC 2.5
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Figure 8: FDO metric for test-analysis correlation with 2% (top), 4% (middle), and 6% (bottom) FBC with augmented correlation bounds (dashed) For the case with 6% FBC all of 50 test cases pass the augmented correlation criteria, while 5 cases from the 4% FBC and 7 cases from the 2% FBC do not meet the correlation criteria. This shows that the updated model is correlated to the entire ensemble of FRF to within 1ı for 6% frequency band correlation. Though the entire ensemble was correlated with in 6% FBC, this does not imply that the errors in the test frequencies are guaranteed to be less than 6%. This is due to the fact that the FDO metric factors in errors due to damping
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parameters and mode shapes, in addition to natural frequency errors. For this ensemble, the errors in the natural frequencies relative to the median test were all less than 8%. Depending on the end use of the model, the user may determine that an 80% passing rate is acceptable. In this case 2% FBC would be sufficient for model correlation. Not requiring all samples to pass the correlation criteria helps reduce the effect of any outlying samples in the model correlation. *HQHUDO3XUSRVH6SDFHFUDIW The second example considers the General Purpose Spacecraft (GPSC) to examine the modified FDO metric on a more complex structure. While this is not a mid-frequency application, it does contain a level of complexity beyond the simple spring-mass system. The ensemble of FRFs was generated by making small variations in several model parameters including element thickness, area, and moment of inertia. These parameter changes affected both the mass and stiffness matrices. This example problem only looks at the correlation of the nominal model with the analytically generated test cases. The GPSC model with the input location is shown in Figure 9. The frequency range for this example was 5 to 50 Hz, which contained 15 normal modes. Some of the modes are closely spaced in frequency, which may make them difficult to distinguish in a modal test.
Figure 9: GPSC with input location
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Figure 10: GPSC ensemble FRF (top), FDO metric for the ensemble with 0% FBC (middle) and 5% FBC (bottom) with augmented correlation bounds (dashed) Figure 10 shows the receptance FRF and the FDO errors with the augmented 1ı bound for 0% FBC and 5% FBC. For 0% FBC only 9 of the 25 test cases pass the augmented correlation criteria, while 21 of 25 test cases pass with 5% FBC. This shows that a substantial portion of the error and variation in the ensemble can be accounted for by small frequency shifts between the model and the experimental data. &RQFOXVLRQ Frequency response based correlation criteria were developed for use in the presence of uncertainty related to an ensemble of test data. It was shown that the FDO metric can parallel validation criteria that are used in modal orthogonality and natural frequency pairing, while making direct use of the FRFs. The FDO metric was shown to be an alternate form of modal orthogonality, and a link can be formed to frequency error by using FBC inside the FDO metric. For a family of test data, the correlation criteria are augmented to allow for greater errors in frequency regions with large variation in the FDO values based solely on the test data. Frequency band correlation is used to determine how greatly the frequency uncertainty affects the error contained in the FDO metric. Correlation in the
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presence of an ensemble is determined by the level of FBC needed to get a sufficient amount of the test cases inside the error bounds generated from the variation in the test data. When dealing with an ensemble of test data, it is difficult to determine what portion of the samples must pass the correlation criteria for the model to be considered valid. While each application will vary, a more defined passing rate for model correlation still needs to be developed. Test-analysis correlation based on FDO was shown to be effective in two simple modally sparse applications. Future work will focus on evaluating the effectiveness of FBC on an example problem that contains high modal density, and determining if the correlation criteria outlined in this paper are still applicable. Additional future work may examine ways to further relax correlation criteria to pass test cases where the testanalysis correlation may only fail at a very small number of frequency points. Assuming the FDO error is normally distributed, one would only expect roughly 68% of the samples to pass the 1ı criterion at any given frequency point. While one member of the ensemble may show excellent correlation in a particular frequency band, it may be uncorrelated elsewhere. This makes it difficult for any member of the ensemble to pass correlation across the entire frequency band of interest. $FNQRZOHGJPHQWV The authors are grateful to Dr. Tom Paez, recently retired from Sandia Laboratory, for his time discussing the uncertainty parts of this work. 5HIHUHQFHV
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
MILͲHDBKͲ340A,TestRequirementsforLaunch,UpperͲstage,andSpaceVehicles.1999. HasselmanT.K.,C.R.N.,ZimmermanDC.,CriteriaforModelingAccuracy:AStateͲofͲtheͲPracticeSurvey, inIMAC18.2000,IMAC:SanAntonio,TX. Babuska, V., Delano, C., Lane, S., Lacy, S., FRF Correlation and Error Metrics for Plant Identification, in 46thAIAAStructures,StructuralDynamics,&MaterialsConference.2005:Austin,TX. He,J.a.F.,Z.,ModalAnalysis.2001,Woburn,MA:ButterworthͲHeinemann. Larsson, P., Sas, P., Model Updating Based on Forced Vibration Testing Using Numerically Stable Formulations,in10thIMAC.1992:SanDiego,CA. Lin, R., Ewins, D., Modal Updating Using FRF Data, in 15th International Seminar on Modal Analysis. 1990:K.U.Leuven. Rad, S., Methods for Updating Numerical Models in Structural Dynamics in Department of Mechanical Engineering.1997,UniversityofLondon:London,England. Visser, W.J., Updating Structural Dynamics Models Using Frequency Response Data, in Department of MechanicalEngineering.1992,ImperialCollegeofScience,Technology,andMedicine:London,UK. Friswell, M.I., Motershead, J.E, Finite Element Model Updating in Structural Dynamics. 1995: Kluwer AcademicPublishers. Nimityongskul, S., Kammer, D., Frequency Domain Model Reduction Based on Principal Component Analysis,inIMAC27.2009:Orlando,FL. Kammer, D.C., Nimityongskul, S., Energy Based Comparison of Test and Analysis Response in the FrequencyDomain,in26thInternationalModalAnalysisConference.2008:Orlando,FL. Kozak, M., Comert, M., Ozguven, H., A model updating routine based on the minimization of a new frequencyresponsebasedindexforerrorlocalization,in25thIMAC.2007:Orlando,FL.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Error quantification in calibration of AFM probes due to non-uniform cantilevers Hendrik Frentrup1 , Matthew S. Allen2 1
Graduate Student, Universität Stuttgart, Germany Institute of Applied and Experimental Mechanics (IAM), Pfaffenwaldring 9, 70550 Stuttgart
[email protected] 2
Assistant Professor, University of Wisconsin-Madison 535 ERB, 1500 Engineering Drive, Madison, WI 53706
[email protected]
Abstract For more than two decades, the Atomic Force Microscope (AFM) has provided valuable insights in nanoscale phenomena, and it is now widely employed by scientists from various disciplines. AFMs use a cantilever beam with a sharp tip to scan the surface of a sample both to image it and to perform mechanical testing. The AFM measures the deflection of the probe beam so one must first find the spring constant of the cantilever in order to estimate the force between the sample and the probe tip. Commonly applied calibration methods regard the probe as a uniform cantilever, neglecting the tip mass and any nonuniformity in the thickness along the length of the beam. This work explores these issues, recognizing that dynamic calibration boils down to finding the modal parameters of a dynamic model for a cantilever from experimental measurements and then using those parameters to estimate the static stiffness of a probe; if the modes of the cantilever are not what was expected, for example because the non-uniformity was neglected, then the calibration will be in error. This work explores the influence of variation in the thickness of a cantilever probe along its length on its static stiffness as well as its dynamics, seeking to determine when the uniform beam model that is traditionally employed is not valid and how one can ascertain whether the model is valid from measurable quantities. The results show that the Sader method is quite robust to non-uniformity so long as only the first dynamic mode is used in the calibration. The thermal method gives significant errors for the non-uniform probe studied here. Nomenclature αn b E F Γ h I ks kB L m
nth frequency parameter cantilever width Young’s modulus force hydrodynamic function cantilever thickness second moment of area spring constant Boltzmann’s constant cantilever length mass
M ψn ωn Q ρ ρf T w W χ x
Moment nth mode shape natural frequency quality factor cantilever density density of the surrounding fluid temperature deflection static deflection optical lever factor coordinate along beam axis
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_40, © The Society for Experimental Mechanics, Inc. 2011
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1 Introduction Originally designed to measure conductors and insulators on an atomic scale, the inventors of the Atomic Force Microscope (AFM) envisioned a device that can be applied to measure forces and examine surfaces in many fields of science [1]. One great advantage of the AFM is the fact that the sample of interest need not be coated, measured in vacuum or conduct an electrical current. AFMs can operate under ambient conditions. Hence, the range of applications for an AFM are enormous and allow quantitative research on the nanoscale where other microscopic methods are far beyond their limits. Today, the AFM is indeed useful to scientists in the fields of medicine, biotechnology, chemistry, engineering and many more. A particularly spectacular example for the potential of AFM is the imaging of single atoms within a pentacen molecule [2]. It is necessary to perform the imaging at nearly absolute zero and sharpen the tip of the AFM by picking up a CO molecule to achieve this resolution level. Nonetheless, these dimensions were only explored in a theoretical realm before. Recently, the AFM also rendered possible the manipulation of single atoms on a semiconductor surface at room temperature [3]. Sugimoto et al. [3] describe how they implemented co-called dip-pen nanolithography with the AFM. The tip apex is wetted with atoms which could then be individually deposited to write patterns on the semiconductor surface. These examples show how important of a role the AFM plays in the development of nanoscale electronics and chemistry. Moreover, AFM techniques are commonly employed in microbiology for their advantage over electron microscopy when measuring living organisms. Only under conditions where the organisms prosper is it possible to directly observe their cell growth [4]. Hence, measurements on living cells have to be done in aqueous solutions in order to observe dynamic events on this scale, such as the interaction between cell membranes and drugs [5]. Structural imaging being one usage, force spectroscopy is also increasingly being used in microbiology to measure the nanoscale chemical and physical properties of cells. The practical potential of force spectroscopy is demonstrated in a study on the nanomechanical properties of cancer cells [6]. AFM indentation on metastatic cancer cells discovered a significantly lower stiffness compared to benign cells despite their morphological similarity, suggesting that the AFM might be more effective for cancer screening than visual inspection of the cells. The AFM uses a sharp tip mounted on a small cantilever beam to scan over the surface of a sample. While scanning the sample, the deflection of the microcantilever is measured by pointing a laser at its free end and recording the motion of the reflected laser spot with a photodiode [5]. With help from a calibration sample, it is possible to determine the relationship between the voltage output of the diode and the cantilever’s deflection, called deflection sensitivity. A second calibration must be performed if one wishes to relate the measured deflection to the tip-sample force. This second calibration is the focus of this work. Since the deflections of the cantilever are linear, the spring constant, i.e. the static stiffness of the cantilever, is the parameter that must be determined to find the forces of interest. In classical beam theory, the spring constant1 of a uniform cantilever ks = Ebh3/4L3 depends on its Young’s modulus E and it’s geometry, with b being the cantilever’s width, L its length [7]. The thickness of the beam, h, is typically assumed to be uniform along the length. Due to considerable variations in microfabrication, the properties of the cantilevers, especially Young’s modulus, thickness and mass distribution, cannot be determined very easily. Calibration is therefore of crucial importance in force spectroscopy, and even in routine contact-mode or friction force imaging if one needs to know the force applied by the probe during imaging.
1.1 Calibration methods To be practical, AFM calibration methods must not damage the probes in the calibration process and one must be able to perform them quickly and without the need for additional complex equipment other than the AFM. Moreover, effective calibration methods must be applicable under ambient conditions as 1
3
In detail, ks = 3EI/L3 , where I is the beam’s second moment of area I = bh /12.
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the AFM usually operates at such. The focus of this work is on two non-destructive, dynamic calibration methods, the Sader method and the Thermal Tune method. Sader developed his method by taking the effects of a surrounding fluid on the oscillating cantilever into account [8]. He uses the natural frequency and Q factor of the first mode of the cantilever to determine its static stiffness. Some information on the cantilever’s geometry is required to use the Sader method, namely the planview dimensions of a rectangular cantilever, i.e. the width and length, which govern the effects of the fluid for flexural oscillations. The planview dimensions can be easily determined using optical microscopy and therefore this does not pose a severe limitation for the calibration. However, Sader assumed a uniformly thick cantilever in his derivation and did not take a tip mass into account. Allen et al. [9] pointed out that these assumptions are not accurate and, depending on the mode used for the calibration, can lead to errors of considerable extent in the calibration process. They quantified the error in the stiffness estimated by the Sader method due to an unmodeled rigid tip and proposed a method that can be used to estimate the tip mass from measurements of the natural frequencies of the probe. The tip was found to cause considerable calibration error for some AFM probes if not correctly accounted for. This work uses a similar approach to explore the effect of a non-uniform thickness along the beam’s length on the dynamic properties of the cantilever, which influence the accuracy of the calibration. The Thermal Tune method, first proposed by Hutter and Bechhoefer [10] exploited the equipartition theorem to determine the spring constant. The theorem states that the mean kinetic or potential energy of each mode of a cantilever when excited by only thermal noise is equal to 1/2 kB T [11], with T being the temperature and kB being Boltzmann’s constant. This relationship is used to relate the mean thermal oscillation amplitude with the spring constant. The method is quite simple, although there are a number of important details that must be accounted for, as described by Cook et al. [12].
2 Effects of Non-uniform Thickness on Dynamic Properties Many commercially available AFM probes are significantly non-uniform along their lengths. For example, the SEM images shown below were obtained from a CSC38-B cantilever manufactured by Mikromasch. The nominal thickness of the beam is given by the manufacturer as 1 μm. Scanning Electron Microscope (SEM) images indicate that the thickness of the cantilever is not uniform but has a considerable taper toward the tip. It is almost three times as thick as nominal at the point where the tip starts. The profile of the beam was estimated using these SEM images and will be used to quantify the effect of this thickness non-uniformity on calibration for this particular probe, with a view to extending the methodology to other probes.
(a)
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Figure 1: SEM images of a silicon nitride cantilever beam. On the left, the tip and a portion of the cantilever are shown. In (b), a detail near the tip is shown where it can be observed how the thickness of the cantilever increases from 1.830 μm to 3.472 μm toward the tip.
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Figure 2: Different thickness profiles of the cantilever beam. The estimate of the actual profile and two fitted counterparts are shown The mechanics of the measured cantilever profile will also be compared to two different profiles that approximate it with a small number of parameters. The first models the non-uniformity with a section that has constant thickness, hnom with a taper starting at x0 , and reaches the same thickness as the real cantilever at the free end. This profile will be called the “partially linear” profile. The second models the non-uniformity as a linearly increasing thickness. This profile cannot mimic the prominent increase in thickness that the real profile has at the end of the cantilever, so that part is regarded as an extra mass without rotary inertia and lumped at the end of the cantilever. This latter profile shall be referred to as “linear and lumped”. Both profiles are defined by only three parameters, which are given in Table 1. The “linear and lumped #2” profile is a result of the sensitivity analysis shown in Figure 4. Its parameters were chosen to match the experimental result of the frequency spacing more closely, whereas the parameters for the "linear and lumped" profile were chosen based on the SEM images shown previously. More complicated profile models could be envisioned, but then one would have more free parameters that must be either assumed or determined.
Partially linear Linear and lumped Linear and lumped #2
Profile parameters hnom = 1.0 μm h(1) = 3.876 μm h0 = 1.218 μm h1 = 0.275 μm h0 =1.193 μm h1 = 0.302 μm
x0 = 0.83 Δm = 5.792 · 10-12 kg Δm = 6.372 · 10-12 kg
Table 1: The parameters of the two chosen profiles fitted to the real cantilever profile. The definition of the parameters is given in section 6.1 of the appendix. A Ritz model will be created for each of these probes, so the equations of motion governing the motion of the cantilever are given as follows, where q denotes the frequency domain amplitude of the generalized
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coordinates [13]. The inertia and stiffness of the beam as well as the virtual mass added by the fluid and the dissipative effect of the fluid are included in the calculations (see section 6.2 in the appendix for details). −ω 2 [M]q + iω[C]q + [K]q = 0
(1)
The Ritz method can be used to obtain approximate solutions for the mode shapes and natural frequencies of continuous structures. Here this approach is used to find the approximate mode shapes of non-uniform cantilevers. However, later we shall make use of the fact that a single term Ritz model is an exact model for the contribution of a single mode to a structure’s response if the mode shape used in the Ritz series is exact. The analytical model presented in [9] gives the mode shapes of a uniform cantilever [13] with a rigid tip on its free end. Those analytical mode shapes will be used in this work as basis functions for the Ritz method in order to find the modes of the cantilever for each of the thickness profile models in Table 1. φn (x) = sin(αn x) + sinh(αn x) + Rn (cos(αn x) − cosh(αn x)) Rn =
(2)
sin(αn ) + sinh(αn ) cos(αn ) − cosh(αn )
The parameters αn depend on the boundary conditions which include the effect of the tip. In order to account for non-uniformity, the model used here allows for a variable thickness along the cantilever’s length, h = h(x).
(a)
(b)
Figure 3: Mode shapes of the cantilever. The dashed lines depict the analytical mode shapes of a uniform cantilever with tip - solid lines are Ritz estimates of the mode shapes for the measured profile and for the “linear and lumped #2” profile. The odd modes are shown in (a), even modes in (b). Figure 3 shows the mode shapes of a cantilever with a tip mass of mr = 0.088 and a uniform nominal thickness compared to those with the same tip and the thickness profile estimated from the SEM images. The third set of mode shapes shown are for the "linear and lumped #2" model described previously. The mode shapes in Figure 3 indicate lower amplitudes of vibration at the end of the beam for the “linear and lumped #2” cantilever profile than for the other two because the additionally lumped mass increases the
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inertia of the beam at the free end. Also, the higher the order of the mode, the more the respective mode shapes are curved, in particular close to the free end of the beam, so thickness non-uniformity has a more prominent effect on the higher modes.
Experimental [9] Uniform + Tip Measured profile Partially linear Linear and lumped Linear and lumped #2
ω2 /ω1
δexp
ω3 /ω1
δexp
ω4 /ω1
δexp
7.806 6.475 7.807 7.813 7.658 7.812
17.05% 0.01% 0.09% 1.89% 0.08%
23.572 18.507 23.665 23.698 23.056 23.642
21.49% 0.39% 0.53% 2.19% 0.30%
48.490 36.826 48.409 48.964 46.771 48.042
24.05% 0.17% 0.98% 3.54% 0.92%
Table 2: Frequency spacings of the cantilever beam. δexp is the deviation of the respective frequency spacing compared to the experimental value. The spacings between the natural frequencies for all of these models are shown in Table 2. By comparing the frequency spacings rather than the frequencies, the comparison is not dependent on the modulus, density, and other physical properties of the probe. The analytical model underestimates the spacings between the modes considerably and with increasing deviation for higher modes. The model that uses the measured profile reproduces the frequency spacings with a very high accuracy. The partially linear profile is almost as accurate as the measured profile while the “linear and lumped” profile presents higher deviations, but both are significantly more accurate than the uniform model with a tip. The parameters for the “linear and lumped #2” model were chosen to minimize the difference in the frequency spacings, so the agreement is excellent. One can achieve good agreement in terms of the frequency spacings for both ways of parameterization. However, as shown below, the partially linear profile was found to erroneously represent the static properties of the measured profile, so the “linear and lumped” profile is of primary interest. Figure 4 depicts the sensitivity of the “linear and lumped” profile. The point of origin for the analysis was the initial set of parameters for this profile given in Table 1 and all three parameters were varied between 90 to 120% of their initial value. The starting thickness h0 of the profile has proportionally higher influence on the frequency spacings than the other two parameters and is therefore varied to a lesser extent. The graph shows that it is not possible to match the frequency spacing of the measured profile in all four modes exactly, but it is very well possible to lower the deviation considerably. The second set of parameters for the “linear and lumped” profile given in Table 1 were found by using this sensitivity information to achieve better agreement. In order to determine how well each of these models captures the static stiffness of the cantilever, an analytical solution was derived for the spring constant of a cantilever with an arbitrary thickness variation along its length. The deflection curve of the cantilever is given by w = M (x)/EI(x). Loaded with a static tip force, the bending moment is linear. Thus, we can express the spring constant as follows ks = h3es
1 =− 3
Ebh3es , 4L3
1 x 0
0
(x − 1) dxdx h(x)3
(3) −1 .
(4)
The static stiffnesses computed for each of the probe models using this approach are shown in Table 3. The manufacturer estimated the spring constant of the cantilever at 0.03 N/m, presumably based on the
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Figure 4: Variations in h0 , h1 and Δm parameters of the “Linear and Lumped” profile showing the sensitivity of the frequency spacing with respect to these parameters. Manufacturer’s specification: ks = 0.03 (0.01 − 0.08) N/m ks [N/m] Uniform + Tip Measured profile Partially linear Linear and lumped Linear and lumped #2
0.0278 0.0580 0.0278 0.0585 0.0560
Table 3: Static stiffness, ks , estimated for each of the models of the thickness profile. nominal thickness of 1 μm, and the analytical model agrees well with this. However, the manufacturer also states that the spring constant could be anywhere in the range from only a third to more than twice the nominal value. In fact, the profile from the SEM images has a spring constant of 0.058 N/m which is almost twice the nominal value. The other parameterizations of the model have a completely different static behavior. The considerable taper toward the free end of the beam has no noticeable effect on the beam’s static stiffness as the “partially linear” spring constant is equal to that of the uniform cantilever. On the other hand, the static stiffness of the "linear and lumped" profile agrees very well with that of the measured profile.
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3 Corrections of Calibration Methods 3.1 Sader Method Modeling the cantilever with single mode, its first natural frequency, ω1 , depends on the mass m and the stiffness of the oscillator, ω12 = ks/m ⇒ ks = Me ρhbLω12 [8]. With h, b, and L being the cantilevers thickness, width and length. The geometry of the cantilever and its density ρ are used to determine the mass. The expression presented by Sader treats the mass distribution as uniform along the length of the cantilever, so the effective normalized mass, Me is equal to 0.2427 when L/b > 5 [14]. When vibrating in vacuum, this assumption is very accurate. However, an oscillating cantilever immersed in a fluid moves the surrounding fluid as it vibrates, creating an inertial loading on the beam, a virtual mass. The resonance frequencies of an immersed cantilever are lower compared to a cantilever in vacuum due to the inertial loading. The surrounding fluid also has a damping effect on the cantilever, which lowers the cantilever’s quality factor Q, reduces the peak amplitude and broadens its resonance curve [15]. Sader accounted for the effects of the fluid by including a hydrodynamic force as part of the loading on the cantilever [16], based on the hydrodynamic function, Γ. By modeling the effect of the fluid, Sader was able to determine the mass and spring constant of the cantilever from the Q-factor and resonance frequency according to ks = 0.1906ρf b2 LΓi Q ω12 [14], where ρf denotes the density of the surrounding fluid. Following the approach in [9], the Sader method can be modified to include the effect of a massive tip and a non-uniform thickness. To this end, we introduce an equivalent thickness with respect to stiffness and inertia, h3ek and hem which are the following, h3ek,n
1 =
0
1
h(x)3 (ψn )2 dx 1 2 0 (ψn ) dx
hem =
0
h(x) (ψn )2 dx 1 2 0 (ψn ) dx
(5)
so that we can keep the original coefficients mnn and knn . The prime in equation (5), denotes differentiation with respect to x. The equivalent thicknesses, hem and hek , are the thicknesses that a uniform cantilever must have to produce the same mnn and knn terms in a single-term Ritz model. Therefore, we can replace h3 with h3ek and h with hem in the stiffness and inertial coefficients in section 6.2 of the appendix, and after some algebra one obtains the following equation for the static stiffness of the probe. ks,Sader
3π = ρf b2 LΓi 4
hes hek,n
3
mnn Q ωn2 knn
(6)
This expression is also valid for a uniform probe with a rigid tip, as presented in [9], and for a uniform probe without a tip it reduces to the formula given by Sader. If the thickness profile of a probe is known, then one can compute hes , and with the mode shapes one can compute mnn , knn , and hek,n . Then, this expression would give an accurate estimate of the spring constant, even in the presence of nonuniformity. Unfortunately, these quantities are not known in practice, so it is difficult to determine the needed constants. One must, instead, make a simplifying assumption regarding the cantilever, such as uniform thickness, in order to obtain an estimate for ks . Our primary purpose for deriving eq. (6) is to estimate the error incurred by such an assumption. Consider an exact solution for the thickness profile and mode shapes. Denote the parameters for that model as hes , hek,n , mnn , and knn . These parameters would produce the true static stiffness if used in conjunction with the measured Q and ωn . An approximate model gives different values for these parameters, which shall be distinguished with hats (ˆ ). Then, the error incurred in the Sader method due to the simplifying assumption is given by the following. ˆ kˆs,model − ks,true hes hek,n 3 m ˆ nn knn = −1 (7) δks,Sader = ˆ ks,true hes hek,n mnn kˆnn The Ritz mode shapes found previously for the measured profile were found to correspond well with the experimentally measured mode shapes shown in [17], so that model will be assumed to be exact
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δks
Mode 1
Mode 2
Mode 3
Mode 4
Uniform No Tip - Measured Uniform + Tip - Measured Linear and Lumped - Measured Linear and Lumped #2 - Measured
-2.22% -2.71% -0.06% -0.22%
2.62% 4.14% 5.91% 6.00%
6.24% 15.91% 10.52% 9.66%
14.23% 25.38% 15.43% 14.09%
Table 4: Error in the static stiffness estimated by the method of Sader for various profiles as a function of the mode number used in the calibration. The model based on the measured profile was taken to be exact. and used to compute the error in approximating the probe with each of the models listed in Table 1. The resulting relative errors for each of the models are given in Table 4. One observes that each of the models is capable of estimating the static stiffness of the probe to within 3% if the first mode is used in the calibration, and to within 6% if the second is used, although the errors become significantly larger if modes 3 or 4 are used. These modes are more sensitive because the tip mass and the stiffened section of the probe near the tip begins to have an important effect on the mode shapes. It is also interesting to note that the original original Sader model of a uniform beam without tip has smaller deviations from the measured profile than the model with a tip. Considering that the tip adds inertia to the model, Sader’s original uniform cantilever is less stiff and lacks the inertial effect of the tip at the same time, leading to a mutual compensation in the calibration process and less deviation.
3.2 Thermal Tune Method The Thermal Tune method is based on equivalence between the mean-square potential energy of the cantilever and 1/2 kB T . 2 2 L 1 1 ∂ w Evib = kB T = EI(x) dx (8) 2 2 ∂x2 0 The kinetic energy is related to the motion of the tip of the cantilever, which is related to the output signal of the photodiode. However, the sensitivity of the photodiode is measured under static conditions and the shape of a cantilever under static loading is different from its shape when vibrating freely. This is accounted for using the method described by Cook et al. [12], which multiplies the deflection measured by the photodetector, d∗c , with the factor χ to yield the actual deflection needed for the calibration, dc . The slope of the cantilever at xl , where the laser is reflected on the beam, determines the laser spot’s position on the photodetector [12]. Hence, χ depends on the ratio of the end-loaded slope Wend to the freely oscillating slope Wfree χ(xl ) =
Wend (xl ) . Wfree (xl )
(9)
W is the cantilever’s normalized shape2 , i.e. the normalized deflection curve in case of the end-loaded cantilever [12] and the normalized mode shape of a freely vibrating cantilever is given by Wfree (x) = ψ(x)/ψ(1). Following the same approach used to derive the modified Sader method, the Thermal Tune relationship between the static stiffness and the probe parameters is [17] 2
Normalized shape implies W (1) ≡ 1.
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ks,Thermal
3kB T = (d∗c )2
hes hek,n
3
ψn (1)2 2 knn Wend (1)
(10)
Likewise, the error induced by non-uniformity in the Thermal Tune method is δks,Thermal
ˆ kˆs,model − ks,true hes hek,n 3 = = ˆ ek,n ks,true hes h
Wend ˆ W
end x=1
2
ψˆn (1) ψn (1)
2
knn − 1, kˆnn
(11)
where the hats (ˆ) once again denote the entities for the model of interest. The normalized deflection curve and its derivative for a uniform cantilever are known [12]. The derivative of the deflection curve for a cantilever with an arbitrary thickness profile is calculated using the analytical model for the static deflection of a non-uniform model, which was also used to derive eq. (4). x
Wend (x)
(a)
=
(x−1) 0 h(x)3 dx 1 x (x−1) 0 0 h(x)3 dxdx
(12)
(b)
Figure 5: Deflection curves of uniform and non-uniform cantilevers under static loading in (a) and the derivative with respect to x, i.e. the slope of the cantilever, in (b). The deflection curves and their derivatives with respect to x are given in Figure 5 for both the uniform and the measured thickness profiles. It can be observed that the slopes are considerably different near the free end, and this difference is squared in eqs. (10) and (11), so it may be important. The rotation of the mode shapes, ψn (1), i.e. the shape of the freely vibrating cantilever, also induces error in the Thermal Tune method. The slope at the free end of the cantilever is influenced by the profile of the beam, as was illustrated in Figure 3. Table 5 presents the errors in the Thermal Tune estimated spring constant once again using the thickness profile from the SEM images as the “true” model. In contrast with the Sader method, the thermal method is significantly in error if a uniform probe model is used, even if the first mode is used in the calibration. The error becomes extremely unacceptable if the second or higher modes are used. In contrast, the linear and lumped models are acceptably accurate for the first mode but even they become very inaccurate if the third or higher modes are used.
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δks
Mode 1
Mode 2
Mode 3
Mode 4
Uniform No Tip - Measured Uniform + Tip - Measured Linear and Lumped - Measured Linear and Lumped #2 - Measured
15.31% 22.53% 6.32% 4.64%
72.79% 71.88% 13.82% 8.87%
160.93% 134.93% 35.79% 28.21%
287.70% 214.03% 73.20% 63.07%
Table 5: Error in the static stiffness estimated by the Thermal Tune method for various profiles as a function of the mode number used in the calibration. The model based on the measured profile was taken to be exact.
4 Conclusion The thickness profile of a cantilever has a decisive influence on its static stiffness as well as its mode shapes. Two common atomic force microscope calibration methods were analyzed to see what influence this has on the calibration. The Sader method was found to accurately estimate the static stiffness of the probe under study, but only if the first mode was used in the calibration. We presume that similar trends would hold for other probes so long as the non-uniformity is not too drastic, so the Sader method seems to be quite a robust choice. On the other hand, the thermal method depends on the rotation of the cantilever under static and dynamic loading and so it was found to be very sensitive to the probe’s non-uniformity. These results suggest that the thermal method should not be trusted unless the probe of interest is known to be uniform. A few simple models were explored in an effort to capture the effect of thickness nonuniformity. The models were found to reproduce the natural frequencies and mode shapes of the probe reasonably well, but calibration is sensitive to the model. The “linear and lumped” model did reduce the error in the Thermal Tune calibration somewhat, but even then the results showed that one may only be able to trust the calibration based on either the first or perhaps the second mode of the probe. Also, future works should seek to characterize other probes as was done here for the CSC38 probe, to see whether any of the models proposed here can account for the range of commonly encountered nonuniformities with sufficient fidelity.
5 Acknowledgments The authors would like to express their gratitude to the College of Engineering at the University of Wisconsin-Madison, the Institute of International Education, the association “Global Education for European Engineers and Entrepreneurs” (GE4), Office of International Affairs at the Universität Stuttgart, and the foundation “Landesstiftung Baden-Württemberg” for giving Mr. Frentrup the opportunity to study and conduct his research at the University of Wisconsin-Madison. The authors also wish to acknowledge Peter C. Penegor for his contribution by obtaining the cantilever profile from the SEM images.
6 Appendix 6.1 Parameterized profiles h(x) = h0 + h1 x , Δm = ρbL
1 0
0≤x≤1
hmeasured (x) − h(x)dx
(13) (14)
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h(x) =
hnom hnom + h1 (x − x0 )
0 ≤ x < x0 x0 ≤ x ≤ 1
with h1 =
h(1) − hnom 1 − x0
(15)
6.2 Ritz coefficients 2 Itip ρf b2 LΓr (ω) mnn + mtip ψn (1)2 + 2 ψn (1) 4 L π Cnn = ρf ωb2 LΓi (ω) mnn 4
Mnn = ρbL(hem mnn ) +
π
Knn = knn =
1 0
2 ψn dx,
Eb (h3 knn ) 12L3 ek,n
(17) (18)
mnn =
(16)
0
1
(ψn )2 dx
(19)
References [1] G. Binnig, C. F. Quate, and C. Gerber. Atomic force microscope. Physical Review Letters, 56:930– 933, 1986. [2] L. Gross, F. Mohn, N. Moll, P. Liljeroth, and G. Meyer. The chemical structure of a molecule resolved by atomic force microscopy. Science, 325:1110–1114, 2009. [3] Y. Sugimoto, P. Pou, O. Custance, P. Jelinek, M. Abe, R. Perez, and S. Morita. Complex patterning by vertical interchange atom manipulation using atomic force microscopy. Science, 322:413–417, 2008. [4] A. Touhami, M. H. Jericho, and T. J. Beveridge. Atomic force microscopy of cell growth and division in staphylococcus aureus. Journal of Bacteriology, 186:3286–3295, 2004. [5] Y. F. Dufrene. Towards nanomicrobiology using atomic force microscopy. Nature Review Microbiology, 6:674–680, 2008. [6] S. Cross, Y. S. Jin, and J. Rao andf J. K. Gimzewski. Nanomechanical analysis of cells from cancer patients. Nature Nanotechnology, 2:780–783, 2007. [7] J. E. Sader and L. White. Theoretical analysis of the static deflection of plates for atomic force microscope applications. Journal of Applied Physics, 74:1–5, 1993. [8] J. E. Sader, I. Larson, P. Mulvaney, and L. White. Method for the calibration of atomic force microscope cantilevers. Review of Scientific Instruments, 66:3789–3798, 1995. [9] M. S. Allen, H. Sumali, and P. C. Penegor. Experimental/analytical evaluation of the effect of tip mass on atomic force microscope calibration. Journal of Dynamic Systems, Measurement, and Control, Accepted, April 2009, DOI: 10.1115/1.4000160. [10] J. L. Hutter and J. Bechhoefer. Calibration of atomic force microscope tips. Review of Scientific Instruments, 64:1868–1873, 1993.
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[11] N. A. Burnham, X. Chen, C. S. Hodges, G. A. Matei, E. J. Thoreson, C. J Roberts, M. C. Davies, and S. J. B. Tendler. Comparison of calibration methods for atomic force microscopy cantilevers. Nanotechnology, 14:1–6, 2003. [12] S. M. Cook, T. E. Schaeffer, K. M. Chynoweth, M. Wigton, R. W. Simmonds, and K. M. Lang. Practical implementation of dynamic methods for measuring atomic force microscpoe cantilever spring constants. Nanotechnology, 17:2135–2145, 2006. [13] J. H. Ginsberg. Mechanical and Structural Vibrations: Theory and Applications. John Wiley & Sons, 2001. [14] J. E. Sader, J. W. M. Chon, and P. Mulvaney. Calibration of rectangular atomic force microscope cantilevers. Review of Scientific Instruments, 70:3967–3969, 1999. [15] M. K. Ghatkesara, E. Rakhmatullinab, H. P. Langa, C. Gerbera, M. Hegnera, and T. Brauna. Multiparameter microcantilever sensor for comprehensive characterization of newtonian fluids. Sensors and Actuators B: Chemical, 135:133–138, 2008. [16] J. E. Sader. Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope. Journal of Applied Physics, 84:64–76, 1998. [17] M. S. Allen, H. Sumali, and P.C. Penegor. Effect of tip mass on atomic force microscope calibration by thermal method. In 27th International Modal Analysis Conference, Orlando, Florida, 2009.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Demystifying Wireless for Real-World Measurement Applications Kurt Veggeberg, Business, Development Manager (
[email protected]) National Instruments,11500 N. Mopac C, Austin, TX 78759 ABSTRACT - Wireless technology extends the concept of PC based data acquisition beyond the limits of cables and wired infrastructure for new remote or distributed measurement application. Gain an understanding of wireless networking basics. Learn how to deploy reliable wireless measurement in a variety of outdoor or harsh environments for reliable and secure data acquisition systems. Examples of distributed outdoor noise and structural monitoring systems will illustrate networking layers and topologies for specific applications.
Introduction Understanding technology capabilities and application requirements is important when selecting a wireless technology for your application. The reasons to choose wireless include reduced installation costs, installation and deployment flexibility, and the ability to address new applications. Before selecting wireless, you first need to ensure the bandwidth available with wireless meets your application requirements
Choosing the Right Technology
Although the ability to eliminate cabling costs with wireless installations presents potential cost savings, wireless technology must address the application requirements. Two of the main reasons to select a wired protocol are bandwidth and reliability. Standard wired 100BASE-TX Ethernet is faster than both ® wireless IEEE 802.11g, or Wi-Fi, and IEEE 802.15.4, which provides the basis for Zigbee . When gigabit Ethernet at 1 Gbit/s is included, the bandwidth advantage for Ethernet is clear. If you do not require a bandwidth above 100 Mbit/s, then the cost savings combined with installation flexibility make wireless an effective option. Ethernet, Copper Wireless Ethernet, Fiber Wireless (100BASE-TX) (100BASE-FX) (IEEE 802.11n) (IEEE 802.15.4) Physical Wire or RF Frequency Copper Fiber Optic 2.4 GHz 2.4 GHz Bandwidth (max bit 100 Mbit/s 100 Mbit/s 100 Mbit/s 250 kbit/s rate) Range (without repeaters) 100 m 400 m ~100 m ~300 m Power Requirements High High Medium Low Typical Battery Lifetime – – 1–2 days 3–5 years Table 1. Bandwidth, Range, and Power Comparison for Ethernet and Wireless Technology
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_41, © The Society for Experimental Mechanics, Inc. 2011
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434 Networks based on 2.4 GHz Spectrum – Bandwidth Range and Power Requirements The frequency spectrum covers a broad range of signals. Today we will focus on the 2.4 GHz unlicensed section within the general frequency range of radio wave. Within this “radio spectrum” there are different frequencies used for different applications from long wave radio to satellite broadcasting. Wi-Fi in the 2.4 GHz range is a sweet spot for distance and bandwidth. (Figure 1)
Figure 1.Radio Spectrum “Sweet Spot.” There are three key factors to consider when evaluating wireless technologies: bandwidth, range, and power requirements. When you compare wireless protocols based on IEEE 802.11 and IEEE 802.15.4, Wi-Fi has the advantage in bandwidth with a maximum bit rate of 100 Mbit/s, while 802.15.4 has the advantage in distance and power requirements. This is a typical trade-off made in wireless protocols. WiFi offers significantly higher data rates, which require additional encoding; extra data requires additional radio traffic resulting in increased power consumption by the radio. This bandwidth and power trade-off is obvious in systems such as laptops or smart phones with integrated Wi-Fi that typically operate for a matter of days between recharging and provide high-speed data transfer, compared to a wireless sensor network based on IEEE 802.15.4 technology that might operate for years on standard AA batteries and transfer reduced data between sleep states. (Figure 2)
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Figure 2.Tradeoffs in wireless technologies. For technologies based on IEEE 802.15.4, this trade-off in bandwidth also results in up to a 10X improvement in distance. At a maximum distance of 300 m and a bandwidth trade-off from 54 Mbit/s to 250 kbit/s, protocols based on IEEE 802.15.4 are ideal for low-speed, long-distance remote monitoring applications, while Wi-Fi is ideal for shorter-distance, higher-power, and higher-bandwidth applications. Significant time and money has been invested into researching the use of wireless technology for remote monitoring. Yet, significant wireless deployments are just beginning to materialize in industry such as construction site management and building acoustics where running wires can be difficult. There are many advantages to eliminating cables in remote monitoring applications, but there are also many challenges. As standards such as Wi-Fi (IEEE 802.11) continue to mature, those challenges are being addressed. IEEE 802.11 has a variety of advantages for remote data acquisition and data streaming for dynamic ® signal acquisition as compared to other standards such as IEEE 802.15 (Zigbee ) including range and security. IEEE 802.11 typically operates on 2.4 GHz. It is typically specified with a range from 30 to 100 meters with data rates from 54 to 600 Mbps. The range depends on a variety of factors and can be extended significantly through a variety of network topologies and high gain antennas. IEEE 802.11 divides the band from 2400 to 2483.5 GHz into channels, analogously to how radio and TV broadcast bands are carved up but with greater channel width and overlap. For example the 2.4000– 2.4835 GHz band is divided into 13 channels each of width 22 MHz but spaced only 5 MHz apart, with th channel 1 centered on 2.412 GHz and 13 on 2.472 GHz to which Japan adds a 14 channel 12 MHz above channel 13. (Figure 3)
Figure 3.Wi-Fi Channels.
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436 Availability of channels is regulated by country, constrained in part by how each country allocates the frequency to various services. Japan permits the use of all 14 channels (with the exclusion of 802.11g/n from channel 14), while Spain allowed only channels 10 and 11 and France allowed only 10, 11, 12 and 13 (now both countries follow the European model of allowing channels 1 through 13). Most other European countries are almost as liberal as Japan, disallowing only channel 14, while North America and some Central and South American countries further disallow 12 and 13. IEEE 802.15.4-2006 specifies the physical layer and media access control for low-rate wireless personal area networks (LR-WPANs). It is maintained by the IEEE 802.15 working group. In the 2.4 GHz band ® there are 16 Zigbee channels, with each channel requiring 5 MHz of bandwidth. The center frequency for each channel can be calculated as, FC = (2405 + 5 * (ch - 11)) MHz, where ch = 11, 12, ..., 26.
Figure 4.IEEE 802.15 channels. In order to plan wireless networks it is important to remember the channels that are used by 802.11 and 802.15.4. These channels do overlap on the frequencies ranges between 2400 and 2480 MHz (2.4 GHz). With knowledge about these channels it is possible to plan and install systems that avoid channel overlap. It is also worth noting that two wireless devices, one based on 802.15.4 and the other based on 802.11 use much different data rates so even if they are transmitting on the same frequency they will function. In the case where either is trying to transmit at maximum data rate it is a good idea to architect systems to avoid channel overlap. One way to ensure co-existence of Wi-Fi and WSN channels includes setting Wi-Fi Access Points and ® Zigbee channels to avoid overlap. The other includes creating spatial distance between systems with same channel. (Figure 5)
®
Figure 5.Setting Wi-Fi and Zigbee channels to avoid overlap. Figure 5 is an example of channels that are commonly used with Wi-Fi to avoid the overlap between different 22 MHz Wi-Fi channels, in this case channels 1, 6, and 11 are used on the wireless access point. These channels are actually very common channels in Wi-Fi systems
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437 Another way to avoid channel overlap is to plan systems with enough distance between networks that the channels that are within range are different and then channels outside of network range can repeat. This is channel spacing. Typically this would be greater than 30 m for 802.11 and greater than 300 m. for 802.15.4 (Figure 6)
Figure 6.Channel Spacing Security Security is a top concern for many engineers and scientists considering wireless. The reasoning behind this is due in large part to the failings of early wireless standards such as wired equivalent privacy (WEP), which did not prevent unauthorized access well. There are two main components of network security that must be addressed before wireless is widely adopted: authentication and encryption. A wireless network is inherently more accessible than a wired network (such as Ethernet) because it is not a closed system: data travels through the air. IEEE 802.11X has evolved to provide authentication on wireless networks based on the Extensible Authentication Protocol (EAP). Clients on the network must identify themselves before being granted access to the network. There are other less sophisticated strategies for preventing unauthorized network access as well. Good security practice for wireless networks includes MAC and/or Internet Protocol (IP) address filtering and service set identifier (SSID) suppression. Even if data is accessible to an unauthorized user, it is not necessarily intelligible. Data encryption on wireless networks has evolved significantly over the last decade from clear-text broadcasts to 128-bit cryptography. The Advanced Encryption Standard (AES) is now an NIST standard and a requirement for all U.S. government installations. (Figure 7)
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Figure 7. Time required for exhaustive key search (brute force attack) Network Topology In addition to total distance, protocols based on IEEE 802.15.4 offer a couple of options for network topologies. A Wi-Fi system is typically configured in a star topology with a center access point and clients up to 30 m from the access point. If you need additional distance, a tree topology for which you can use either Wi-Fi repeaters or IEEE 802.15.4 routers helps extend your distance While standard Wi-Fi installations support repeaters or routers to extend distance and can be configured in a cluster or tree, they do not support meshing, which is the ability for a node or device to route packets back to the gateway. If network reliability is important, then with an IEEE 802.15.4 mesh network an end node can route packets through multiple routers to a gateway. This provides network reliability in case a router fails. Many 802.15.4-based wireless sensor networks (WSNs) support star, cluster tree, and mesh networking topologies. (Figure 8).
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Measurement Speeds and Wireless Throughput Wi-Fi offers higher bandwidth and as the IEEE 802.11 wireless protocol can support much higher sample rates than IEEE 802.15.4 based protocols. Measurement type, number of measurement channels, and measurement speed will determine the throughput requirements.
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439 For high-speed measurements such as dynamic signal acquisition, Wi-Fi offers additional bandwidth. For instance, 24-bit high-speed sound and vibration data is sent in 32-bit packets and for 4 channels at 51.2 kS/s the required throughput is 6.6 Mbit/s. There is some additional overhead for Wi-Fi packets, but clearly the sample rate of 51.2 kS/s requires the bandwidth of Wi-Fi.
IEEE 802.15.4 based protocols are well suited for higher channel count applications. As an example an NI WSN application with 8 nodes and 4 analog and 4 digital channels per node at 1 second sample interval requires 5.2 kbit/s. The 82 Bytes per sample packet includes packet header information, 4 analog input channels, 4 DIO channels, and channel information such as link quality and battery voltage.
For larger topologies such as a network with four routers and 32 end nodes the total throughput is 44.6 kbit/s. An important note is that in this topology the 32 end nodes communicate through one of the four routers so the network traffic is doubled from these end nodes. To calculate throughput in this extended topology multiply the number of nodes in this case, connected directly to the gateway by 1 hop and the number of end nodes connected to a router and then gateway by two hops and add the results.
Distance Requirements An important consideration is the distance from your measurement to your network access. If the distance is greater than 30 meters line of sight, then you may need repeaters for Wi-Fi. Even if distances are less than 100 m, RF interference sources including trees or buildings can reduce the achievable distance. To ensure a reliable system, a site survey is recommended for all wireless installations. If required distances exceed 100 m, then IEEE 802.15.4 offers an option with a maximum distance of 300 m line of sight, and with routers the total distances can be extended. Power Availability The final consideration when deciding between wireless technologies is power availability. For two- to three-year battery deployments at lower bandwidths, IEEE 802.15.4 is ideal. The central gateway and embedded PC require either 9 to 30 VDC power or solar power; however, end nodes function for several years on standard AA batteries. In Wi-Fi, an access point generally requires power while the end devices are typically powered by DC or solar power for extended operation. After considering the issues of throughput, range and power, you can more easily select the wireless technology that is right for your application. Addressing your application requirements is the first step. For any wireless installation, you should analyze the RF performance at the deployment site. Site surveys conducted by professionals ensure adequate coverage, network performance, and the ability to scale as you add more sensors.
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440 Applications for Wi-Fi-based Wireless Data Acquisition The higher bandwidth of Wi-Fi at up to 100 Mbit/s enables wireless data acquisition systems to address high speed waveform measurements such as strain and acceleration. The trade-off for higher bandwidth is power. An example wireless data acquisition application is short term load and strain tests. UT Ferguson Structural Engineering Lab is researching economical methods for inspection and monitoring of steel-girder highway bridges (temperature, strain, and acceleration). In the example below, a load cell was attached to the crane load line above the lift bucket to obtain accurate weight measurements of road fill being used to collapse a bridge segment being tested. A Wi-Fi transmitter was connected to the load cell so that the load data could be easily read and recorded from across the work-site. (Figures 9,10,11)
Figure 9. Bridge collapse test crane load measurements.
Figure 10.Remote load cell monitoring.
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Figure 11. Wi-Fi DAQ module with battery in housing box for instrumenting load cell.
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442 Applications for IEEE-802.15.4 based WSNs The low power and longer distance available with IEEE 802.15.4-based networks fits well for longer-term remote measurement applications. One example is structural health monitoring. The ability to easily distribute several nodes up to 300 m from a gateway and further extend this distance through mesh routers, makes WSN ideal for monitoring large structures like bridges or buildings. The system can easily measure strain. The battery operated end nodes are easily installed close to critical areas of the bridge without the requirement of local power or communication wiring. Then data is sent wireless to a gateway with a real-time PC for storage and connectivity to IT infrastructure. (Figures 12 & 13)
Figure 12.Wireless Sensor Modules being tested on a bridge on I-35 in Austin, Texas
Figure 13. Wireless range testing on bridge girders. Wireless DAQ and Wireless Sensor Networks If wireless meets your application requirements, you then need to decide between two wireless technologies: Wi-Fi or IEEE 802.15.4-based networks. The trade-off between wireless protocols typically comes down to bandwidth, distance, and power. Wi-Fi has the bandwidth advantage while IEEE 802.15.4 based networks perform better in applications that require longer-distance coverage and lower power. IEEE 802.15.4-based protocols often deliver additional network flexibility with a mesh network topology, which routes packets from end nodes to the gateway through the shortest path available.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Data Merging for Multi-Setup Operational Modal Analysis with Data-Driven SSI 1 ¨ M. Dohler , P. Andersen2 , and L. Mevel1 1 INRIA, 2 Structural
Centre Rennes - Bretagne Atlantique, Campus de Beaulieu, F-35042 Rennes, France
Vibration Solutions A/S, NOVI Science Park, Niels Jerners Vej 10, DK-9220 Aalborg East, Denmark
Abstract In Operational Modal Analysis (OMA) of large structures we often need to process sensor data from multiple non-simultaneously recorded measurement setups. These setups share some sensors in common, the so-called reference sensors that are fixed for all the measurements, while the other sensors are moved from one setup to the next. To obtain the modal parameters of the investigated structure it is necessary to process the data of all the measurement setups and normalize it as the unmeasured background excitation of each setup might be different. In this paper we present system identification results using a merging technique for data-driven Stochastic Subspace Identification (SSI), where the data is merged and normalized prior to the identification step. Like this, the different measurement setups can be processed in one step and do not have to be analyzed separately. We apply this new merging technique to measurement data of the Heritage Court Tower in Vancouver, Canada.
1
Introduction
Subspace-based linear system identification methods have been proven efficient for the identification of the eigenstructure of a linear multivariable system in many applications. Our main motivation in this paper is output-only structural identification in vibration mechanics. This problem consists in identifying the modal parameters (natural frequencies, damping ratios and mode shapes) of a structure subject to ambient unmeasured vibrations, by using accelerometer measurements or strain gauges. This is output-only system identification, as the excitation input is unknown and not measured. Examples are, amongst others, offshore structures subject to swell, bridges subject to wind and traffic, etc. We wish to analyze how the data-driven Stochastic Subspace Identification (SSI) with the Unweighted Principal Component algorithm [7] can be adapted when several successive data sets are recorded, with sensors at different locations in the structure. For doing this, some of the sensors, called the reference sensors, are kept fixed, while the others are moved. Like this, we mimic a situation in which lots of sensors are available, while in fact only a few are at hand. However, there is one unpleasant feature of structural identification of structures subject to ambient excitation, namely that excitation is typically turbulent in nature and nonstationary. For example, fluid/structure interaction in offshore structures results in shock effects causing nonstationary excitation, and the same holds for wind and traffic on bridges. Like this, the excitation factor can change from setup to setup. The relevance of merging successive records, and its implementation in the case of nonstationary excitation, are the subject of this paper. We describe a new merging algorithm for data-driven SSI and test it on vibration data of the Heritage Court Tower in Vancouver, Canada. 2 2.1
Reference-based Data-driven Stochastic Subspace Identification (SSI) Single setup
We consider a linear multi-variable output-only system described by a discrete-time state space model ⎧ ⎪ = F Xk + Vk+1 ⎨ Xk+1 (ref) Yk = H (ref) Xk ⎪ (mov) ⎩ Y = H (mov) Xk k
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_42, © The Society for Experimental Mechanics, Inc. 2011
(1)
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with • Xk the state vector at time instant k, (ref)
• Yk
the observed output vector of the reference sensors,
(mov)
• Yk • H
(ref)
the observed output vector of all the sensors minus the reference sensors (the remaining sensors),
the observation matrix with respect to the reference sensors,
• H (mov) the observation matrix with respect to the remaining sensors, • F the state transition matrix, • Vk the unmeasured stationary Gaussian white noise. Let furthermore
(ref) Yk • Yk = all the observed output at time instant k, (mov) Yk (ref) H • H= the full observation matrix, H (mov) • N the number of measurements (k = 1, . . . , N ), • r the total number of sensors and r (ref) the number of reference sensors. The classical reference-based data-driven subspace identification of the eigenstructure (λ, φλ ) of the system (1) consists of the following steps for the Unweighted Principal Component algorithm [7]: We choose p and q as variables with p + 1 ≥ q that indicate the quality of the estimations (a bigger p leads to better estimates) and the maximal system order (≤ qr (ref) ). Normally, we choose p = q − 1, but in the case of measurement noise p = q − 1 + l should be chosen, where l is the order of the noise. We build the data matrices ⎛ ⎞ ⎛ ⎞ .. (ref) (ref) .. (ref) Y Yq+1 . YN −p−1 Yq+2 . YN −p ⎟ ⎜ Yq+1 ⎜ q ⎟ ⎜ ⎟ ⎜ (ref) ⎟ .. (ref) .. (ref) ⎜ ⎟ ⎜ ⎟ Y Y . Y Y Y . Y def ⎜ q q+2 q+3 N−p+1 ⎟ + − def ⎜ q−1 N −p−2 ⎟ Yp+1 = ⎜ (2) ⎟ , and Yq = ⎜ ⎟ . . . . . . . . .. .. .. .. .. .. .. .. ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ .. (ref) (ref) .. (ref) Y Y . Y Y Y . Y q+p+1
q+p+2
N
1
1
and the weighted Hankel matrix
+ Hp+1,q = Yp+1 Yq−
T
2
N −p−q
T −1 − Yq− Yq− Yq .
With the factorization Hp+1,q = Op+1 Xq into matrix of observability and Kalman filter state sequence with ⎛ ⎞ H ⎜ HF ⎟ ⎜ ⎟ def ⎜HF 2 ⎟ Op+1 = ⎜ ⎟ ⎜ .. ⎟ ⎝ . ⎠ HF p
(3)
(4)
we can retrieve the matrices H as the first block row of Op+1 and F from the least squares solution of ⎛ ⎞ ⎛ ⎞ H HF ⎜ HF ⎟ ⎜HF 2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ .. ⎟ F = ⎜ .. ⎟ . ⎝ . ⎠ ⎝ . ⎠ HF p−1 HF p 1 As H p+1,q is usually a very big matrix and difficult to handle, we continue the calculation in practice with the R part from an RQ-decomposition of the data matrices, see [7] for details. This will lead to the same results as only the left part of the decomposition of Hp+1,q is needed.
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Finally we obtain the eigenstructure (λ, φλ ) of the system (1) from det(F − λ I) = 0, F ϕλ = λ ϕλ , φλ = Hϕλ . + In the following we will skip the subscripts of the matrices Hp+1,q , Yp+1 , Yq− and Op+1 .
2.2
Multiple setups
Instead of a single record for the output (Yk ) of the system (1), Ns records
(1,ref) (2,ref) (N ,ref) Yk Yk Yk s ... (1,mov) (2,mov) (N ,mov) Yk Yk Yk s ! ! ! Record 1 Record 2 Record Ns
(5)
(j,ref)
are now available collected successively. Each record j contains data Yk from a fixed reference sensor pool, (j,mov) and data Yk from a moving sensor pool. To each record j = 1, . . . , Ns corresponds a state-space realization in the form ⎧ (j) (j) (j) ⎪ = F Xk + Vk+1 ⎨ Xk+1 (j,ref) (j) (6) Yk = H (ref) Xk (reference pool) ⎪ ⎩ (j,mov) (j) o (j,mov) Yk = H Xk (sensor pool n j) with a single state transition matrix F . Note that the unmeasured excitation V (j) can be different for each setup j as the environmental conditions can slightly change between the measurements. However, during each setup j the noise V (j) is assumed to be stationary. Note also that the observation matrix H (ref) is independent of the specific measurement setup if the reference sensors are the same throughout all measurements j = 1, . . . , Ns . For each setup j we obtain a “local” weighted Hankel matrix −1 + − T − − T − H(j) = Y(j) Y(j) Y(j) Y(j) Y(j) (7) + − according to equations (2)-(3), where Y(j) is filled with data from all the sensors and Y(j) with data from the reference sensors of this setup (see Equation (2)). The question is now how to adapt the subspace identification from Section 2.1 to
• merge the data from the multiple setups j = 1, . . . , Ns to obtain global modal parameters (natural frequencies, damping ratios, mode shapes), and to • normalize or re-scale the data from the multiple setups as the background excitation may differ from setup to setup. In the following section we present two approaches for this problem: the common practice approach PoSER that processes all the setups separately and merges them at the end, and the new approach PreGER, that processes all the setups together. 3 3.1
Merging strategies PoSER approach with UPC
When having multiple measurement setups that share some reference sensors, it is common practice to perform the subspace identification algorithm of Section 2.1 on each setup separately in order to obtain the (local) modal parameters. To obtain the natural frequencies and damping ratios of the whole structure, the appropriate values of all setups are averaged. In order to merge the obtained (partial) mode shapes, we have to re-scale them as they were calculated on excitation factors that were possibly different from setup to setup. This is done in a least-square sense on the reference sensor part of each partial mode shape. This approach is also called PoSER (Post Separate Estimation Re-scaling), see also e.g. [6]. Especially when the number of setups is large, this approach can be tiresome as many stabilization diagrams have to be analyzed. Some modes may be less excited in some of the setups, and it might be difficult to distinguish closely spaced modes in the diagrams.
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Figure 1: Merging partial mode shape estimates φ(j) , j = 1, ..., Ns into a global mode shape estimate φ(all) in the PoSER approach.
3.2
PreGER approach with UPC
Another merging approach that was described for covariance-driven SSI in [4, 5, 2] makes use of a factorization of the Hankel matrix of each setup and normalizes them with a common right factor to introduce the same excitation factor to all the setups. We adapt this idea to the data-driven SSI with the UPC algorithm. We also call this method PreGER (Pre Global Estimation Re-scaling). For each setup j = 1, . . . , Ns we build the weighted Hankel matrix (7) that has the factorization property H(j) = (j) (j) O X . In order to merge the data we first take the different excitation factors of each setup into account, which are present in the Kalman filter state sequence X (j) since the matrix of observability is only dependent of the observation matrix H (j) and state matrix F that are not affected. In the first step, all the Hankel matrices H(j) are ∗ re-scaled with a common Kalman filter state sequence X (j ) of one fixed setup j ∗ , then the resulting matrices are merged and a global modal parameter estimation is finally done on the merged matrix.
Figure 2: Merging Hankel matrices of each setup to obtain a global Hankel matrix and global mode shape estimate φ(all) in the PreGER approach.
In detail, we separate the weighted Hankel matrices H(j) into matrices H(j,ref) and H(j,mov) by taking the appro+ + priate rows of H(j) that correspond to the reference resp. moving sensor data from Y(j,ref) resp. Y(j,mov) , see also Equation (7). As the weighted Hankel matrices fulfill the factorization property H(j) = O(j) X (j) we now have H(j,ref) = O(ref) X (j) ,
H(j,mov) = O(j,mov) X (j) ,
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with the matrices of observability ⎛ ⎞ ⎛ ⎞ H (j) H (ref) ⎜ H (j) F ⎟ ⎜ H (ref) F ⎟ ⎜ (j) 2 ⎟ ⎜ (ref) 2 ⎟ ⎜ ⎟ ⎜ F ⎟ O(j) = ⎜H F ⎟ , O (ref) = ⎜H ⎟, ⎜ . ⎟ ⎜ ⎟ . .. ⎝ .. ⎠ ⎝ ⎠ H (j) F p
⎛
O(j,mov)
H (ref) F p
⎞ H (j,mov) ⎜ H (j,mov) F ⎟ ⎜ (j,mov) 2 ⎟ ⎜ F ⎟ = ⎜H ⎟. ⎜ ⎟ .. ⎝ ⎠ . H (j,mov) F p
Note also, that O(j) consists of the rows of O(j,ref) and O(j,mov) . For the normalization we need the matrices X (j) in the same state basis. We juxtapose the matrices H(j,ref) , j = 1, . . . , Ns to H(all,ref) = H(1,ref) H(2,ref) . . . H(Ns ,ref) and decompose this matrix to
H(all,ref) = O(ref) X (1)
X (2)
...
X (Ns ) ,
from where we obtain the matrices X (j) . We choose one setup j ∗ ∈ {1, . . . , Ns } and re-scale the matrices H(j,mov) to ¯ (j,mov) = H(j,mov) X (j) T (X (j) X (j) T )−1 X (j ∗ ) , H ¯ (j,mov) = O (j,mov) X (j ∗ ) holds. In the last step we interleave the block rows of the matrices H ¯ (j,mov) , so that H (j ∗ ,ref) (all) ¯ j = 1, . . . , Ns , and the matrix H , to obtain the merged matrix H with the factorization property ⎛ ⎞ ⎛ ⎞ H (all) H (ref) ⎜ H (all) F ⎟ ⎜ H (1,mov) ⎟ ⎜ (all) 2 ⎟ ⎜ (2,mov) ⎟ ∗ ⎜ ⎟ ⎟ ¯ (all) = O(all) X (j ) with O(all) = ⎜H F ⎟ and H (all) = ⎜ H ⎜H ⎟ ⎜ ⎟ ⎜ ⎟ .. .. ⎝ ⎠ ⎝ ⎠ . . H (all) F p
H (Ns ,mov)
and perform the subspace system identification on it to obtain the global modal parameters. 4 4.1
Modal analysis of the Heritage Court Tower Description of the Heritage Court Tower and vibration measurements
The building considered in this study is the Heritage Court Tower (HCT) in downtown Vancouver, British Columbia in Canada. It is a relatively regular 15-story reinforced concrete shear core building. In plan, the building is essentially rectangular in shape with only small projections and setbacks. Typical floor dimensions of the upper floors are about 25 m by 31 m, while the dimensions of the lower three levels are about 36 m by 30 m. The footprint of the building below ground level is about 42 m by 36 m. Typical story heights are 2.70 m, while the first story height is 4.70 m. The elevator and stairs are concentrated at the center core of the building and form the main lateral resisting elements against potential wind and seismic lateral and torsional forces. The tower structure sits on top of four levels of reinforced concrete underground parking. The parking structure extends approximately 14 meters beyond the tower in the south direction forming an L-shaped podium. The parking structure and first floors of the tower are basically flush on the remaining three sides. The building tower is stocky in elevation having a height to width aspect ratio of approximately 1.7 in the east-west direction and 1.3 in the north-south direction. Because the building sits to the north side of the underground parking structure, coupling of the torsional and lateral modes of vibration was expected primarily in the EW direction. As reported in [8], a series of ambient vibration tests was conducted on April 28, 1998 by researchers from the University of British Columbia [3]. It was of practical interest to test this building because of its shear core, which concentrates most of lateral and torsional resisting elements at the center core of the building. Additional structural walls are located close to the perimeter of the building but are arranged in such a way that they offer no additional torsional restraint. Shear core buildings may exhibit increased torsional response when subjected to strong earthquake motion depending on the uncoupled lateral to torsional frequency ratio and of the amount of static eccentricity in the building plan [1]. The dynamic characteristics of interest for this study were the first few lateral and torsional natural frequencies and the corresponding mode shapes. The degree of torsional coupling between the modes was also investigated.
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The vibration measurements were conducted using an eight-channel system (with force-balanced accelerometers) and were recorded in four different measurement setups. The accelerometers were typically located in the northwest and northeast corners of the building on every other floor starting from the roof down to the ground floor. Details of the field testing of this structure are given in [3].
Figure 3: HCT Building and setup close up
4.2
Modal analysis with the PoSER and PreGER approaches
The modal analysis for both the PoSER and PreGER approach was tuned with the same parameters. The maximal considered model order was 80 and the number of samples was 6560 and hence relatively low, amounting to 328s at a sampling rate of 20Hz. The parameters for the modal extraction from the stabilization diagrams are shown in Figure 4.
Figure 4: Tuning of the SSI approach
In the PoSER approach all the four setups are processed separately, as described in Section 3.1. In the PreGER approach all the four setups are processed at once, as described in Section 3.2. The resulting stabilization diagrams are shown in Figures 5 and 6.
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Figure 5: PoSER stabilization diagram.
Figure 6: PreGER stabilization diagram.
In the frequency range of interest [0 – 6 Hz], 6 modes could be identified. Their natural frequencies and damping ratios are displayed in Tables 1 and 2. A comparison to the natural frequencies obtained by [8] together with the characteristics of the corresponding mode shapes is given in Table 3. Mode 1 2 3 4 5 6
Frequency [Hz] 1.228 1.286 1.453 3.859 4.260 5.350
Damping Ratio 2.035 1.898 1.348 1.260 1.497 1.840
Table 1: Identified modes with the PoSER approach.
Mode 1 2 3 4 5 6
Frequency [Hz] 1.229 1.295 1.449 3.855 4.258 5.369
Damping Ratio 2.914 2.466 1.516 1.491 2.242 2.928
Table 2: Identified modes with the PreGER approach.
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Table 3: EMA modes and mode shape descriptions [8].
For all modes, the damping ratios of the PreGER estimates are somewhat higher than the PoSER estimates. This might be due to the fact, that the natural frequencies in each measurement setup are slightly different. Then, the resulting frequency for each mode obtained by the PreGER approach is associated to a higher damping ratio, consequence from the merging of overlapping frequencies. The mode shapes obtained by the PoSER and PreGER approaches are shown in Figure 8 and a MAC comparison between them is shown in Figure 7. The MAC values are very close to 1, indicating very similar mode shapes.
Figure 7: MAC between mode shapes estimated by PoSER and PreGER approaches.
5
Summary of results
This paper focuses on obtaining the full mode shapes from a structure, under the assumption that sensor measurements were collected in different sessions. Two approaches were considered, both based on the data-driven SSI framework. The first approach, PoSER, merges the mode shapes obtained on the different setups after the SSI of each setup, while the second approach, PreGER, merges the correlation of the data before performing the SSI on the full set of data. All modes were recovered, damping estimates were consistent, albeit a bit higher for the PreGER approach, which is expected. As for the mode shapes, good MAC coherency was obtained between the two methods. The PreGER approach shows good prospect because it does not need any post processing of the estimation results in order to get the full mode shapes of the structure and no threshold based matching of modes between setups is needed.
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mode 1
mode 2
mode 3
mode 4
mode 5
mode 6
Figure 8: First 6 mode shapes obtained with the PoSER and the PreGER approach.
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References [1] J. de la Llera and A. Chopra. Accidental and natural torsion in earthquake response and design of buildings. Earthquake Engineering Research Center, University of California, 1994. ¨ ˜ [2] M. Dohler, E. Reynders, F. Magalhaes, L. Mevel, G. D. Roeck, and A. Cunha. Pre- and post-identification merging for multi-setup OMA with covariance-driven SSI. In Proceedings of IMAC 28, the International Modal Analysis Conference, Jacksonville, FL, 2010. [3] C. Dyck and C. Ventura. Ambient Vibration Measurements of Heritage Court Tower. EQ LAB, University of British Columbia, Earthquake Engineering Research, 1998. [4] L. Mevel, M. Basseville, A. Benveniste, and M. Goursat. Merging sensor data from multiple measurement setups for nonstationary subspace-based modal analysis. Journal of Sound and Vibration, 249(4):719–741, 2002. [5] L. Mevel, A. Benveniste, M. Basseville, and M. Goursat. Blind subspace-based eigenstructure identification under nonstationary excitation using moving sensors. IEEE Transactions on Signal Processing, SP-50(1):41– 48, 2002. ˜ [6] E. Reynders, F. Magalhaes, G. D. Roeck, and A. Cunha. Merging strategies for multi-setup operational modal analysis: application to the Luiz I steel arch bridge. In Proceedings of IMAC 27, the International Modal Analysis Conference, Orlando, FL, 2009. [7] P. Van Overschee and B. De Moor. Subspace Identification for Linear Systems: Theory, Implementation, Applications. Kluwer, 1996. [8] C. Ventura, R. Brincker, E. Dascotte, and P. Andersen. FEM updating of the heritage court building structure. In Proceedings of IMAC 19, the International Modal Analysis Conference, volume 1, pages 324–330, 2001.
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Proceedings of the IMAC-XXVIII
February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc. 2QWKH2SHUDWLRQDO0RGDO$QDO\VLVRI6ROLG5RFNHW0RWRUV
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Sebastiaan Fransen1, Daniel Rixen2, Torben Henriksen1, Michel Bonnet3 European Space Agency, ESTEC, P.O. Box 299, 2200 AG Noordwijk, The Netherlands, EU 2 Delft University of Technology, 3mE, Mekelweg 2, 2628 CD Delft, The Netherlands, EU 3 European Space Agency, ESRIN, P.O. Box 64, 00044 Frascati, Italy, EU
$%675$&7 ESA’s new small launcher – VEGA – has been designed as a single body launcher with three solid rocket motor stages and an additional liquid propulsion upper module used for attitude and orbit control, and satellite release. In order to verify the performance of the solid rocket motors, all of the motors are tested in static firing tests on a test bench. In the frame of the correlation of the solid rocket motor mathematical models, an operational modal analysis tool was developed that is based on the Least Squares Complex Exponential method. The tool allows the computation of experimental poles and modeshapes from the accelerometer data recorded during a firing test. Convergence can be verified by means of the classical stabilization diagram and by the reconstruction of the correlation functions on the basis of the stable poles. ,QWURGXFWLRQ In order to verify the thrust performance of the solid rocket motors of ESA’s new small launcher VEGA, static firing tests are conducted. In such test the motor is suspended on a test bench and ignited. Besides the measurement of the thrust by a loadcell, also other performance parameters are recorded such as temperatures, pressures and vibratory accelerations of the motor case. In figure 1 the firing test of VEGA’s third stage is depicted, for which the test bench in Sardinia (Italy) is used.
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T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_43, © The Society for Experimental Mechanics, Inc. 2011
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As the finite element models of the motors shall be dynamically correlated, before using them in a launchersatellite coupled dynamic analysis, it is essential to extract the modal information from the test measurements as accurately as possible. For this purpose an operational modal analysis tool was developed in the MATLAB environment. The implemented methodology is based on the Least Squares Complex Exponential (LSCE) method [1] and enables the generation of stabilization diagrams and the recovery of wireframe modes. Besides these standard features, a solution was implemented to perform operational modal analysis in the presence of harmonic excitations [2]. Harmonic excitations in combustions chambers of solid rocket motors are a well-known phenomenon that could jeopardize the convergence of the poles in the vicinity of those harmonic frequencies. Finally, the tool was completed with a feature that enables the detection of the most dominant modes and the reconstruction of the original correlation functions on the basis of those modes. In order to demonstrate the capabilities of the tool, the operational modal analysis of one of the solid rocket motor firing tests is discussed. /6&(0HWKRG In the LSCE method [1], the correlation functions between the various accelerometer outputs are used as an input for the computation of the modal characteristics of the structure. The correlation function Rij (t ) between the response signals i and j at an instant of time t is equal to the response of the structure at at j , see figure 2.
i due to an impulse
white noise input correlation function or impulse response Rij
i i
i i
j
i i
)LJXUH &RUUHODWLRQIXQFWLRQEHWZHHQVLJQDOVLDQGM LPSXOVHUHVSRQVH Assuming the damping to be small, the correlation function is given by a summation of N decaying sinusoids: N
Iri Arj
¦mZ
Rij (t )
r 1
r
d r
e ] rZrt sin(Zrd t T r )
(1)
Each modal contribution is characterized by the multiplication of several modal parameters indexed by the modal subscript r: Modal parameter I ri is the modeshape displacement at DOF i , Arj is the input intensity at DOF j ,
9r
is the modal damping ratio,
and
Zrd
Z
d r
Zr
is the natural frequency,
Tr
is the phase angle, m r is the generalized mass,
is the damped eigenfrequency given by:
Zr 1 9 r2
(2)
It is evident that the modal parameters can be computed from eq.(1), once experimental correlation functions Rij (t ) between various accelerometer outputs on the structure are available. In order to explain how the modal parameters are exactly solved, we will first write eq.(1) in terms of complex modes:
BookID 214574_ChapID 43_Proof# 1 - 23/04/2011
455 N
N
r 1
r 1
¦ e srk't Crij ¦ e srk't Crij*
Rij (k't )
*
(3)
where the complex eigenvalue s r is given by,
sr
Z r 9 r r iZ r 1 9 r2
(4)
Expressing eq.(3) in terms of complex conjugate forms we obtain: 2N
¦e
Rij (k't )
s r k't
C rij
(5)
r 1
A polynomial of the order 2 N ( k 1 2 N ) – known as Prony’s equation – exists of which e (number of timesteps equals the number of roots):
E 0 E1e s 't E 2 e s 2 't E 2 N 1e s ( 2 N 1) 't e s ( 2 N ) 't r
r
r
s r 't
are roots
0
r
(6)
or, 2
E 0 E1e s 't E 2 e s 't E 2 N 1e s 't r
r
r
2 N 1
e sr 't
2N
0
(7)
or,
E 0 E1Vr1 E 2Vr2 E 2 N 1Vr2 N 1 Vr2 N
0
(8)
Before we can solve the roots Vr we first need to determine the coefficients
Ek
(note that
purpose we multiply the correlation functions at time instant k with the coefficient values for k
Ek
E 2 N 1 ).
For this
and superimpose these
0 2 N (a summation over the time co-ordinate):
2N
¦ E k Rij (k't ) k 0
§ 2 N s r k't · ¨ E k ¦ e C rij ¸ ¦ k 0© r 1 ¹ 2N
§ 2N k · ¨ E k ¦Vr C rij ¸ ¦ k 0© r 1 ¹ 2N
2N § k· C ¨ rij ¦ E kVr ¸ ¦ r 1© k 0 ¹ 2N
0
(9)
Note that at least 2 N equations shall be written to solve E 0 E 2 N 1 . By the application of a time shift strategy, i.e. by starting at successive time samples, a linear system of equations can be build to solve the coefficients
E 0 Rijn E1 Rijn1 E 2 N 1 Rijn2 N 1 where Rij ( k't )
>R@ij ^E `
Rijn2 N
Ek :
(10)
Rijk and n 1 L . The above equations can also be written as:
^R'`ij
> @
(11)
Where R is known as the Hankel matrix. Assuming we have stations, the number of successive time samples
p response stations of which q are reference
L required to solve the coefficients E k follows from:
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q (q 1) · § L¨ qp ¸ t 2N 2 ¹ © Having only one reference station, i.e. q
Lt
(12)
1 , the required number of successive time samples L is given by:
2N p
(13)
In order to stay within the time window T with sample time dT , the number of successive time samples L shall be limited to:
Ld
T 2N dT
(14)
)LJXUH 6WDELOL]DWLRQ'LDJUDP $QDO\VLV6HWWLQJV Time span of correlation functions Sample time Max Polynomial Order Prony’s Equation Number of time steps in timeshift window Number of sensors Number of reference sensors 7DEOH $QDO\VLVVHWWLQJV
9DOXH
T =2s dT =0.0005 s N =130 L =3400 (1.7 s) p =18 q =1
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The system of equations takes the following form for
p response stations of which q are reference stations:
^R '`11 ½ ª >R @11 º °^ ` ° » «>R @ « 12 »^E ` °® R ' 12 °¾ « » ° ° « » °¯^R '`qp °¿ «¬>R @qp »¼ Having solved the coefficients
(15)
E
from the system of equations given by eq.(14) in a least square sense using
pseudo-inverse techniques, we can now solve the roots Vr from the polynomial (Prony’s equation) given by eq.(8). By variation of the polynomial order of Prony’s equation, a so-called stabilization diagram can be constructed, which helps to identify the stable modes. As an example case throughout this paper we will discuss the operational modal analysis conducted in the frame of the qualification firing test of the 1st stage of the VEGA launcher. This test was conducted at Kourou Spaceport in French Guiana in December 2007. The settings for this operational modal analysis are listed in table 1. The resulting stabilization diagram is shown in figure 3. +DUPRQLFV In the LSCE method the excitation is assumed to be random white noise. In case the excitation also includes harmonics, then those harmonics will be identified as modes with negligible damping. As those harmonic modes potentially could disturb the identification of the structural modes, especially when the harmonic and structural mode are close in frequency, one could include them as predefined poles in the solution sequence. This will improve the quality of the true poles of the structural modes [2]. From eq.(4) we can see that for pure harmonics, i.e. zero damping, the complex eigenvalue equals s r
riZ r .
r iZ r 't
As such we have two extra roots of Prony’s polynomial, namely Vr e e cos(Z r 't ) r i sin(Z r 't ) , which are roots of eq.(8). This means we can write two extra time signals (correlation functions) in accordance with eq.(10): s r 't
E0 ½ ° ° ª0 sin(Z r 't ) sin Z r (2 N 1)'t º ° E1 ° «1 cos(Z 't ) cos Z (2 N 1)'t » ® ¾ ¬ r r ¼° ° °¯E 2 N 1 °¿
sin(Z r 2 N't ) ½ ® ¾ ¯cos(Z r 2 N't )¿
(16)
These two independent linear equations for the coefficients E must be satisfied in order to represent the harmonics in the time signals. Let us assume that m harmonic frequencies in the frequency range of interest exist. Adding the linear equations (16) to the linear system defined by eq.(15), and assuming only one reference station (i.e. q 1 ), we get:
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458
ª R10 « « « R pL 1 « « 0 « 1 « « « 0 « «¬ 1
R12 m 1
A
R
L 2m 2 p
sin Z1 (2m 1)'t cos Z1 (2m 1) 't B
R
R12 m
C
L 2 m 1 p
sin Z1 (2m)'t cos Z1 (2m)'t
D
sin Zm (2m 1)'t sin Zm (2m)'t cos Zm (2m 1)'t cos Zm (2m)'t
º » » E0 ½ L2N 2 » ° b1 ° Rp ° »° sin Z1 (2 N 1)'t » °° E 2 m 1 °° ® ¾ cos Z1 (2 N 1)'t » ° E 2 m ° » » ° b2 ° ° ° sin Zm (2 N 1)'t » ¯°E 2 N 1 ¿° » cos Zm (2 N 1)'t »¼ ½ R12 N ° ° E ° ° ° ° R pL 2 N 1 ° ° ° sin Z1 (2 N't ) ° ® ¾ ° cos Z1 (2 N't ) ° ° ° F ° ° ° sin Zm (2 N't ) ° °cos Z (2 N't ) ° m ¯ ¿ R12 N 1
(17) In symbolic form we can write :
>A@ ^b ` >C @
1 ( Lpx 2 m ) ( 2 mx1)
^b `
^E`
(18)
^b `
^E`
(19)
2 ( Lpx 2 N 2 m ) ( 2 N 2 mx1)
Lpx1
and
>B@ ^b ` >D@
1 ( 2 mx 2 m ) ( 2 mx1)
2 ( 2 mx 2 N 2 m ) ( 2 N 2 mx1)
2 mx1
From eq.(18) we can solve ^b1 ` :
^b1 ` >B@1 >^F ` >D@^b2 `@
(20)
Substituting in eq.(17) yields :
>>C @ >A@>B@
1
>D@@^b2 ` ^E` >A@>B@1 ^F `
(21)
From eq.(21) ^b2 ` can be found as a least square solution. The coefficients ^b1 ` can then be solved from
eq.(20). Together ^b1 ` and ^b2 ` provide the coefficients of Prony’s polynomial. The roots of the polynomial, solved again from eq.(8), will include the harmonic frequencies since the procedure presented here enforces them exactly.
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)LJXUH 6WDELOL]DWLRQ'LDJUDPZLWKSUHGHILQHGSROHVDWKDUPRQLFIUHTXHQFLHV In figure 4 the stabilization diagram is shown when predefined harmonics are included as poles of Prony’s polynomial. Those predefined poles are located at the 50, 100, 150 and 200 Hz and coincide with the known acoustic harmonic frequencies of the combustion chamber. In the stabilization diagram, the harmonics are indicated with a + sign. If we compare figures 3 and 4, we can see that the stabilization of the low frequency modes below the first harmonic at 50Hz is significantly improved when using predefined harmonics. The same applies to the modes found around the second harmonic at 100Hz. 5HFRYHU\RI0RGH6KDSHV Once the roots Vr are known from Prony’s polynomial, we can find the residues Crij are the complex mode shapes times the modal participation factors
Rij (k't )
Rijk
2N
¦ e srk't C rij r 1
¦ e C ¦ V C 2N
s r 't
2N
k
rij
r 1
k
r
Iri Arj
from eq.(5), which
Arj :
rij
(22)
r 1
where,
i 1 p j 1 q k 0 L 1 Assuming that the number of reference stations q
(23)
1 , eq.(22) can be written as follows in matrix form:
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460
1 ª 1 «V V2 « 1 « V12 V22 « « «¬V1L1 V2L1
º ª C111 »« C » « 211 2 V2 M » « C311 »« »« V2LM1 »¼ «¬C2 M 11
1 1 V2 M 1 V2 M V22M 1 V2LM11
( Lx 2 M )
C1 p1 º » C2 p1 » C3 p1 » » » C2 Mp1 »¼
( 2 Mxp )
ª R110 « 1 « R11 « R112 « « « R L1 ¬ 11
R p01 º » R p01 » R p01 » » » R p01 »¼
(24)
Lxp
Or,
1 ª 1 « V V2 « 1 2 « V1 V22 « « L 1 «¬V1 V2L 1
1
V2 M 1 V22M 1 L 1 V2 M 1
( Lx 2 M )
º ª I1T A1 º » »« T » « I2 A2 » » « I3T A3 » » »« »« » V2LM1 »¼ «¬I2TM A2 M »¼ 1 V2 M V22M
( 2 Mxp )
ª R 0T º « 1T » « R » « R 2T » « » « » « L 1T » ¬R ¼ Lxp
)LJXUH :LUHIUDPHPRGHRIVWVWDJHDW+]±)RUZDUGGRPHPRGH
)LJXUH $FFHOHURPHWHUSRVLWLRQV UHG DQGZLUHIUDPHPRGHOZLWKUHVSHFWWR)(0
(25)
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461
The modeshapes
ImT Am
in eq.(25) can be solved in a least squares sense. Eq.(25) presumes M poles were found
from Prony’s polynomial. In practice only those poles will be used that are in the frequency range of interest, i.e. M<
t
¦Vrl Irp Arp r 1
*
l *dT for a specific sensor p equals the sum of the modal gains: 2M
¦V
*
rp
R lp
(26)
r 1
A result of such computation is shown in figure 7. If we take the max modal gain of each mode over all sensors, we find the bar chart shown in figure 8. Now we can select those modes that contribute more than a certain percentage to the maximum value of any time signal (see again figure 8). On the basis of the selected dominant modes we can reconstruct the time signals (correlation functions) and compare them against the original time signals given by the right-hand side of eq.(25). In order to ease the comparison it is more convenient to convert the signals to the frequency domain by FFT. An example for one of the time signals is shown in figure 9. We can see that at the dominant modes, the match with the original correlation function is as good as the recovered time signal based on all poles.
)LJXUH 1RUPDOL]HGPRGDOJDLQVIRUHDFKVHQVRUVLJQDODWPD[UHVSRQVH
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462
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)LJXUH 0D[LPDOQRUPDOL]HGPRGDOJDLQVSHUPRGHRYHUDOOVHQVRUV
)LJXUH &RUUHODWLRQIXQFWLRQVHQVRU±RULJLQDOYHUVXVUHFRYHUHG
BookID 214574_ChapID 43_Proof# 1 - 23/04/2011
463
&RQFOXVLRQV The Least Squares Complex Exponential method has been used in the frame of modal characterization of solid rocket motors. In order to avoid the detection of strong harmonics that are associated to the forcing function rather than the structure of the solid rocket motor, the method of predefined poles was used. It was shown that stable modes in the frequency range of interest can be identified easily from a stabilization diagram. In this diagram the predefined poles can be highlighted as well. In order to find the dominant modes amongst the identified stable modes, the modal gains can be used which are computed anyway as part of the modeshape recovery procedure. On the basis of the dominant modes, one should be able to reconstruct the correlation functions without significant loss of accuracy. 5HIHUHQFHV [1] Brown, D.L. et al., Parameter Estimation Techniques for Modal Analysis. SAE Technical Paper Series, (790221), 1979. [2] Mohanty, P. and Rixen, D.J., Operational Modal Analysis in the Presence of Harmonic Excitations, Journal of Sound and Vibration, 270(Issues1-2):93-109, Feb. 2004. [3] Fransen S. et al., Damping Methodology for Condensed Solid Rocket Motor Structural Models, IMAC 2010, Jacksonville, USA, 2010
BookID 214574_ChapID FM_Proof# 1 - 23/04/2011
BookID 214574_ChapID 44_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Frequency-domain modal analysis in the OMAX framework
Tim De Troyer, Mark Runacres Department of Industrial Sciences & Technology, Erasmushogeschool Brussel, Nijverheidskaai 170, B-1070 Brussels, Belgium e-mail:
[email protected]
Patrick Guillaume Department of Mechanical Engineering, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium
ABSTRACT Ambient noise is generally seen as an unwanted excitation that disturbs the estimation of vibration parameters. Averaging techniques are then used to decrease as much as possible the influence of the noise. However, this noise also excites the mechanical structure and thus increases the vibration response level. Moreover, it is possible that (broadband) noise excites vibration modes that are not well excited by the artificially applied forces. Those modes are missed by classical estimation methods. Recently, classical EMA and OMA were combined into the so-called OMAX framework. In this framework both the artificial force and the ambient excitation are considered useful in determining the modal parameters. In this paper it is shown that the classical frequency-domain modal parameter estimators (rational fraction polynomial based and state space based) can be used without changing them, if the correct non-parametric preprocessing is applied to calculate the frequency response function (FRF) and the power spectrum (PSD). Special attention is paid to the case of structure-exciter interaction, where a direct OMAX approach would result in erroneous results. Also the importance of scaling the FRF and PSD is discussed. The approach is demonstrated on a typical OMAX case: flight flutter test of an airplane wing.
NOMENCLATURE Hm (Ω f ) H(Ω f ) N(Ω f ) D(Ω f ) No (Ω f ) Ωf θ θD θNo ε(Ω f , θ, H) W(ω f ) J Γo Υ (.)T
FRF right matrix-fraction model measured FRF numerator matrix polynomial denominator matrix polynomial Numerator matrix polynomial for output o polynomial basis function coefficients matrix denominator coefficients matrix numerator coefficients matrix error equation weighting matrix in error equation jacobian matrix jacobian submatrix corresponding to numerator coefficients jacobian submatrix corresponding to denominator coefficients matrix transpose
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_44, © The Society for Experimental Mechanics, Inc. 2011
465
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466 M Ro So To I ΔH ΔθD ΔM ΔΥ Ξ Cov(.) δi j Re(.) Im(.) Dcom Vr λr Lr P .m γ2 Syy Sf f
1
reduced normal equations reduced normal equations submatrix reduced normal equations submatrix reduced normal equations submatrix identity matrix by definition perturbation on the FRF perturbation on the denominator coefficients perturbation on the reduced normal equations perturbation on the Υ jacobian submatrix covariance matrix submatrix covariance matrix kronecker delta (1 if i = j, 0 elsewhere) taking real part taking imaginary part companion matrix eigenvector of companion matrix eigenvalue of companion matrix modal participation vector weighting matrix for covariance matrix mth iteration multiple coherence function autopower spectrum of outputs autopower spectrum of outputs
INTRODUCTION
The combined experimental-operational estimation of vibration data emerged from typical engineering applications like the testing of bridges and aeroplanes. In the flight flutter testing [1] of aeroplanes e.g., the wings are excited both by an applied force and by the atmospheric turbulence. Another example is the modal analysis of bridges: the structure is excited by ambient forces (wind, trafic) and by applied forces [2] . In classical Experimental modal analysis [3, 4] , the ambient excitation is considered as disturbing noise that one should get rid off. In Operational modal analysis however [5−8] , one only uses the response of the system caused by the unknown ambient forces. Modal parameter estimators in the OMAX framework consider both the deterministic contribution caused by the applied forces and the stochastic contribution caused by the unmeasurable forces [9−11] . Both contributions contain useful information of the system. As in classical frequency-domain EMA and OMA, the input data can be (averaged) spectral functions (i.e. FRF and PSD) or the input and output (I/O) fourier coefficients. The approach described in this paper focuses on ABS functions as input data. The FRF and PSD models considered are frequency-domain rational fraction polynomial models. Both the single-reference common-denominator model and the polyreference matrix-fraction description can be used. The use of the frequency domain allows easy pre-filtering of the data and fast yet accurate algorithms to estimate the modal parameters. The FRF and PSD are constructed from the measured input and output signals using fast FFT and correlation techniques. If the artificial excitation is harmonic, the deterministic component can be calculated without leakage by averaging over an integer number of periods. If the excitation is random, or if only short data sequences are measured, other techniques are necessary to obtain the FRF and the PSD. The main drawback of these techniques is the possibility of leakage errors. The advantage is that it is is quite easy to determine the stochastic part of the PSD, as will be discussed in Section 2.2. An alternative approach is to feed the fourier transform of the measured input and output signals directly into the modal parameter estimator. The main drawback of this method is that it results in a non-linear optimization. This requires an iterative procedure, this algortihm is slower than the one-step FRF based estimators. Another major drawback is that it is not possible to compensate for the correlation between the applied and unmeasured inputs if the exciter interacts with the structure. It is not yet clear if the parametric estimator can be adapted to account for this correlation. The subspace identification methods can be formulated in a OMAX framework for the input/output based estimator, the so-called combined deterministic-stochastic algorithms. The main advantage of the state space model is that it is readily extended to MIMO measurements. The FRF and PSD can be combined in the same way as for the rational-fraction modelled estimators.
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"NQMJUVEF E#
"NQMJUVEF E#
¦
¦
¦
¦
¦ ¦
¦ ¦
'SFRVFODZ )[
'SFRVFODZ )[
Figure 1: Low (left) and high (right) noise level analytical (thick blue line) and (non-parametric) FRF estimate (thin red line), crosspower spectrum (green dots), applied force spectrum (black dotted line) and standard deviation of the unknown ambient force (black dashed line)
OMAX is really advantageous when some modes are not (well) excited by the applied force. This can be because it is difficult to inject sufficient energy at specific frequencies (e.g. bridges). It can also be that the force is applied in the node of a mode. In this case, an operational (broadband) input can reveal these modes. Figure 1 shows the FRFs obtained from shaker measurements compared with the analytical FRF for low and high noise. The shaker is applied in the node of the second, third and fourth mode. The FRFs of two different response locations are plotted. The angles (not shown) follow a similar behaviour. It is clear that the second, third and fourth mode are not (always) present in the FRF. The PSD clearly shows five peaks. In Section 2 the required pre-processing steps are adapted for the OMAX approach with exciter-structure interaction. The comparison of the different frequency-domain estimators is done on a theoretical basis in Section 3. Finally some conclusions are drawn.
2
NON-PARAMETRIC IDENTIFICATION
The FRF and PSD are calculated from the Fourier coefficients of the measured input and output signals. The system is assumed linear.
2.1
Deterministic part
Several classical approaches are used everyday to determine FRFs. Which one is best suited depends on the nature of the ap plied force. Starting from the fourier transform of the measured outputs X(ωk ) = √1N nN−1 x(n) exp(− j2πkn/N) and forces F(ωk ) =
BookID 214574_ChapID 44_Proof# 1 - 23/04/2011
468 √1 N
N−1 n
f (n) exp(− j2πkn/N), the crosspower spectra are obtained as S XF (ωk ) =
Nb 1 Xb (ωk )FbH (ωk ) Nb b=1
S FF (ωk ) =
Nb 1 Fb (ωk )FbH (ωk ) Nb b=1
Nb 1 S FX (ωk ) = Fb (ωk )XbH (ωk ) Nb b=1
S XX (ωk ) =
(1)
Nb 1 Xb (ωk )XbH (ωk ) Nb b=1
with Nb the number of (possibly overlapping) blocks. The time samples can also be weighted with a suitable window (Hanning, Hamming, . . . ) before fourier transforming. This approach is also known as Welch procedure [12] . The H1 and H2 estimates are then found as
H1 (ωk ) = S XF (ωk )S FF (ωk )−1 H2 (ωk ) = S XX (ωk )S FX (ωk )−1
(2)
If the force is periodic (e.g. a multisine), the use of the errors-in-variables estimator is proposed Hev [13] . This is a special version of the instrumental variables estimator that yields consistent estimates of the FRFs without requiring any a priori noise information or the need for additional instrumental variables.
⎛ ⎞⎛ ⎞−1 Np Np ⎜⎜⎜ 1 ⎟⎟⎟ ⎜⎜⎜ 1 ⎟⎟⎟ Hev (ωk ) = ⎜⎜⎜⎝ X p (ωk )⎟⎟⎟⎠ ⎜⎜⎜⎝ F p (ωk )⎟⎟⎟⎠ N p p=1 N p p=1
(3)
with N p the number of periods.
2.2
Stochastic part
The responses contain both the deterministic part and the stochastic part.
X(ωk ) = H(ωk )F(ωk ) + G(ωk )E(ωk )
(4)
with X and F the measured responses and forces respectively. H is the FRF of the system (to be estimated), G is the filter of the (unknown) operational excitation E . This excitation is assumed zero mean normally distributed (white noise) with unkown variance. In order to apply the classical OMA preprocessing algorithms (Welch periodogram and the correlogram method [14] ), the deterministic component H(ωk )F(ωk ) must be removed from the responses. If the force–and thus the response–is periodic, the mean value could be calculated over the period as for the Hev estimator. The mean of the response could then be subtracted from the response to obtain only the stochastic part.
⎛ ⎞ Np ⎜⎜⎜ 1 ⎟⎟⎟ Xs,p (ωk ) = X p (ωk ) − ⎜⎜⎜⎝ X p (ωk )⎟⎟⎟⎠ , N p p=1
p = 1 . . . Np
(5)
This approach is wrong in the case of structure-exciter interaction. The responses are then corrupted due to the mass-loading effect of the shaker. In order to remove the deterministic part, an estimate is needed of the FRF of the decoupled system. The responses can then be compensated for the deterministic part; only the response due to the operational forces on the decoupled system remains.
X s,p (ωk ) = X p (ωk ) − H(ωk )F p (ωk ),
p = 1 . . . Np
(6)
The mathematical derivation is shown in Appendix A. If the responses are not corrected, or if only the mean is subtracted, the mass-loading effect is disregared. The obtained spectra are characteristic of the whole system-shaker combination. The resonance frequencies of this whole system obviously differ from the
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469
25
Amplitude (dB)
0 −25 −50 −75 −100 −125
25
50
75 Frequency (Hz)
100
125
Figure 2: Crosspower spectrum, corrected with a non-parametric estimate of the FRF (green crosses) and corrected with the mean of the responses (blue circles); the black vertical lines illustrate the location of the resonance frequencies of the uncoupled system
frequencies of the decoupled system, so an OMAX combination of uncorrected or wrongly corrected spectra would decrease rather than increase the accuracy of the estimates. Figure 2 illustrates this. An important conclusion is that the correlogram method is not suited as the obtained spectra would be those of the system-shaker combination. The elimination of the deterministic part directly in the time-domain signal would require deconvolving the response with the impusle response function.
2.3
Positive power spectra
The auto- and cross-power spectra have a four quadrant symmetry: they contain both the stable and unstable poles. It is better to calculate the so-called positive power spectra (PSD+ ); these have the same structure as an FRF and they can be directly fed into the classical modal parameter estimators. The positive power spectra are obtained by inverse fourier transforming the power spectra, and then fourier transforming only the positive lags of the correlation function [10] . To avoid leakage the correlation function can be weighted by a rectangular or an exponential window.
2.4
Summary of the non-parametric pre-processing
Since the combined FRF/XP+ approach allows to formulate all current modal parameter estimators in the OMAX framework, we briefly summarize the proposed pre-processing procedure as follows:
1. Measure Nb periods of the force fb (n) and response yb (n) of the signal ( N points per period), 2. Fourier tranform the data: F p (ωk ) and Yb (ωk ), 3. Calculate the Hev estimate through cyclic averaging for periodic signals (or H1 or H2 for other signals), 4. Calculate the coherence function mγo2 (ωk ) and the variance Var(H)(ωk ) on the FRF,
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470 5. Determine the stochastic part of the response Ysto,b (ωk ) by Eq. (6), or measure Ysto,b (ωk ) directly by removing the applied force and considering the system in operational conditions (2N points), see e.g. the approach followed in Section 4, 6. Calculate the power spectra and positive power spectra S+YY (ωk ) from the stochastic part of the response, 7. Calculate the variance on the positive power spectra Var(S+YY )(ωk ), 8. Feed [Hev (ωk ), S+YY (ωk )] and [Var(H)(ωk ), Var(S+YY )(ωk )] into the parametric estimator.
3
PARAMETRIC IDENTIFICATION
A polynomial FRF or state-space (SS) model can be fitted through the measured data. Several frequency-domain modal parameter estimators exist that accept both FRF and PSD as input. As the PSD+ have the same modal structure as the FRF, they can be estimated without having to reformulate the estimator [10] . This is true for the state space, common-denominator and matrix fraction models that use the FRF as primary input data.
3.1
FRF and PSD+ driven
The general OMAX parametric frequency-domain model can be written as
ˆ k )F(ωk ) + G(ω ˆ k )E(ωk ) X(ωk ) = H(ω n(ωk ) c(ωk ) X(ωk ) = F(ωk ) + E(ωk ) d(ωk ) d(ωk )
(7)
with n(ωk ) the FRF model numerator, d(ωk ) the FRF model denominator and c(ωk ) the noise filter numerator. The denominator of the noise filter must be equal to the denominator of the FRF. This is the core of the OMAX approach: as both the deterministic (FRF: H ) as stochastic (PSD+ : G) models share the same denominator, they have the same poles (resonance frequencies and damping ratios). Thus, by including the PSD+ , the accuracy of the estimated poles will increase because more data is used to estimate them. Different FRF models exist that lead to different estimation algorithms.
3.1.1
COMMON-DENOMINATOR MODEL
A scalar matrix-fraction description, better known as a common-denominator model, models the FRF between output o(1, · · · , No ) and input i(1, · · · , Ni ) as
Noi (Ωk , θ) Hˆ oi (Ωk , θ) = d(Ωk , θ)
(8)
at frequency line k. Noi (Ωk , θ) is the numerator polynomial and d(Ωk , θ) the common-denominator polynomial, defined by
Noi (Ωk , θ) =
n
Noi, j Ωkj
j=0
d(Ωk , θ) =
n
(9) d j Ωkj
j=0
The coefficients Noi, j and d j are the parameters to be estimated. These coefficients are grouped together in one parameter vector θ. The linearized (weighted) equation error Eoi (ωk ) is obtained by replacing in Eq. (8) the model Hˆ oi (Ωk , θ) by the measured FRF Hoi (ωk ) and multiplying with the denominator polynomial d(Ωk , θ)
Eoi (ωk ) = Woi (ωk ) Noi (Ωk ) − Hoi (ωk ) d(Ωk )
(10)
Woi (ωk ) is a frequency-dependent weighting which can be used to improve the estimator. The estimates of the coefficients are then found by minimizing the least-squares cost function l= |Eoi (ωk )|2 (11) o,i
k
BookID 214574_ChapID 44_Proof# 1 - 23/04/2011
471 The main advantages of the common-denominator model is that it is flexible and that it allows a fast estimator: the least-squares complex frequency-domain estimator (LSCF) [16] . This estimator also yields very clear stabilization charts. The main drawback of this estimator is that it does not implicitely enforce a rank one constraint on the residues. A singular value decomposition (SVD) is then required to calculate the mode shapes and modal participation factors. This SVD reduces the accuracy of the FRF fit.
3.1.2
RIGHT MATRIX-FRACTION DESCRIPTION MODEL
The PolyMAX estimator [17, 18] is the polyreference counterpart of the LSCF estimator. This estimator uses a right-matrix fraction description (RMFD) model. The relationship between No outputs and Ni inputs can be modeled in the frequency domain, at frequency line Ωk
H(Ωk ) = N(Ωk )D−1 (Ωk )
(12)
with N(Ωk ) the No × Ni numerator matrix polynomial and D(Ωk ) the Ni × Ni denominator matrix polynomial, defined by
N(Ωk ) =
n
N j Ωkj
j=0
D(Ωk ) =
n
(13) D j Ωkj
j=0
The matrix coefficients N j and D j are the parameters to be estimated. These coefficients are grouped together in one parameter matrix θ. The least-squares cost function becomes
l=
o
with the 1 × Ni row vector Eo (ωk )
trace Eo (ωk )H Eo (ωk )
(14)
k
Eo (ωk ) = Wo (ωk ) No (Ωk ) − Ho (ωk ) D(Ωk )
(15)
The PolyMAX estimator is fast and yields very clear stabilization charts, just as the LSCF. Moreover, a rank one constraint is implicitely imposed on the residues. The major drawback of the PolyMAX is that it is not consistent. An important limitation for applications in the OMAX framework is that the number of reference responses must be equal to the number of inputs Ni of the FRF. This is due to the fact that the PSD+ must have the same dimensions as the FRF to fit into the estimator.
3.1.3
LEFT MATRIX-FRACTION DESCRIPTION MODEL
The left matrix-fraction description (LMFD) considers all responses simultaneously. The FRF model is now
H(Ωk ) = D−1 (Ωk )N(Ωk )
(16)
with N(Ωk ) the No × No numerator matrix polynomial and D(Ωk ) the No × Ni denominator matrix polynomial. The least-squares cost function is now
l=
i
with the 1 × No row vector Ei (ωk )
trace Ei (ωk )H Ei (ωk )
(17)
k
Ei (ωk ) = Wi (ωk ) Ni (Ωk ) − D(Ωk ) Hi (ωk )
(18)
The main drawback is that the size of the matrices is now proportional to the number of outputs No . Also the number of reference responses to fit in the OMAX framework must be equal to No . In typical modal analytical applications, the number of outputs is much higher than the number of inputs, so a RMFD is often more suited.
BookID 214574_ChapID 44_Proof# 1 - 23/04/2011
"JMFSPOEFGMFDUJPO P
"JMFSPOEFGMFDUJPO P
472
¦ ¦
5JNF T
¦ ¦
5JNF T
5JNF T
"DDFMFSBUJPO H
"DDFMFSBUJPO H
¦ ¦
¦
¦
5JNF T
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Figure 3: Measurement of aileron deflection (top) and response acceleration at the wing (bottom), full measurement record processed (left) and detail (right).
3.1.4
SUBSPACE BASED IDENTIFICATION
An excellent overview of the frequency-domain combined deterministic-stochastic subspace estimator can be found in [10] . The subspace methods are beyond the scope of this paper. One comparison is however appropriate. The FRF models discussed above ˆ k )). These coefficients are disregarded as they do not have a physical estimate extra coefficients (the numerator coefficients of G(ω meaning. They only increase the uncertainty on the estimates, but it is the price one has to pay to use the classical estimators in the OMAX framework. The state-space model, however, is inherently suited to handle data in an OMAX framework.
3.2
Input-output driven
The frequency-domain estimators described in Section 3.1 can be reformulated to use directly the input and output fourier coefficients. The least-squares cost function is now
|Eo,b (ωk )|2
l=
o
and the error equation is reformulated as
Eo,b (ωk ) =
b
(19)
k
D(Ωk )Xo,b (ωk ) − Noi (Ωk )Fi,b (ωk ) C o (Ωk )
(20)
with C o (Ωk ) the numerator coefficients of the noise transfer function G(Ωk ), Nb is the number of blocks. This is a non-linear optimization problem that must be solved iteratively. The I/O approach will thus be slower than the one-step FRF based estimators. As the (nonparametric) FRF must be estimated first to eliminate the deterministic part from the responses, one is better to continue with the FRF and to calculate the PSD+ than to return to the original spectra and minimize the non-linear cost function.
4
APPLICATION TO FLUTTER TEST DATA
We have applied the OMAX procedure described in Section 2.4 to flight flutter test data. The test has been performed on a commercial aircraft equipped with fly-by-wire control. This primary flight control system was also used as excitation device: the electric force signal was superposed on the (auto)pilot’s control signal and fed to the ailerons. The excitation signal was a sine-sweep that was sent to both ailerons, but with a phase lag op π rad., resulting in a anti-symmetrical excitation. Multiple accelerometers were used to capture the response of the aircraft, but we limited our OMAX test to four outputs. Figure 3 shows the force signal for the port aileron, and one
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473
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Figure 4: Typical FRF (left) and XP+ (right), non-parametric estimate (red dots) and standard deviation (black dashed line), EMA FRF fit (thin green line—left) and OMA XP+ fit (thin green line—right), OMAX FRF and XP+ combined fit (thick blue line—left and right); the manufacturer’s identified resonant frequencies are the vertical dotted lines.
response at the port wing tip. These measurements can be split up in two parts: in the first part, the aircraft is excited with the sine sweep (EMA); in the second part no extra excitation was applied but the response of the aircraft (due to turbulence, vibration from the engines,. . . ) was still measured (OMA). We processed both parts separately to determine the FRF (from the first part) and the XP+ (from the second part), since the stochastic part of the response was immediately available. The FRF and the coherence function are estimated using the H1 approach with 9 averages and 50% overlap. A Hanning window is applied to the time data to prevent leakage errors. The FRF and the corresponding standard deviation for one output are shown in Figure 4, left. This FRF is curve fitted using a MLE with model order 20. This overmodelling is necessary to account for model errors and unmodelled dynamics. The thin green line in Figure 4 shows the fit of the MLE. Note that we have omitted the frequency scale in the figure as was requested by the aircraft’s manufacturer. In our opinion, this does not affect the interpretation of the results shown in this Section. The XP+ are calculated using the Welch periodogram method with 9 blocks and 50% overlap, weighted with a Hanning window. Exactly the double of the number of data points is used compared with the FRF, to account for the reduction in points when calculating the positive power spectra. The variance on the XP+ can then be determined [10] . The XP+ is also curve fitted using a MLE with model order 20. The XP+ together with the standard deviation and the MLE fit are shown in Figure 4, right. Eventually both FRF and XP+ are estimated simultaneously by the same MLE estimator in the OMAX framework. The relative scaling of FRF and XP+ is thus done implicitly by weighting the error equation with the standard deviation. The results are also plotted on Figure 4 in blue. It can be seen that the FRF fit hardly changes, but that the fit of the XP+ is altered more compared with the FRF only (EMA) and XP+ only (OMA) respectively. Figure 5 shows the relative error between the estimated resonant frequencies and the values provided by the manufacturer. Also the 95% confidence interval on the frequency is shown. The fourth and the seventh mode are not well estimated by the FRF, while OMAX has a smaller error for the seventh mode; but OMAX estimate of the fourth mode is very close to that of the EMA estimate. It can be verified in Figure 4 that the seventh mode is present in the XP+ but not in the FRF. The fourth mode is also not present in the FRF. OMAX misses this fourth mode because the relative uncertainty of the XP+ is higher than the uncertainty of the FRF, and thus puts more weight on the EMA part of OMAX.
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15
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1
2
3
4 5 6 Mode number
7
8
1
2
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7
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8
Figure 5: Relative error of the resonant frequencies (left) and damping ratios (right) with 95% confidence bounds, FRF (blue crosses, left), OMAX (red dots, middle) and XP+ (green triangles, right) based estimates.
5
CONCLUSIONS
This paper describes the OMAX framework in modal analysis: the combined use of experimental data (with artificial excitation) and operational data (due to ambient excitation, noise) to identify a system’s modal parameters. The main advantage of the OMAX approach is that modes can be found even if they are not (well) excited by the applied force. A procedure is proposed to process experimental and operational data in such a way that the current FRF-based parametric estimators can be used directly in an OMAX framework. It is verified that the OMAX framework can identify modes missed by the FRF if they are present in the spectrum. It is also shown that this is possible even if the inputs are correlated due to an exciter-structure interaction. This is demonstrated by using Monte Carlo simulations and flight flutter test data.
ACKNOWLEDGMENTS The first author is research assistant of the department of Industrial Sciences & Technology of the Erasmushogeschool Brussel. The financial support of the Fund for Scientific Research (FWO Vlaanderen); the Institute for the Promotion of Innovation of Science and Technology in Flanders (IWT); The Concerted Research Action ’OPTIMech’ of the Flemish Community; the Research Council (OZR) of the Vrije Universiteit Brussel (VUB) and the department of Industrial Sciences & Technology of the Erasmushogeschool Brussel (EHB) are gratefully acknowledged.
A
CALCULATION OF THE STOCHASTIC PART OF A RESPONSE
We perform the mathematical derivation for a simple 2-DOF system with operational forces E1 and E2 , and a shaker on the second mass. The load cell is modelled as a spring with stiffness klc , and the (electrical) force on the shaker is denoted by Fe . The system is schematically represented in Figure 6. Based on Eq. (??), the equation of motion of this 2-DOF system, combined with a shaker and load cell, is given by
⎡ ⎢⎢⎢Z11 ⎢⎢⎢ ⎢⎢⎣Z21 0
Z12 Z22 + klc −klc
⎤⎧ ⎫ ⎧ ⎫ ⎪ 0 ⎥⎥⎥ ⎪ Y⎪ E ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 1⎪ ⎬ ⎪ ⎨ 1⎪ ⎬ ⎥⎪ −klc ⎥⎥⎥⎥ ⎪ Y2 ⎪ E2 ⎪ =⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎦⎪ ⎩ ⎭ ⎩ Ze + klc Ye Fe ⎭
(21)
with Y, E, Fe the DFT of the response y(t), unknown forces e(t) and applied force fe (t) respectively, and Zoi the dynamic stiffness relating the response of mass o to an excitation at mass i. We have dropped the dependency on ωk for clarity. It is common to use a load cell to measure the force that the shaker injects into the mechanical system via the stinger. The load cell is
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475 Fe ke
klc
me ce
k1
k2
k3
m2
m1 c1
c3
c2
E1
E2
Y1
Y2
Figure 6: 2-DOF mechanical system with exciter (and load cell) mounted at mass 2. Applied force Fe (ωk ) and unknown ambient excitation E1 (ωk ) and E2 (ωk ).
a piezo-electric crystal, the force is measured via the relative displacement between the shaker mass and mass 2:
Flc = −klc (Y2 − Ye ) Expanding the left hand side of Eq. (21) as
reveals the load cell force:
which yields, combined with Eq. (22)
⎡ ⎢⎢⎢Z11 Y1 ⎢⎢⎢ ⎢⎢⎣Z21 Y1
+ +
Z12 Y2 (Z22 + klc )Y2 −klc Y2
⎡ ⎢⎢⎢Z11 Y1 ⎢⎢⎢ ⎢⎢⎣Z21 Y1
+ +
Z12 Y2 Z22 Y2 −klc (Y2 − Ye )
⎡ ⎢⎢⎢Z11 ⎢⎢⎢ ⎢⎢⎣Z21 0
(22)
+ +
⎤ ⎧ ⎫ ⎥⎥⎥ ⎪ E1 ⎪ ⎪ ⎪ ⎨ ⎪ ⎬ ⎥ ⎪ −klc Ye ⎥⎥⎥⎥ = ⎪ E2 ⎪ ⎪ ⎪ ⎪ ⎦ ⎪ ⎩ (Ze + klc )Ye Fe ⎭
(23)
+ +
⎤ ⎧ ⎫ ⎥⎥⎥ ⎪ E ⎪ ⎪ ⎪ ⎨ 1⎪ ⎬ ⎥ ⎪ klc (Y2 − Ye )⎥⎥⎥⎥ = ⎪ E2 ⎪ ⎪ ⎪ ⎪ ⎦ ⎪ ⎩ Z e Ye Fe ⎭
(24)
⎤⎧ ⎫ ⎧ ⎫ ⎪ 0 ⎥⎥⎥ ⎪ Y1 ⎪ E1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎬ ⎪ ⎨ ⎬ ⎥⎥⎥ ⎪ 0 ⎥⎥ ⎪ Y2 ⎪ E2 + Flc ⎪ =⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎦⎩ ⎭ ⎩ Ze Ye Fe − Flc ⎭
Z12 Z22 0
(25)
It is clear from Eq. (25) that the dynamics of the 2-DOF system are now decoupled from the dynamics of the exciter. Since we are only interested in the 2-DOF system, we use only the first two rows:
Z11 Z21
Z12 Z22
Y1 E1 = Y2 E2 + Flc
The FRFs are found by inverting the dynamic stiffness, which gives
Y1 H = 11 Y2 H21
or, after rearranging terms:
Y1 H = 11 Y2 H21
H12 H22
H12 H22
E1 E2 + Flc
E1 H + 11 E2 H21
(26)
H12 H22
(27)
0 Flc
(28)
The first term of the right hand side of Eq. (28) describes the response of the (decoupled) system due to the ambient excitation (the stochastic part of Y ), while the second term gives the deterministic response due to the effectively applied force Flc . If Flc is measured and an estimate of H is available, then the stochastic part of Y can be calculated by subtracting the deterministic response:
Y1 H − 11 Y2 H21
which gives Eq. (6).
H12 H22
0 Flc
=
H11 H21
H12 H22
E1 E2
(29)
BookID 214574_ChapID 44_Proof# 1 - 23/04/2011
476 REFERENCES [1] Kehoe, M. W., A Historical Overview of Flight Flutter Testing, Nasa technical memorandum 4720, NASA, Dryden Flight Research Center, Edwards, October 1995. [2] Deckers, K., De Troyer, T., Reynders, E., Guillaume, P., Lefeber, D. and De Roeck, G., Applicability of low-weight Pneumatic Artificial Muscle Actuators in an OMAX framework, Proceedings of the International Conference on Noise and Vibration Engineering ISMA, pp. 2445–2456, Leuven, Belgium, September 2008. [3] Heylen, W., Lammens, S. and Sas, P., Modal Analysis Theory and Testing, Katholieke Universiteit Leuven, Department Werktuigkunde, Heverlee, 1997. [4] Maia, N. and Silva, J., Theoretical and Experimental Modal Analysis, Research Studies Press, Hertfordshire, 1997. [5] Brincker, R., DeStefano, A. and Piombo, B., Ambient data to analyse the dynamic behaviour of bridges: A first comparison between different techniques, Proceedings of the 14th International Modal Analysis Conference (IMAC), pp. 477–482, Dearborn, 1996. [6] Ibrahim, S., Brincker, R. and Asmussen, J., Modal parameter identification from responses of general unknown random inputs, Proceedings of the 14th International Modal Analysis Conference (IMAC), pp. 446–452, Dearborn, 1996. [7] Parloo, E., Application of frequency-domain system identification techniques in the field of operational modal analysis, Ph.d., Vrije Universiteit Brussel, department MECH, Pleinlaan 2, 1050 Brussel, Belgium, 2003. [8] Peeters, B. and de Roeck, G., Reference-based stochastic subspace identification for output-only modal analysis, Mechanical Systems and Signal Processing, Vol. 13, No. 6, pp. 855–878, nov 1999. [9] Guillaume, P., Verboven, P., Cauberghe, B. and Vanlanduit, S., Frequency-domain system identification techniques for experimental and operational modal analysis, Proceedings of 13th Symposium on System Identification SYSID, Rotterdam, August 2003. [10] Cauberghe, B., Applied frequency-domain system identification in the field of experimental and operational modal analysis, Ph.d, Vrije Universiteit Brussel, department MECH, Pleinlaan 2, 1050 Brussel, Belgium, 2004. [11] Reynders, E. and De Roeck, G., Reference-based combined deterministic-stochastic subspace identification for experimental and operational modal analysis, Mechanical Systems and Signal Processing, Vol. 22, No. 3, pp. 617–637, 2008. [12] Welch, P. D., The use of fast fourier transform for the estimation of power spectra: A method based on time averaging over short modified periodograms, IEEE Transactions on Audio and Electroacoustics, Vol. 15, No. 2, pp. 70–73, June 1967. [13] Verboven, P., Frequency-domain system identification for modal analysis, Ph.d, Vrije Universiteit Brussel, department MECH, Pleinlaan 2, 1050 Brussel, Belgium, 2002. [14] Blackman, R. B. and Tukey, J. W., The measurement of power spectra from the point of view of communication engineering, Dover Publications, 1958. [15] Bendat, J. S. and Piersol, A. G., Random Data: Analysis and Measurement Procedures, John Wiley & Sons, New York, 1971. [16] Guillaume, P., Verboven, P. and Vanlanduit, S., Frequency-Domain Maximum Likelihood Identification of Modal Parameters with Confidence Intervals, International Conference on Noise and Vibration Engineering ISMA23, pp. 359–366, Leuven, 1998. [17] Peeters, B., Lowet, G., Van Der Auweraer, H. and Leuridan, J., A new procedure for modal parameter estimation, Sound and Vibration, Vol. 38, No. 1, pp. 24–29, 2004. [18] Peeters, B., Van Der Auweraer, H., Guillaume, P. and Leuridan, J., The Polymax frequency-domain method: A new standard for modal parameter estimation?, Shock and Vibration, Vol. 11, No. 3–4, pp. 395–409, 2004.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Application of Principal Component Analysis Methods to Experimental Structural Dynamics
Randall J. Allemang, PhD Professor
Allyn W. Phillips, PhD Research Associate Professor
Matthew R. Allemang Research Assistant
Structural Dynamics Research Laborator y Depar tment of Mechanical Engineering University of Cincinnati Cincinnati, Ohio 45221-0072 U. S. A.
ABSTRACT Principal Component Analysis (PCA) methods have been variously developed and applied within the experimental modal community for some time based upon the underlying linear/superposition nature of structural dynamics. While historically the use of these techniques has been restricted to the areas of model order determination (Complex Mode Indicator Function [CMIF]), enhanced frequency response function estimation, and parameter identification, increasingly they are being applied to the areas of test/model validation, experimental model correlation/repeatability and experimental/structural model comparison. With the increasing volume of data being collected today, techniques which provide effective extraction of the significant data features for quick, easy comparison are essential. This paper explores the general development and application of PCA to experimental modal analysis and its ability to provide the analyst with an effective global trend visualization tool. As examples, the results from a laborator y test structure (circular plate), a civil infrastructure (bridge), and a comparative study (automotive) are presented. Nomenclature Ni = Number of inputs. No = Number of outputs. Nf = Number of frequencies. NS = Shor t matrix dimension. NL = Long matrix dimension. ω = Frequency (rad/sec). λ r = Complex modal frequency [B] = Generic data matrix.
[H(ω )] = Frequency response function matrix (No × Ni )). [T(ω )] = Transmissibilty function matrix (No × Ni )). [GFF (ω )] = Force cross power spectra matrix (Ni × Ni )). [U] = Left singular vector matrix. [S] = Principal value matrix (diagonal). [Σ] = Singular value matrix (diagonal). [Λ] = Eigenvalue matrix (diagonal). [V] = Right singular vector, or eigenvector, matrix.
1. Introduction The concept of indentifying the various underlying linear contributors in a set of data is needed in many fields of science and engineering. The techniques have been independently developed and/or discovered by many authors in many completely different application areas. Many times the procedure has acquired a different name depending upon the individual or specific application focus. This has resulted in a confusing set of designations for fundamently similar techniques: Principal Component Analysis (PCA), Independent Component Analysis (ICA), Complex Mode Indicator Function (CMIF), Principal Response Functions, Principal Gains, and others. Without detracting from the insight and ingenuity of each of the original developers, each of these techniques relies upon the property of the Singular Value Decomposition (SVD) to represent a set of functions as a product of weighting
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_45, © The Society for Experimental Mechanics, Inc. 2011
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factors and independent linear contributions. Today, this is known more widely as the SVD approach to principal component analysis (PCA). 2. Background Principal component analysis (PCA) is the general description for a number of multivariate data analysis methods that typically provide a linear transformation from a set of physical variables to a new set of virtual variables. The linear transformation is in the form of a set of orthonormal vectors (principal vectors) and associated scaling for the orthonormal vectors (principal values). These scaling terms are often thought of as the weight or importance of the associated orthonormal vectors since the orthonormal vectors have no overall scaling (unity length) but are or thogonal to one another (have no projection or relationship to one another). While there are many approaches to computing the principal components, most modern implementations utilize eigenvalue decomposition (ED) methods for the square matrix case and singular value decomposition (SVD) for the rectangular matrix case. Historically, the primar y governing equation is as follows: [ B ]N×N = [ U ]N×N ⎡ S ⎦N×N [ U ]TN×N
(1)
In this relationship, the common form star ts with a data matrix ([B]) that is frequently real-valued and square which is transformed by the orthonormal vectors ([U]) and the diagonal principal value matrix (⎡ S ⎦). Since the or thonormal vectors have no scaling (unity length), the principal values contain the physical scaling of the original data matrix ([B]). The scaling nature of the principal values gives rise to the term principal gain that is occasionally found in structural dynamics. With respect to structural dynamics, the PCA is frequently performed time by time or frequency by frequency across a data matrix that is square or rectangular at each time or frequency. The common form of the PCA concept when developed via the singular value decomposition of a complex-valued data matrix becomes: [ B ]NL×NS = [ U ]NL×NS ⎡ S ⎦NS ×NS [ V ]H NS ×NS
(2)
In this version of the relationship, the common form star ts with a data matrix ([B]) that is frequently complexvalued and rectangular which is transformed by the orthonor mal vectors ([U] and [V]) and the diagonal principal value matrix (⎡ S ⎦). If the data matrix ([B]) is complex valued, it is important to note that the phasing between the left principal vector ([U]) and the associated right principal vector ([V]) is generally not unique. If the principal vectors are used independently (left or right) this arbitrar y phase issue must be accounted for. Plotting the principal values, largest to smallest across the time or frequency range results in a 2-D scaling function that is often used to determine specific time dependent or frequency dependent features in the data without looking at individual measurements. This can be tremendously helpful when the number of terms in the data matrix (rows and columns) is large. The PCA techniques were first developed in the early 1900s but did not come into common use in structural dynamics until the 1970s and 1980s with the advent of math packages like EISPACK ® and LINPACK ® and subsequent development of user friendly software like Mathmatica ® and Matlab ®. A number of textbooks are now available that discuss these methods but many of the available texts do not include complex valued, spectral analysis that is common to structural dynamics data [1-8] . PCA methods have found common usage in many science and engineering areas. The most common applications involve multidimensional scaling, linear modeling, data quality analysis and analysis of variance. These are the same applications that makes the general PCA methods so attractive to structural dynamics. Since much of structural analysis is built upon linearity, super-position and linear expansion, the PCA methods have found wide and increasing use, par ticularly in the experimental data analysis areas of structural dynamics. Common uses are the evaluation of force independence in the multiple input, multiple output (MIMO) estimation of frequency response functions (FRFs), the evaluation of close and repeated modal frequencies in the MIMO FRF
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matrix using a method that is known as the complex mode indicator function and the development of virtual FRF functions known as principal response functions. 3. Common Applications With respect to structural dynamics, at least three PCA based methods are relatively well know and frequently utilized. The principal force analysis associated with MIMO FRF estimation, the close or repeated mode analysis of the complex mode indicator function (CMIF) and the virtual FRF estimation related to principal response functions have all been used for ten or more years and are summarized in the following sections. 3.1 Principal (Virtual) Forces The current approach used to evaluate correlated inputs for the MIMO FRF estimation problem involves utilizing principal component analysis to determine the number of contributing forces (principal or virtual forces) to the [GFF ] matrix [9-11] . The [GFF ] matrix is the cross power spectra matrix involving all of the multiple, simultaneous force inputs applied to the structure during the MIMO FRF estimation. In this approach, the matrix that must be evaluated is: ⎡ GFF11 ⎢ . ⎢ H [GFF ] = [GFF ] = ⎢ . ⎢ . ⎢ ⎣ GFFN 1 i
.
.
.
.
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This application of PCA involves an eigenvalue decomposition of the [GFF ] matrix at each frequncy of the measured power spectra of the forces. Since the eigenvectors of such a decomposition are unitary, the eigenvalues should all be of approximately the same size if each of the inputs is contributing to the excitation of the structure equally. If one of the eigenvalues is much smaller at a particular frequency, one of the inputs is not present or one of the inputs is correlated with the other input(s) at that frequency. [ GFF (ω ) ] = [ V(ω ) ] ⎡ Λ(ω ) ⎦ [ V(ω ) ]H
(4)
[ Λ ] in the above equation represents the eigenvalues of the [GFF ] matrix. If any of the eigenvalues of the [GFF ] matrix are zero or insignificant, then the [GFF ] matrix is singular. Therefore, for a three input test, the [GFF ] matrix should have three eigenvalues at each frequency. (The number of significant eigenvalues is the number of uncorrelated inputs). Figure 1 shows the principal force plots for the three input case. These principal force cur ves are no longer linked to a specific physical exciter location due to the linear transfor mation involved. These cur ves are sometimes referred to as vir tual forces. Note that the overall difference in the three curves is typically one or two orders of magnitudes and there will be some fluctuation in the curves in the frquency region where there are lightly damped modes due to the exciter-structure interaction. Figure 2 shows a case where one exciter is not turned on (but the load cell is still active and measuring force). The distinct change in amplitude of the third cur ve indicates that there are only two independent forces contributing to the force power spectra matrix.
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Principal Forces
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Figure 2. Principal (Vir tual) Force Spectrum - One Exciter Off 3.2 Complex Mode Indicator Function Another application of the SVD approach to PCA, applied to multiple reference FRF measurements and identified as the Complex Mode Indication Function (CMIF), was first developed for traditional MIMO FRF data in order to
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identify the proper number of modal frequencies, par ticularly when there are closely spaced or repeated modal frequencies [12-15] . The CMIF indicates the existence of real normal or complex modes and the relative magnitude of each mode. The CMIF is defined as the economical (function of the short dimension) singular values, computed from the MIMO FRF matrix at each spectral line. The CMIF is the plot of these singular values, typically on a log magnitude scale, as a function of frequency. The peaks detected in the CMIF plot indicate a normalized response dominated by one or more significant contributions; therefore, the existence of modes, and the corresponding frequencies of these peaks give the damped natural frequencies for each mode. In this way, the CMIF is using the PCA approach to take advantage of the superposition principal commonly know as the expansion theorem. In the application of the CMIF to traditional modal parameter estimation algorithms, the number of modes detected in the CMIF determines the minimum number of degrees-of-freedom of the system equation for the algorithm. A number of additional degrees-of-freedom may be needed to take care of residual effects and noise contamination. [H(ω )] = [U(ω )] [Σ(ω )] [V(ω )]H
(5)
Most often, the number of input points (reference points), Ni , is less than the number of response points, No . In the above equation, if the number of effective modes at a given frequency is less than or equal to the smaller dimension of the FRF matrix, ie.Ne ≤ Ni , the singular value decomposition leads to approximate mode shapes (left singular vectors) and approximate modal participation factors (right singular vectors). The singular value is then equivalent to the the scaling factor Qr divided by the difference between the discrete frequency and the modal frequency jω − λ r . For a given mode, since the scaling factor is a constant, the closer the modal frequency is to the discrete frequency, the larger the singular value will be. Therefore, the damped natural frequency is the frequency at which the maximum magnitude of the singular value occurs. If different modes are compared, the stronger the modal contribution (larger residue value), the larger the singular value will be. The peak in the CMIF indicates the location on the frequency axis that is nearest to the pole. The frequency is the estimated damped natural frequency, to within the accuracy of the frequency resolution.
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Figure 3. Complex Mode Indicator Function (CMIF) Since the mode shapes that contribute to each peak do not change much around each peak, several adjacent
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spectral lines from the FRF matrix can be used simultaneously for a better estimation of mode shapes. By including several spectral lines of data in the singular value decomposition calculation, the effect of the leakage error can be minimized. If only the quadrature (imaginary) part of the FRF matrix is used in CMIF, the singular values will be much more distinct. 3.3 Principal Response Functions Similar in concept to the CMIF is the development of the Principal Response Functions (PRF) [16-18] . In this case, however, instead of performing the decomposition of the FRF data matrix frequency by frequency, the entire data matrix is arranged in two dimensions such that each column contains a single FRF and the singular value decomposition is then performed on this new data matrix. ⎡⎧ ⎫ ⎧ ⎫ ⎧ ⎫⎤ [A]Nf ×NS NL = ⎢⎨H11 (ω )⎬ . . . ⎨Hpq (ω )⎬ . . . ⎨HNS NL (ω )⎬⎥ = [U] [Σ] [V]H ⎣⎩ ⎭ ⎩ ⎭ ⎩ ⎭⎦
(6)
It was then recognized that the left singular vectors [U] are a function of frequency and contain the primar y spectral characteristics and the singular values [Σ] the relative contribution of those vectors to the global data matrix. By selecting only the ’r’ most significant singular values and associated left singular vectors, the principal response functions are then defined as: [PRF] = [H] = [Ur ] [Σr ] = [A] [Vr ]
(7)
The principal response functions reveal the nature of the noise and noise floor contained in the data and, contrary to normal intuition, demonstates that more data is not always better. Once the real infor mation in the data matrix is identified, the addition of more measurements can only contribute to the noise and hence effectively raises the noise floor and makes parameter identification more difficult. The following two plots reveal for the circular plate that although there are 252 individual measurements (36x7), only about 25 PRFs are necessary to explain the spectral information. The left figure is the plot of the top 25 PRFs; the right is the top 50 PRFs. It becomes clear by comparison that the additional PRF’s contribute primarily noise and establish the measurement noise floor.
Principal Response Functions [25]
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Figure 4. C-Plate, First 25 (left) and First 50 (right) Principal Response Functions 4. Other Structural Dynamics Applications A number of examples in the following sections are common ways of utilizing the PCA methodology in structural dynamics testing situation. The first examples are taken from MIMO FRF data from a simple circular plate (CPlate). This C-Plate is made of steel, approximately 3/ inch thick and 36 inches in diameter. Seven reference
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acclerometers were held in fixed locations while a hammer was roved to each of 36 input locations. The latter examples are taken from a series of experimental tests performed on four different, fully trimmed automotive structures (Vehicle B through Vehicle E). The data utilized for these examples was MIMO transmissability functions ([T(ω )]) taken with a Four Axis Road Simulator. The MIMO transmissibility data is in the form of acceleration normalized by displacement and the data matrix is 4 by 150 representing the 4 ver tical displacments of the inputs of the Road Simulator and the 150 accelerometers located all over the vehicles. Note that in these cases, when different vehicles are compared, the response acclerometers were in approximately the same locations on the different vehicles. These tests were conducted to compare and contrast vehicles that were in the same general class (in terms of size) but with clearly different characteristics, identified based upon owner feedback [19-20] . 4.1 Modal Parameter Validation The following set of figures presents a comparison of the measurement synthesis for the cplate test structure. For this case, two different parameter identification runs are presented; the first where all modes were identified properly [green curve - cplate synthesis (good)] and the second where one mode was missed [red curve - cplate synthesis (poor)]. The first PCA curve (red) provides an indication that the quality of fit around the 780 Hz mode for the second run is suspect; the second PCA curve (red) clearly indicates that one mode was missed.
Principal Component Analysis
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Figure 5. C-Plate, Measured versus Synthesized Data, All Principal Components The following curves show the comparisons in the above figure on a curve by curve basis and include the correlation information between the curves, using a measurement correlation coefficient, calculated in the same way as the synthesis correlation coefficient (SCC) or the frequency response analysis correlation (FRAC) [21-22] .
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Principal Component #1 1.000 0.997 0.915 0.997 1.000 0.918 0.915 0.918 1.000
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Figure 6. C-Plate, Measured versus Synthesized Data, First and Second Principal Components Principal Component #3 1.000 0.621 0.609 0.621 1.000 0.982 0.609 0.982 1.000
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Figure 7. C-Plate, Measured versus Synthesized Data, Third and Four th Principal Components The next set of figures represents a modal synthesis for a civil infrastructure test on a small bridge. The set of FRF data had 55 inputs and 15 fixed response locations. This data is ver y noisy, has relatively high modal density with moderate to heavy damping on several modes. For these reasons, the modal parameter estimation is much more difficult. The following example shows the results for a relatively quick fit of the data and demonstrates some problems with the overall modal scaling and some missing modes.
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Principal Component #1 1.000 0.957 0.957 1.000
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4.2 Evaluation of Linearity/Time Variance/Repeatability Comparing the principal components of a set of MIMO FRF or transmissibility data from different test levels, different tests conducted at different times or tests conducted to evaluate changes in the test object are good examples of the application of principal component analysis methods. The following examples use the SVD approach to PCA to evaluate changes in the transmissiblity data matrix to different reference input levels. It is much easier to compare the principal values as a function of frequency for the entire data set rather than to compare specific measurement DOFs.
Principal Component Analysis
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Figure 11. Vehicle B, Three input levels, All Data, All Principal Components Principal Component #1 1.000 0.985 0.966 0.985 1.000 0.985 0.966 0.985 1.000
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Figure 12. Vehicle B, Three input levels, All Data, Primar y and Secondary Principal Components
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Figure 13. Vehicle E, Three input levels, All Data, All Principal Components Principal Component #1 1.000 0.843 0.812 0.843 1.000 0.963 0.812 0.963 1.000
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Figure 14. Vehicle E, Three input levels, All Data, Primar y and Secondary Principal Components
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4.3 Evaluation of Testing Parameters During testing of a series of vehicles, the straps used to restrain the vehicle with respect to the four actuators of the Four Axis Road Simulator were damaged and could not be used in further testing. The straps were replaced but the new straps were in a different configuration as well as being of different sizes. The SVD approach to PCA was utilized to evaluate both the before and after variation of the MIMO transmissibilty data as well as to take a fur ther look at the effects of using one or more straps to restrain the vehicle.
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Figure 15. Vehicle D, Various Strap Conditions, All Data, All Principal Components Principal Component #1 1.000 0.902 0.891 0.895 0.973 0.902 1.000 0.995 0.964 0.840 0.891 0.995 1.000 0.966 0.833 0.895 0.964 0.966 1.000 0.867 0.973 0.840 0.833 0.867 1.000
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Figure 16. Vehicle D, Various Strap Conditions, All Data, Primar y and Secondary Principal Components
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4.4 Directional Scaling Frequently, the dominant direction of response is not clear when working with fully trimmed vehicles and par ticularly vehicle subcomponents. The SVD approach to PCA can be used to compare the overall scaling of the response from X to Y to Z direction in the data. In the following examples, the scaling of the Z direction can be seen to be at a much lower level that the X and Y directions. This can be due to a lack of response in the Z direction or possibly incorrect calibration applied to the X and Y directions or to the Z directions.
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Figure 17. Vehicle B, Direction Comparisons, All Data, All Principal Components Principal Component #1 1.000 0.994 0.341 0.999 0.994 1.000 0.377 0.998 0.341 0.377 1.000 0.358 0.999 0.998 0.358 1.000
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Figure 18. Vehicle B, Direction Comparisons, All Data, Primar y and Secondary Principal Components
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Figure 19. Vehicle E, Direction Comparisons, All Data, All Principal Components Principal Component #1 1.000 0.991 0.664 0.999 0.991 1.000 0.679 0.996 0.664 0.679 1.000 0.669 0.999 0.996 0.669 1.000
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Figure 20. Vehicle E, Direction Comparisons, All Data, Primar y and Secondary Principal Components
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Figure 21. All Vehicles, X Direction Data, All Principal Components Principal Component #1 1.000 0.815 0.739 0.830 0.815 1.000 0.707 0.888 0.739 0.707 1.000 0.645 0.830 0.888 0.645 1.000
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Figure 23. All Vehicles, Y Direction Data, All Principal Components Principal Component #1 1.000 0.797 0.742 0.843 0.797 1.000 0.784 0.889 0.742 0.784 1.000 0.699 0.843 0.889 0.699 1.000
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Figure 25. All Vehicles, Z Direction Data, All Principal Components Principal Component #1 1.000 0.959 0.893 0.682 0.959 1.000 0.893 0.719 0.893 0.893 1.000 0.574 0.682 0.719 0.574 1.000
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4.5 Structure Comparisons One of the primary reasons to compare different vehicles is to determine overall structure characteristics that affect some performance criteria (squeak and rattle performance, for example) in order to clearly see that one vehicle is different from other similar vehicles. In this case, the vehicles are clearly different and looking at individual measurements or modal properties may not easily give global or overall (similar or dissimilar) structure characteristics.
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Figure 27. All Vehicles, All Data, All Principal Components Principal Component #1 1.000 0.760 0.808 0.833 0.760 1.000 0.741 0.680 0.808 0.741 1.000 0.887 0.833 0.680 0.887 1.000
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4.6 Structural Component/Sub-System Comparison This case is similar to the previous case but is limited to a subset of the data in the region of the dashboard of each vehicle. Again, clear differences can be seen in the trend of the data with some vehicles.
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Figure 29. All Vehicles, Dash SubSystem, All Principal Components Principal Component #1 1.000 0.891 0.857 0.668 0.891 1.000 0.925 0.682 0.857 0.925 1.000 0.547 0.668 0.682 0.547 1.000
−2
10
10
−3
10
−2
10
−3
Amplitude
Amplitude
Principal Component #2 1.000 0.944 0.927 0.696 0.944 1.000 0.966 0.594 0.927 0.966 1.000 0.656 0.696 0.594 0.656 1.000
−4
10
10 B_CBB C_CBB D_CBB E_CBB
−5
10
−4
5
10
15
20 25 Frequency [Hz]
30
35
40
10
B_CBB C_CBB D_CBB E_CBB
−5
5
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15
20 25 Frequency [Hz]
30
Figure 30. All Vehicles, Dash SubSystem, Primar y and Secondary Principal Components
35
40
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497
5. Conclusions This paper has explored the general background, development, application and usage of Principal Component Analysis within the experimental modal analysis community. It has been noted that techniques have been discovered and re-discovered over time as various analysis problems have been encountered. Regardless of their original development, these various ingenious solution techniques to revealing underlying linear contributors all now share a common numerical foundation in the singular value decomposition. Nonetheless, the various intuitive physical interpretations of the data matrix are highly valuable contributions to the community. With their broad applicability and ability to extract effective global trend infor mation, the various PCA techniques provide the analyst with robust numerical tools for making informed parameter identification decisions.
6. Acknowledgements The authors would like to acknowledge the support for a portion of the experimental work described in this paper by The Ford Foundation University Research Program (URP) and The Ford Motor Company. The authors would also like to acknowledge several direct and indirect, written and verbal communications concerning the use of PCA methods with test engineers at Lockheed Martin Space Systems (Mr. Ed Weston, Dr. Alain Carrier and Mr. Thomas Steed) and ATA Engineering, Inc. (Mr. Ralph Brillhar t and Mr. Kevin Napolitano) using the Principal Gain terminology. 7. References [1]
Wilkinson, J.H., The Algebraic Eigenvalue Problem, Oxford University Press, Oxford, U.K., 1965, pp. 12-13.
[2]
Strang, G., Linear Algebra and Its Applications, Third Edition, Harcour t Brace Jovanovich Publishers, San Diego, CA, 1988, 505 pp.
[3]
Lawson, C.L., Hanson, R.J., Solving Least Squares Problems, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1974, 340 pp.
[4]
Jolliffe, I.T., Principal Component Analysis Springer-Verlag New York, Inc., New York, NY, 1986, 271 pp.
[5]
Ljung, Lennar t, System Identification: Theory for the User, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1987, 519 pp.
[6]
Deprettere, E.F.A., SVD and Signal Processing: Algorithms, Applications and Architectures, Elsevier Science Publishing Company, Inc., New York, NY, 1988, 477 pp.
[7]
Hyvarinen, A., Karhunen, J, Oja, E., Independent Component Analysis, John Wiley & Sons, Inc., New York, NY, 2001, 481 pp.
[8]
Jackson, Edward J., A User’s Guide to Principal Components, John Wiley & Sons,Inc., Hoboken, NJ, 2003, 569 pp.
[9]
Allemang, R.J., Rost, R.W., Brown, D.L., "Multiple Input Estimation of Frequency Response Functions: Excitation Considerations", ASME Paper Number 83-DET-73, 1983, 11 pp.
[10] Allemang, R.J., Brown, D.L., Rost, R.W., "Multiple Input Estimation of Frequency Response Functions for Experimental Modal Analysis", U.S. Air Force Report Number AFATL-TR-84-15, 1984, 185 pp. [11] Allemang, R.J., Brown, D.L., Rost, R.W., "Measurement Techniques for Experimental Modal Analysis", Experimental Modal Analysis and Dynamic Component Synthesis, USAF Technical Report, Contract Number F33615-83-C-3218, AFWAL-TR-87-3069, Volume 2, 1987.
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[12]
"Complex Mode Indication Function and Its Application to Spatial Domain Parameter Estimation", Shih, C.Y., Tsuei, Y.G., Allemang, R.J., Brown, D.L., Journal of Mechanical Systems and Signal Processing, Academic Press Limited, Volume 2, Number 4, pp. 367-372, 1988.
[13]
"Complex Mode Indicator Function and its Application to Spatial Domain Parameter Estimation", Shih, C.Y., Tsuei, Y.G., Allemang, R.J., Brown, D.L., Proceedings, 13th International Seminar on Modal Analysis, Katholieke Universiteit Leuven, Belgium, 15 pp., 1988.
[14] "Complex Mode Indication Function and its Application to Spatial Domain Parameter Estimation", Shih, C.Y., Tsuei, Y.G., Allemang, R., Brown, D.L., Proceedings, International Modal Analysis Conference, pp. 533-540, 1989. [15]
"A Complete Review of the Complex Mode Indicator Function (CMIF) with Applications", Allemang, R.J., Brown, D.L., Proceedings, International Conference on Noise and Vibration Engineering (ISMA), Katholieke Universiteit Leuven, Belgium, 38 pp., 2006.
[16]
"Estimating the Rank of Measured Response Data Using SVD and Principal Response Functions", Pickrel, C.R., Proceedings of the Second International Conference on Structural Dynamics Modeling, Test Analysis and Correlation DTA/NAFEMS, pp. 89-100, 1996.
[17]
"Numerical Assessment of Test Data Using SVD", Pickrel, C.R., Proceedings, Inter national Modal Analysis Conference, pp. 1577, 1997.
[18]
"Numerical Assessment of Test Data Using SVD", Pickrel, C.R., Proceedings of SPIE, the International Society for Optical Engineering, Vol. 3089 (2), pp. 1577-1587, 1997.
[19] "Squeak and Rattle Detection : A Comparative Experimental Data Analysis", Mantrala, Ravi, MS Thesis, University of Cincinnati, 134 pp., 2007. [20] "Using Group Transmissibility Concepts to Compare Dissimilar Vehicle Platfor ms", Yee, Abbey, MS Thesis, University of Cincinnati, 59 pp., 2009. [21]
"FRAC: A Consistent way of Comparing Frequency Response Functions", Heylen, W., Lammens, S., Proceedings, International Conference on Identification in Engineering, Swansea, pp. 48-57, 1996.
[22] "The Modal Assurance Criterion (MAC): Twenty Years of Use and Abuse", Allemang, R.J., Proceedings, International Modal Analysis Conference, pp. 397-405, 2002. Sound and Vibration Magazine, Vol. 37, No. 8, pp. 14-23, August, 2003.
BookID 214574_ChapID 46_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Application of Modal Scaling to the Pole Selection Phase of Parameter Estimation
Allyn W. Phillips, PhD Research Associate Professor
Randall J. Allemang, PhD Professor
Structural Dynamics Research Laborator y Depar tment of Mechanical Engineering University of Cincinnati Cincinnati, Ohio 45221-0072 U. S. A.
ABSTRACT Modern modal parameter estimation algorithms are frequently presented as two stage solution processes where the first stage identifies the system poles and unscaled modal vectors (participation factors) of either long or short dimension, and where the second stage identifies the scaled modal vectors (residue vectors) of generally long dimension and modal scaling. This paper explores the value of having the long dimension, scaled modal information available during the pole selection process. Among the advantages of this approach is the availability of the full length residue vector for visualization and the modal scaling in order to evaluate relative contribution and physical significance. A comparison of the residue quality for this solution approach and the dominant traditional approaches is presented. The methods and results are compared using mean phase (MP), mean phase deviation (MPD), and vector scatter plots.
Nomenclature Ni = Number of inputs. No = Number of outputs. Nf = Number of data lines (frequencies). Nt = Number of data lines (times). Ne = Number of effective modal frequencies. N∞ = Number of theoretical modal frequencies. NL = Size of long dimension. NS = Size of shor t dimension. N = Number of modal frequencies. λ r = Complex modal frequency (rad/sec). λr = σr + j ωr σ r = Modal damping. ω r = Damped natural frequency. s = S-Domain (Laplace) frequency variable (rad/sec).
si = Generalized frequency variable (rad/sec).
ω i = Frequency (rad/sec).
Apqr = Residue for output DOF p, input DOF q, mode r. Qr = Modal scaling for mode r. Mr = Modal mass for mode r. MAr = Modal A mode r. {ψ }r = Scaled modal vector for mode r. ψ pr = Scaled modal coefficient for output DOF p, mode r. {ψˆ }r = Unscaled modal vector for mode r. {L}r = Modal participation vector for mode r. [I] = Identity matrix. [H(ω )] = Frequency response function matrix (No × Ni )). RIpq = Residual inertia for output DOF p, input DOF q. RFpq = Residual flexibility for output DOF p, input DOF q.
1. Introduction Scaled modal vectors are typically estimated by modern modal parameter estimation algorithms after the modal frequencies are determined. Depending on the parameter estimation method chosen to estimate the modal frequencies, there is often an initial estimate of the unscaled modal vector or a subset of the unscaled modal T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_46, © The Society for Experimental Mechanics, Inc. 2011
499
BookID 214574_ChapID 46_Proof# 1 - 23/04/2011
500 vector. Internal to the software implementation, choices are available to estimate the scaled modal vectors and modal scaling in a number of ways. For experimental data that does not perfectly match the theoretical requirements of the modal parameter estimation algorithms, the choice of what procedure is used to estimate the final scaled modal vectors and associated modal scaling factors will effect the final answer. In some situations, typically for noisy data or data that was difficult to acquire, the final scaled modal vector may look different from the initial unscaled modal vector when presented in a modal vector animation. While this may be confusing to the user, it is perfectly understandable since the estimates are generally least squares or weighted least squares estimates that do not constrain the solution(s) in the same way. There are several acceptable choices for estimating the scaled modal vectors and modal scaling but some may be preferable in order to reduce this apparent inconsistency. 2. Background In order to understand why different modal vectors are estimated at various stages in the modal parameter estimation process for modern algorithms, a general overview of the process, the theory and the practical, experimental implementation is required. Complete details of this development are given in several references [1-7] and is referred to as the Unified Matrix Polynomial Approach (UMPA). In the following explanation, only the frequency response function form of the equations is used to explain the modal vector scaling issues but an equivalent explanation involving impulse response functions (IRFs) parallels this explanation, equation by equation. To begin with, the multiple input - multiple output (MIMO) frequency response function (FRF) model that is commonly used in the initial stage of modal parameter estimation is: m
n
Σ ⎡[α k ] (si )k ⎤⎦ [H(ω i )] = k=0 Σ ⎡⎣[ β k ] (si )k ⎤⎦ [I] k=0 ⎣
(1)
In the above model, the complex-valued modal frequencies can be estimated from the eigenvalue-eigenvector problem formed from the matrix coefficient polynomial equation involving the alpha ([α k ]) coefficient matrices. This results in a matrix coefficient polynomial equation of the following form: m−1 ⎪ [α ] sm + [α + [α m−2 ] sm−2 + . . . . . . . . . + [α 0 ] ⎪ = 0 m−1 ] s ⎪ m ⎪
(2)
When the modal frequencies are estimated from the eigenvalue-eigenvector problem that is associated with solving this matrix coefficient polynomial equation, a unique estimate of the unscaled modal vector is identified at the same time. The length or dimension of this unscaled modal vector is equal to the dimension of the square alpha coefficients which must be equal to the row dimension of the FRF data matrix in order for the matrix coefficient polynomial equation to be conformal. Normally, this row dimension associated with the FRF or IRF data matrix is assumed to be connected with the number of outputs (No ) that were measured. Since the data matrix (FRF or IRF) is considered to be symmetric or reciprocal, the data matrix can be transposed, switching the effective meaning of the row and column index with respect to the physical inputs and outputs. [ H(ω i ) ]No ×Ni = [ H(ω i ) ]TNi ×No
(3)
To eliminate possible confusion, in recent explanations of modal parameter estimation algorithms, the nomenclature of the number of outputs (No ) and number of inputs (Ni ) has been replaced by the length of the long dimension of the data matrix (NL ) and the length of the short dimension (NS ) regardless of which dimension refers to the physical output or input. This means that the above reciprocity relationship can be restated as: [ H(ω i ) ]NL ×NS = [ H(ω i ) ]TNS ×NL
(4)
Note that the reciprocity relationships in Equation 3 and 4 are a function of the common degrees of freedom (DOFs) in the short and long dimensions. If there are no common DOFs, there are no reciprocity relationships. Nevertheless, the importance of Equation 3 and 4 comes from the idea that the dimensions of the FRF matrix can
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501 be transposed and this affects the size of the square alpha coefficients in the matrix coefficient polynomal equation. Finally, once the modal frequencies and unscaled modal vectors are estimated via the eigenvalue-eigenvector problem, the residues (numerators) of the partial fraction model of the FRF data matrix are used to estimate the final, scaled modal vectors and modal scaling. Note that the unscaled modal vector found in the eigenvalueeigenvector problem is available to be used as a weighting vector in the estimation of the residues and, therefore, the final scaled modal vectors and modal scaling. Also note that this weighting vector may be of length equal to the long or short dimension, depending on the modal parameter estimation algorithm being used. [ H(ω i ) ]No ×Ni =
N
Σ r=1
[ Ar ]No ×Ni [ A*r ]No ×Ni 2N [ Ar ]No ×Ni + =Σ jω i − λ r jω i − λ *r r=1 jω i − λ r
(5)
This process means that most modern parameter estimation algorithms are implemented in a two stage procedure that has three steps as follows: Stage 1, Step 1 •
Load Measured Data into Over-Determined Linear Equation Form. •
Utilize Matrix Coefficient Polynomial Based Model (Equation 1).
•
Find Scalar or Matrix Coefficients ([α k ] and [ β k ]).
•
Implement for Various Model Orders (Consistency/Stability Diagram).
Stage 1, Step 2 •
Solve Matrix Coefficient Polynomial for Modal Frequencies (Equation 2). •
Formulate Eigenvalue-Eigenvector Problem.
•
Eigenvalues Determine the Modal Frequencies ( λ r ).
•
Eigenvectors Determine the Unscaled Modal Vectors ({ψ r }) of dimension NS or NL .
Stage 2, Step 3 •
Load Measured Data Into Over-Determined Linear Equation Form (Equation 5). •
Determine Modal Vectors and Modal Scaling from Residues.
The above procedure means that most modern modal parameter estimation algorithms can be summarized by the following Table: Domain Time Freq
Algorithm Complex Exponential Algorithm (CEA) Least Squares Complex Exponential (LSCE) Polyreference Time Domain (PTD) Ibrahim Time Domain (ITD) Multi-Reference Ibrahim Time Domain (MRITD) Eigensystem Realization Algorithm (ERA) Polyreference Frequency Domain (PFD) Simultaneous Frequency Domain (SFD) Multi-Reference Frequency Domain (MRFD) Rational Fraction Polynomial (RFP) Or thogonal Polynomial (OP) Polyreference Least Squares Complex Frequency (PLSCF)
• • • • • • • • • • • •
Matrix Polynomial Order Low High
Coefficients Scalar Matrix Basis
• • •
• •
• • •
• • •
• • • • • •
TABLE 1. Summar y of Modal Parameter Estimation Algorithms
NS × NS NL × NL NL × NL NL × NL NL × NL NL × NL NL × NL NS × NS NS × NS NS × NS
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502 2.1 Theoretical Issues The previous section overviews the overall procedure used by most modal parameter estimation algorithms. This procedure gives rise to two separate estimates of all or portions of the normalized modal vectors that are estimated. In general, only one estimate of each complete scaled modal vector and its associated modal scaling is desired. Since two independent, least squares or weighted least squares processes are involved using potentially different portions of the data matrix, there is no constraint that assures that the different estimates of the modal vectors will be the same. This is highlighted by the following theoretical and experimental discussions. The theoretical frequency response function (FRF) is directly related to the typical mass, damping and stiffness matrix relationships used to describe mechanical systems. This relationship for a large dimension, descretized system is normally represented by Equation 6. −1
[ H(si ) ]N∞ ×N∞ = ⎡ [ M ]N∞ ×N∞ si 2 + [ C ]N∞ ×N∞ si 1 + [ K ]N∞ ×N∞ si 0 ⎤ ⎣ ⎦
(6)
This inverse relationship indicates that the characteristics of the FRF matrix will take on the same properties as the combined mass, damping and stiffness matrices. This means that both matrix representations should yield the same (complex-valued) modal frequencies ( λ r ). Since the mass, damping and stiffness matrices will be symmetric or reciprocal for almost all mechanical systems, this also means that the FRF matrix will have the reciprocity property. Frequently, the damping matrix is assumed to be proportional damping, or Rayleigh damping, (which includes the trivial sub-case of zero damping) which will mean that the modal vectors can be normalized to be a set of real-valued coefficients, referred to as normal modes, which physically indicates that the motion of the masses are limited to in and out-of-phase relationships to one another. If the damping matrix does not follow this mathematical form, then the modal vectors cannot be normalized to a set of real-valued coefficients. For this case, some of the modal coefficients must always be complex-valued, physically indicating the motion of the masses may take on any phase relationship with respect to one another. Rather than relating the FRF matrix to unknown matrix coefficients like mass, stiffness and damping, a different equivalent representation of each term of the FRF matrix can be formulated. One common formulation is a partial fraction, pole-residue model that can be given by Equation 7. This partial fraction representation is ver y common in generalized descriptions of open and closed loop control systems as well as open loop structural dynamics systems. This partial fraction model form is the most frequently used model for estimating the scaled modal vectors once the complex-valued modal frequencies are known. Hpq (si ) =
N∞
Apqr
Σ r=1 si − λ r
+
A*pqr si −
λ *r
=
2N∞
Σ r=1
Apqr si − λ r
(7)
The above par tial fraction model for one FRF can be generalized into the matrix model in the following way: [ H(si ) ]N∞ ×N∞ =
N∞
Σ r=1
[ Ar ]N∞ ×N∞ si − λ r
+
[ A*r ]N∞ ×N∞ si −
λ *r
=
2N∞
Σ r=1
[ Ar ]N∞ ×N∞ si − λ r
(8)
In the above models, the complex-valued modal frequencies ( λ r ) are the same as the those in the original mass, damping and stiffness matrix system. The normalized modal vectors can be found from the numerators of the par tial fraction terms, referred to as the residues (Apqr ). Specifically the relationships between the residues, the normalized modal vectors and the modal scaling can be given by the following: Apqr = Qr ψ pr ψ qr
(9)
Note that in the above equation the scaling term (Qr ) is normally related to one of two different forms of scaling that accompanies the complete normalized modal vector ({ψ }r ). For the special case where the damping is propor tional, the relationship is: Qr =
1 j 2Mr ω r
(10)
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503 For the general case where the damping is either proportional or not, the relationship is: Qr =
1 MAr
(11)
For the above general case, note that the normalized modal vector cannot be a real-valued vector and must contain some complex-valued coefficients. With these relationships in mind, the partial fraction model can be manipulated into several other common forms of the equation that is used to estimate the complex-valued modal vectors once the modal frequencies are known. For example: ⎡ Qr ⎥ [H(si )]N∞ ×N∞ = [ψ ]N∞ ×2N∞ ⎢ [ψ ]T 2N∞ ×N∞ ⎥ s − λr i ⎢ ⎦2N∞ ×2N∞
(12)
2.2 Experimental Issues When experimental data is acquired, generally the data matrix will be rectangular limited by the number of inputs and outputs. Likewise, the data will be taken at discrete frequencies, generally equally spaced, along the frequency axis, not throughout the S − Domain as Equations 6-8 imply. Therefore, the above Equation 7 can be restated as follows: Hpq (ω i ) =
N∞
Σ
r=1
2N∞ A*pqr Apqr Apqr + = Σ * jω i − λ r jω i − λ r r=1 jω i − λ r
(13)
Equation 12 can then be restated as: ⎡ Qr ⎥ [H(ω i )]NL ×NS = [ψ ]NL ×2N ⎢ [ψ ]T 2N×NS ⎥ j ω − λ i r ⎢ ⎦2N×2N
(14)
⎡ Qr ⎥ [H(ω i )]NS ×NL = [ψ ]NS ×2N ⎢ [ψ ]T 2N×NL ⎥ j ω − λ i r ⎢ ⎦2N×2N
(15)
Or, in transposed form:
In order to clearly indicate how the unscaled and scaled modal vectors participate in the solution, the above two equations can be reformulated as follows: ⎡ ⎥ 1 T [H(ω i )]NL ×NS = [ψˆ ]NL ×2N ⎢ ⎥ ⎡ Qr ⎦2N×2N [ψ ] 2N×NS j ω − λ r ⎦ ⎢ i 2N×2N
(16)
⎡ ⎥ 1 T [H(ω i )]NS ×NL = [ψˆ ]NS ×2N ⎢ ⎥ ⎡ Qr ⎦2N×2N [ψ ] 2N×NL j ω − λ i r ⎢ ⎦2N×2N
(17)
The modal scaling (Qr ) and the modal vectors ({ψ }r ) are normally combined into a single scaled modal par ticipation vector ({L}r ) for each mode yielding: ⎥ ⎡ 1 [L ]T 2N×NS [H(ω i )]NL ×NS = [ψˆ ]NL ×2N ⎢ ⎥ j ω − λ i r ⎢ ⎦2N×2N
(18)
⎡ ⎥ 1 [H(ω i )]NS ×NL = [ψˆ ]NS ×2N ⎢ [L ]T 2N×NL ⎥ j ω − λ r ⎦ ⎢ i 2N×2N
(19)
BookID 214574_ChapID 46_Proof# 1 - 23/04/2011
504 In Equations 18 and 19, the number of modal frequencies (N), the modal frequencies ( λ r ) and the unscaled modal vectors ([ψˆ ]) are estimated in the first stage of the modal parameter estimaton procedure resulting from the eigenvalue-eigenvector problem. The modal participation vectors ([L ]) are estimated in the second stage of the modal parameter estimation procedure. Finally, the residues (Apqr ) with appropriate units (displacement/force × rad/sec) are estimated from the following relationship: Apqr = ψˆ pr Lqr
(20)
Modal scaling (Qr ) and the scaled modal vectors ([ψ ]) for the long dimension are then estimated appropriately from the residue vectors ({Ar }) reconstructed for the long dimension using the above equation. Finally, modal mass (Mr ) or modal A (MAr ) are estimated from scaled modal vectors ([ψ ]) and Equations 9 and 10 or Equations 9 and 11, if needed. Therefore, the normalized, scaled modal vectors can be different from the normalized, unscaled modal vectors for several reasons. First of all, if the data that is taken has noise, nonlinearities or is not reciprocal (or in other words has realistic problems), the data does not match the primar y assumptions relating the experimental data to the model. This may result in different vectors since there is a mismatch between the data and the model. Second, the unscaled modal vectors will generally be of different dimension (short versus long or long versus short) when compared to each other. If the unscaled modal vector is of long dimension, all or some subset of the unscaled modal vector can be used as weighting in the second phase when the scaled modal vector is estimated. The two different stages of the modal parameter estimation procedure do not constrain the unscaled and scaled modal vectors to have the same normalized shape. Since the unscaled modal vector may be completely, par tially or not involved in the weighting of the scaled modal vector, different scaled modal vectors may result. At the matching degree of freedoms, the modal coefficients should be the same if the system is reciprocal but this is not a constraint in the estimation process. 2.3 Residuals To account for the modal contribution represented by the modes with natural frequencies below and above the frequency range of interest (FMin to FMax ), it is common for residual terms to be included to account for the effect of these modes within the frequency range of interest. The basic partial fraction equation can be rewritten for a single frequency response function as:
Hpq (ω ) = RFpq + where: •
RFpq = Residual flexibility
•
RIpq (s) = Residual inertia
A*pqr Apqr + + RIpq (ω ) Σ jω − λ *r r = 1 jω − λ r n
(21)
The residual term that compensates for modes below the minimum frequency of interest is called the iner tia restraint, or residual inertia. The residual term that compensates for modes above the maximum frequency of interest is called the residual flexibility. These residuals are a function of each frequency response function measurement and are not global properties of the frequency response function matrix. One example of this common form of residuals is shown graphically in Figure 1 where the frequency band of interst is from 25 Hz. to 175 Hz. The blue curve represents the residual effect of the modes below the frequency band of interest (inertia restraint) and the red curve represents the residual effect of the modes above the frequency range of interest (residual flexibility). Note that in this example, both residuals will be dominantly real-valued.
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Phase (deg)
505
180 90 0 −90 −180 0 10
Magnitude (in/lb)
10
10
10
10
−1
−2
−3
−4
0
50
100
150 Frequency (Hz)
200
250
300
Figure 1. Graphical Example of Residual Contributions Note that residuals must be added for every measurement that participates in the weighted estimation of the scaled modal vectors. For cases where the unscaled modal vector is estimated for the long dimension in the first stage of the modal parameter estimation, this means that a large number of unknowns representing the residuals may need to be added when only a limited number of modal participation coefficients are needed for the short dimension. This can cause memory issues and may degrade the estimates of the scaled modal vectors. It should also be recognized that another minor source of unconstrained variability with respect to the scaled modal vectors comes from the choice of the residual model (one, two or more residual terms) and the frequency range of interest. Since residuals are effectively mathematical terms added to the equation set in an attempt to accommodate potential effects of out-of-band modes, except under specific circumstances, these terms do not have physical significance. It is also important to note that residuals are somtimes computed after the residues are estimated and have no affect on the residues at all. Fur ther, because any of the various residual models can be applied to each of the following scaled modal vector formulations, this paper is not going to explore the independent influence of residuals upon the resultant scaled modal vector quality. The reader interested in the [1,3-8] various residual models is encouraged to read some of the other papers on this topic . Note, however, that since the objective of this paper is to present a qualitative (not quantitative) comparison of the influence of solution technique upon the computed residues, residuals will be included in some solutions in an attempt to provide the best reasonable solution for each technique. 3. Possible Scaled Vector Formulations Based upon the previous discussions, there are at least five common implementations available for estimating the scaled modal vectors from a set of measured FRF data:
BookID 214574_ChapID 46_Proof# 1 - 23/04/2011
506 •
Method 1: Single or multiple reference FRF measurements with least squares estimation of residues, measurement by measurement, with or without residuals.
•
Method 2: Multiple reference FRF measurements with short dimension basis, weighted least squares solution for residues, with or without residuals.
•
Method 3: Multiple reference FRF measurements with short dimension basis (extracted from long dimension), weighted least squares solution for residues, with or without residuals.
•
Method 4: Multiple reference FRF measurements with long dimension basis, weighted least squares solution for residues, with or without residuals.
•
Method 5: Multiple reference FRF measurements with short or long dimension basis, weighted least squares solution for residues, with computational modes utilized as residuals.
Each of the above methods will give a slightly different result for a relatively consistent and noise free set of measured FRF data. If the FRF data has realistic problems caused by random or bias errors (noise), linearity, time var ying and/or reciprocity issues, the results may be significantly different. Another way of describing the various methods is in terms of how each method constrains the solution for the scaled modal vectors. In Method 1, the only constraint is that each FRF measurement is estimated with the same set of modal frequencies. In Method 2, the constraints include a common set of modal frequencies and the unscaled modal vectors of length equal to the short dimension. In Method 3, the constraints include a common set of modal frequencies and a set of unscaled modal vectors of length equal to the short dimension. However, in Method 3, these unscaled modal vectors are obtained by taking the short dimension subset of the DOF(s) from the computed long dimension, unscaled modal vectors. This method is often used to maintain similarity with Method 2 and to reduce the possibility of degradation of the solution when residuals are included. In Method 4, the constraints include a common set of modal frequencies and the unscaled modal vectors of length equal to the long dimension. This method works well when residuals are not needed but has some numerical issues when residuals are included. In Method 5, the unscaled modal vectors and scaled modal vectors are estimated for all possible eigenvalues of the long or short dimension problem before the set of modal frequencies are chosen. The characterisitics of the unscaled modal vectors and the scaled modal vectors along with the consistency/stability of all of the modal parameters are used to select the set of modal parameters that is optimum. While Method 5 is not a totally new approach, with the increase in available memory and compute speed, this method is now more viable. In order to compare the results for unscaled and scaled modal vectors estimated by different methods, a visual plot of the complex valued vectors in the complex plane is ver y useful. Figure 2 is an example of a comparison of the unscaled and scaled modal vectors. In order to quantify the characteristics of the vectors, the computation of [9-13] mean phase and mean phase deviation or mean phase correlation is often utilized .
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507 Mode # 1: 362.30 Hz 0.877% zeta MP :−1.0o MPD :4.87% | MP :88.9o MPD :6.08% U
U
S
S
1 0.8 0.6 0.4
Imaginary
0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1
−0.5
0 Real
0.5
1
Figure 2. Typical Complex Plot of Unscaled and Scaled Modal Vectors Mean Phase (MP or θ MP ): Mean phase is the average angle of the (modal) vector with respect to the real/imaginary axes. A mean phase of 00 means that the vector is oriented around the real axis; a mean phase of ± 900 means that the vector is oriented around the imaginary axis. The mean phase is computed by first solving the following eigenvalue problem: [Im({ψ }) Re({ψ })]T [Im({ψ }) Re({ψ })] {v} = λ {v}
(22)
Then selecting the eigenvector associated with the smallest eigenvalue and recognizing its for m: ⎧v1 ⎫ {v}λ min = ⎨ ⎬ ⎩v2 ⎭ MP = θ MP = tan−1
⎛ v2 ⎞ ⎝ −v1 ⎠
(23)
(24)
Note that the above relation returns the principal angle for the vector, bounded by ± 900 . If it is desired that the angle indicate the direction of largest response, it is necessar y to pay attention to the sign of the numerator and denominator of the inverse tangent function. In general, the mean phase is used to determine whether the modal vector is dominantly real-valued or imaginary-valued so this is not always done. Mean Phase Deviation (MPD): Mean phase deviation represents the scatter of the (modal) vector about the mean phase angle on a fraction or percentage basis. A mean phase deviation of 0.0 percent means that the vector is a normal mode oriented about (rotated to) the mean phase angle. A mean phase deviation larger than 0.0 percent indicates that the vector is a complex mode oriented about the mean phase angle. The mean phase deviation is computed by the following equation: ⎪⎪ ⎧ −j θ ⎫⎪⎪ ⎪⎪Im ⎨e MP {ψ r }⎬⎪⎪ ⎭⎪⎪ × 100 % MPD = ⎪⎪ ⎩ ⎪⎪ ⎧ ⎫⎪⎪ ⎪⎪Re ⎨e−j θ MP {ψ r }⎬⎪⎪ ⎪⎪ ⎩ ⎭⎪⎪
(25)
BookID 214574_ChapID 46_Proof# 1 - 23/04/2011
508 Mean Phase Correlation (MPC): Mean phase correlation is the representation of the mean phase deviation as a zero-to-one correlation coefficient, where a value of 1.0 means that the modal vector is a normal mode. The mean phase correlation is computed by the following equation: MPC = 1. 0 −
MPD 100
(26)
In Figure 2, the green o’s represent the plot of the coefficients of the unscaled modal vector while the blue x’s represent the plot of the scaled modal vectors. The mean phase and mean phase deviation of the unscaled and scaled modal vectors are given in the title of each mode vector plot. The vectors are normalized in every case so that the largest amplitude in each vector has a unity magnitude so that they can be plotted on the same scale. 4. Examples The figures present a set of examples comparing the qualitative impact of residue solution technique on the computed residue solutions. All figures have had the maximum value of the unscaled and scaled modal vector solutions normalized to one in order to present the results on a single graphical vector scatter diagram. In evaluating the results, the concentration is upon the qualitative change in the results; the specific absolute numerical values presented are not of primar y impor tance.
Complex Mode Indicator Function
−5
10
Magnitude
−6
10
−7
10
−8
10
500
1000
1500 Frequency (Hz)
2000
2500
Figure 3. CMIF, C-Plate Data Figure 3 is a Complex Mode Indicator Function (CMIF) plot of the FRF data for a 36 input (long dimension), 7 output (short dimension) matrix of FRF data [14-15] . This laborator y test structure will be used for most of the test case comparisons. The data from this test structure is particularly "clean" and yet actually exhibits some real world test issues like leakage and repeated roots (close modes).
BookID 214574_ChapID 46_Proof# 1 - 23/04/2011
509 Mode # 1: 362.33 Hz 0.876% zeta MP :−0.3o MPD :7.50% | MP :−97.3o MPD :8.03%
Mode # 1: 362.33 Hz 0.876% zeta MP :−0.3o MPD :7.50% | MP :−91.8o MPD :6.97% U
S
U
S
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
Imaginary
Imaginary
U
0
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−0.5
0 Real
0.5
S
S
0
−0.2
−1 −1
U
−1 −1
1
−0.5
0 Real
0.5
1
Figure 4. Vector Plots, PTD (Method 1, Different References), Phase 1 versus Phase 2, Mode 1
MP :0.3
o
Mode # 5: 557.04 Hz 0.524% zeta o MPD :0.69% | MP :88.9 MPD :0.63% U
S
MP :0.3
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−0.5
0 Real
0.5
1
Mode # 5: 557.04 Hz 0.524% zeta o MPD :0.69% | MP :88.8 MPD :0.56% U
S
S
0
−0.2
−1 −1
o
U
S
Imaginary
Imaginary
U
−1 −1
−0.5
0 Real
0.5
Figure 5. Vector Plots, PTD (Method 1, Different References), Phase 1 versus Phase 2, Mode 2
1
BookID 214574_ChapID 46_Proof# 1 - 23/04/2011
510 Mode # 11: 1222.98 Hz 0.335% zeta MP :177.9o MPD :7.11% | MP :86.5o MPD :13.69% U
S
Mode # 11: 1222.98 Hz 0.335% zeta MP :177.9o MPD :7.11% | MP :−89.8o MPD :13.76%
S
U
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
Imaginary
Imaginary
U
0
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−0.5
0 Real
0.5
−1 −1
1
S
S
0
−0.2
−1 −1
U
−0.5
0 Real
0.5
1
Figure 6. Vector Plots, PTD (Method 1, Different References), Phase 1 versus Phase 2, Mode 3
Mode # 13: 1224.06 Hz 0.323%o zeta MPD :16.40% | MP :−77.1 MPD :13.44%
Mode # 13: 1224.06 Hz 0.323% zeta o o MP :179.5 MPD :16.40% | MP :6.3 MPD :23.93%
o
MP :179.5
U
S
S
U
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
Imaginary
Imaginary
U
0
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−0.5
0 Real
0.5
1
S
S
0
−0.2
−1 −1
U
−1 −1
−0.5
0 Real
0.5
1
Figure 7. Vector Plots, PTD (Method 1, Different References), Phase 1 versus Phase 2, Mode 4 For Figures 4 to 7, all seven references were used in the first phase, Polyreference Time Domain (PTD), to identify the modal frequencies. Although this computed an unscaled (short dimension) modal vector, the weighting associated was not used to caluculate the residues. Instead, consistent with Method 1, using only the modal frequency information, the residues were computed from two different references independently. Comparing the results for the two different references shows that sometimes the results are quite similar, other times the results are significantly different in terms of mean phase and modal complexity, and occasionally, the results are totally distor ted and inaccurate.
BookID 214574_ChapID 46_Proof# 1 - 23/04/2011
511 Mode # 1: 362.28 Hz 0.876% zeta MP :−0.9o MPD :4.77% | MP :89.1o MPD :7.77%
Mode # 1: 362.30 Hz 0.877% zeta MP :−1.0o MPD :4.87% | MP :88.9o MPD :6.08% U
S
U
S
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
Imaginary
Imaginary
U
0
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−0.5
0 Real
0.5
−1 −1
1
S
S
0
−0.2
−1 −1
U
−0.5
0 Real
0.5
1
Figure 8. Vector Plots, PTD (Method 5 - left, Method 2 - right), Phase 1 versus Phase 2, Mode 1
o
MP :0.1
Mode # 5: 557.04 Hz 0.522%o zeta MPD :0.20% | MP :−91.2 MPD :0.81% U
S
o
MP :0.1
S
U
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
Imaginary
Imaginary
U
0
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−0.5
0 Real
0.5
1
U
S
S
0
−0.2
−1 −1
Mode # 5: 557.04 Hz 0.522%o zeta MPD :0.20% | MP :−91.2 MPD :0.81%
−1 −1
−0.5
0 Real
0.5
Figure 9. Vector Plots, PTD (Method 5 - left, Method 2 - right), Phase 1 versus Phase 2, Mode 2
1
BookID 214574_ChapID 46_Proof# 1 - 23/04/2011
512 Mode # 11: 1222.99 Hz 0.335% zeta MP :−1.2o MPD :3.81% | MP :−90.1o MPD :13.31% U
S
Mode # 11: 1223.01 Hz 0.335% zeta MP :−1.2o MPD :3.94% | MP :−90.3o MPD :13.52%
S
U
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
Imaginary
Imaginary
U
0
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−0.5
0 Real
0.5
−1 −1
1
S
S
0
−0.2
−1 −1
U
−0.5
0 Real
0.5
1
Figure 10. Vector Plots, PTD (Method 5 - left, Method 2 - right), Phase 1 versus Phase 2, Mode 3
o
MP :−4.8
Mode # 13: 1224.08 Hz 0.323% zeta o MPD :35.24% | MP :−76.4 MPD :13.47% U
S
o
MP :−4.8
S
U
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
Imaginary
Imaginary
U
0
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−0.5
0 Real
0.5
1
U
S
S
0
−0.2
−1 −1
Mode # 13: 1224.08 Hz 0.323% zeta o MPD :35.69% | MP :−76.3 MPD :13.68%
−1 −1
−0.5
0 Real
0.5
1
Figure 11. Vector Plots, PTD (Method 5 - left, Method 2 - right), Phase 1 versus Phase 2, Mode 4 For Figures 8 to 11, three of the potential references were used. Comparing the results of the Polyreference Time Domain (PTD) technique for Method 5 and Method 2, shows that the two techniques produce very similar results as expected since in both cases short dimension, unscaled modal vectors are computed and used to estimate the scaled (long dimension) vectors, however, having the long dimension information available for pole selection is advantageous (Method 5).
BookID 214574_ChapID 46_Proof# 1 - 23/04/2011
513 Auto MAC − pVectors 1
18
0.9 16 0.8 14 0.7
Mode #
12
0.6
10
0.5
8
0.4
6
0.3
4
0.2
2
0.1
2
4
6
8
10 Mode #
12
14
16
18
0
Figure 12. Auto-MAC, PTD, Unscaled Modal Vectors
Auto MAC − sVectors 1
18
0.9 16 0.8 14 0.7
Mode #
12
0.6
10
0.5
8
0.4
6
0.3
4
0.2
2
0.1
2
4
6
8
10 Mode #
12
14
16
18
0
Figure 13. Auto-MAC, PTD, Scaled Modal Vectors For the Polyreference Time Domain (PTD) technique which produces modal frequencies and a short dimension modal vector, comparing Figures 12 and 13 of the MAC for the unscaled (short dimension) and scaled (long dimension) vectors, demonstrates through the clarity of the MAC plot, the advantage of having the long dimension information available.
BookID 214574_ChapID 46_Proof# 1 - 23/04/2011
514 Mode # 1: 362.36 Hz 0.887% zeta MP :−0.3o MPD :9.04% | MP :−91.3o MPD :5.95%
Mode # 1: 362.36 Hz 0.883% zeta MP :0.0o MPD :7.83% | MP :−91.9o MPD :5.83% U
S
U
S
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
Imaginary
Imaginary
U
0
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−0.8
−0.6
−0.4
−0.2
0 Real
0.2
0.4
0.6
0.8
−1 −1
1
S
S
0
−0.2
−1 −1
U
−0.5
0 Real
0.5
1
Figure 14. Vector Plots, ERA (Method 5 - left, Method 4 - right), Phase 1 versus Phase 2, Mode 1
Mode # 5: 557.07 Hz 0.520%o zeta MPD :0.53% | MP :88.8 MPD :0.71%
o
MP :−0.0
U
S
MP :−0.0
S
U
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
Imaginary
Imaginary
U
0
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−0.8
−0.6
−0.4
−0.2
0 Real
0.2
0.4
0.6
0.8
1
U
S
S
0
−0.2
−1 −1
Mode # 5: 557.07 Hz 0.520%o zeta MPD :0.53% | MP :88.8 MPD :0.71%
o
−1 −1
−0.8
−0.6
−0.4
−0.2
0 Real
0.2
0.4
0.6
0.8
Figure 15. Vector Plots, ERA (Method 5 - left, Method 3 - right), Phase 1 versus Phase 2, Mode 2
1
BookID 214574_ChapID 46_Proof# 1 - 23/04/2011
515 Mode # 11: 1222.93 Hz 0.334% zeta MP :0.0o MPD :9.50% | MP :88.4o MPD :5.92% U
S
Mode # 11: 1222.96 Hz 0.338% zeta MP :−0.5o MPD :14.54% | MP :90.2o MPD :12.95%
S
U
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
Imaginary
Imaginary
U
0
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−0.8
−0.6
−0.4
−0.2
0 Real
0.2
0.4
0.6
0.8
−1 −1
1
S
S
0
−0.2
−1 −1
U
−0.5
0 Real
0.5
1
Figure 16. Vector Plots, ERA (Method 5 - left, Method 3 - right), Phase 1 versus Phase 2, Mode 3
o
MP :0.0
Mode # 13: 1224.12 Hz 0.321% zeta o MPD :9.96% | MP :−86.3 MPD :15.79% U
S
o
MP :−1.0
S
U
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
Imaginary
Imaginary
U
0
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−0.8
−0.6
−0.4
−0.2
0 Real
0.2
0.4
0.6
0.8
1
U
S
S
0
−0.2
−1 −1
Mode # 13: 1224.09 Hz 0.321% zeta o MPD :14.21% | MP :−85.1 MPD :16.08%
−1 −1
−0.5
0 Real
0.5
1
Figure 17. Vector Plots, ERA (Method 5 - left, Method 3 - right), Phase 1 versus Phase 2, Mode 4 Figures 14 to 17 show the results of using the Eigensystem Realization Algorithm (ERA) which produces long dimension, unscaled vectors. Comparing the Mean Phase Deviation (MPD) for the unscaled and the scaled modal vectors (right half of plots), shows that the computed (Method 3) scaled shape is slightly different than the first phase unscaled shape. Sometimes the computed shape is more normal, other times the computed shape is more complex. This demonstrates the unconstrained nature of the solution. Making the same comparison on the left half of the plot (Method 5 results), shows the same expected trend. However, since the scaled vector is now along the short dimension, the content of the short dimension vectors are the modal scaling parameters for the long dimension vector. Using this solution allows retention of the
BookID 214574_ChapID 46_Proof# 1 - 23/04/2011
516 viewed/computed first phase solution (long dimension, unscaled modal vector).
Complex Mode Indicator Function
−6
10
−7
Magnitude
10
−8
10
−9
10
−10
10
5
10
15
20 Frequency (Hz)
25
30
35
Figure 18. CMIF, Bridge Data
o
Mode # 1: 7.41 Hz 3.200% zeta o MPDU:1.98% | MPS:−95.8 MPDS:3.98%
o
MPU:−0.0 1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
Imaginary
Imaginary
MPU:0.0 1
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−1 −1
−0.8
−0.6
−0.4
−0.2
0 Real
0.2
0.4
0.6
0.8
1
Mode # 3: 8.25 Hz 2.617% zeta o MPDU:1.38% | MPS:−85.6 MPDS:20.70%
−1 −1
−0.8
−0.6
−0.4
−0.2
0 Real
0.2
0.4
0.6
0.8
1
Figure 19. Vector Plots, Bridge Data, RFPZ (Method 2), Mode 1 and Mode 2 Figures 18 and 19 are taken from a civil infrastructure and represent a complicated, real-world system. The dataset consists of 55 inputs (long dimension) and 15 outputs (short dimension). As can be seen in the CMIF plot, there are numerous modes and in some frequency bands there is high modal density which includes both
BookID 214574_ChapID 46_Proof# 1 - 23/04/2011
517 moderate and heavy damping. Also, Figure 15 shows that even though the short(er) dimension, unscaled modal vectors may be calculated to be nearly normal, when computing the full length, long dimension, scaled vectors, some modes are computed as nearly normal, while other modes are clearly significantly complex. 5. Conclusions This paper has explored the qualitative impact of several estimation procedures on the resulting scaled modal parameters (scaled modal vectors and modal scaling). The four traditional techniques (Methods 1-4) have been compared with a new approach (Method 5) which computes the scaled modal parameters concurrent with the first phase modal frequencies and unscaled modal vectors, while utilizing the first phase computational modes as the residuals. This approach has been shown to be advantageous in providing the analyst with full shape (long dimension) information for traditionally short dimension algorithms (eg. PTD, RFP, et.al.) and complete scaling information for all algorithms. By having such information available, the analyst can make more informed modal parameter selection decisions. The quality of the scaled modal vectors and modal scaling, calculated by this new procedure (Method 5), have been found to be comparable to those computed by traditional approaches (Methods 1-4) and, in most cases, render a secondary, independent estimation of the scaled modal vectors and modal scaling unnecessary. The authors believe that his may be ver y useful when applied to autonomic modal parameter estimation methods (wizard solutions) that are of great interest for modern, commercial modal parameter implementations. 6. References [1]
Allemang, R.J., Brown, D.L., "Modal Parameter Estimation" Experimental Modal Analysis and Dynamic Component Synthesis, USAF Technical Report, Contract No. F33615-83-C-3218, AFWAL-TR-87-3069, Vol. 3, 1987, 130 pp.
[2]
"Vibrations: Analytical and Experimental Modal Analysis", Allemang, R.J., UC-SDRL-CN-20-263-662, Revision 7, http://www.sdrl.uc.edu/academic-course-info/docs/ucme662, 223 pp., 2008.
[3]
"Vibrations: Experimental Modal Analysis", Allemang, R.J., UC-SDRL-CN-20-263-663/664, Revision 8, http://www.sdrl.uc.edu/academic-course-info/docs/ucme663, 421 pp., 2007.
[4] "Chapter 21: Experimental Modal Analysis", Shock and Vibration Handbook, Sixth Edition, Harris, C.M., Piersol, A.G. (Editors), Allemang, R.J., Brown, D.L., McGraw-Hill Book Company, 2009. [5]
"The Unified Matrix Polynomial Approach to Understanding Modal Parameter Estimation: An Update", Allemang, R.J., Phillips, A.W., Proceedings, International Conference on Noise and Vibration Engineering, Katholieke Universiteit Leuven, Belgium, 36 pp., 2004.
[6]
"A Unified Matrix Polynomial Approach to Modal Identification", Allemang, R.J., Brown, D.L., Journal of Sound and Vibration, Volume 211, Number 3, pp. 301-322, April 1998.
[7]
"Modal Parameter Estimation: A Unified Matrix Polynomial Approach", Allemang, R.J., Brown, D.L., Fladung, W., Proceedings, International Modal Analysis Conference, pp. 501-514, 1994.
[8]
"Application of a Generalized Residual Model to Frequency Domain Modal Parameter Estimation", Fladung, W.A., Phillips, A.W., Allemang, R.J., India-US 2001 Symposium, Emerging Trends in Vibration and Noise Engineering, 15 pp., 2001. Journal of Sound and Vibration, Vol. 262, No. 3, pp. 677-705, 2003.
[9] Samman, M.M., "A Modal Correlation Coefficient (MCC) for Detection of Kinks in Mode Shapes", ASME Journal of Vibration and Acoustics, Vol. 118, No. 2, pp. 271-271, 1996. [10] DeClerck, J.P., "Using Singular Value Decomposition to Compare Correlated Modal Vectors", Proceedings, International Modal Analysis Conference, pp. 1022-1029, 1998.
BookID 214574_ChapID 46_Proof# 1 - 23/04/2011
518 [11]
O’Callahan, J., "Correlation Considerations - Par t 4 (Modal Vector Correlation Techniques)", Proceedings, International Modal Analysis Conference, pp. 197-206, 1998.
[12]
Lallement, G., Kozanek, J., "Comparison of Vectors and Quantification of their Complexity", Proceedings, International Modal Analysis Conference, pp. 785-790, 1999.
[13] Fotsch, D., Ewins, D.J. "Applications of MAC in the Frequency Domain", Proceedings, Inter national Modal Analysis Conference, pp. 1225-1231, 2000. [14]
"Complex Mode Indication Function and Its Application to Spatial Domain Parameter Estimation", Shih, C.Y., Tsuei, Y.G., Allemang, R.J., Brown, D.L., -RXUQDORI0HFKDQLFDO6\VWHPVDQG6LJQDO3U RFHVVLQJ Academic Press Limited, Volume 2, Number 4, pp. 367-372, 1988.
[15]
"A Complete Review of the Complex Mode Indicator Function (CMIF) with Applications", Allemang, R.J., Brown, D.L., Proceedings, International Conference on Noise and Vibration Engineering (ISMA), Katholieke Universiteit Leuven, Belgium, 38 pp., 2006.
BookID 214574_ChapID 47_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Quantification of Aleatoric and Epistemic Uncertainty in Computational Models of Complex Systems Angel Urbina Sandia National Laboratories PO Box 5800, MS 0828 Albuquerque, NM 87185 Sankaran Mahadevan Vanderbilt University Box 1831, Station B Nashville, TN 37235
Nomenclature Klin Knon npow E
Linear stiffness of mechanical joint Non linear stiffness of mechanical joint Degree of non linearity of energy dissipation vs force relationship in a mechanical joint Modulus of elasticity, ksi
a max W
Peak absolute acceleration, g
X
f X x FX x M 1 ,M 1
N ai, bi
Standard deviation of the error term(s), where the error term is assumed normally distributed with a mean of zero and standard deviation, W Random variable representing a measure of behavior of the system Probability density function estimator Cumulative density function estimator st
Upper and lower bound on 1 moment of distribution Number of given intervals th Lower and upper limits, respectively, of i interval (i = 1…N)
ABSTRACT For complex engineering systems, testing-based assessment is increasingly sought to be replaced by simulations using detailed computational models. This is due to a lack of experimental data and/or resources to conduct these experiments at the system level. Components, which are part of the system, are usually cheaper to build and test relative to the system itself. The availability of component data coupled with the lack of system data and the complexity of the system being model leads to a need to build models in a building-block or hierarchical manner. This approach takes advantage of data at the component level by guiding the development of each component model. These models are then coupled to form the system model. Quantification of uncertainty in a system response is required to establish the confidence in representing the actual system behavior. To be accurate, this quantification needs to include both aleatoric uncertainty (due to natural variability) and epistemic uncertainty (due to lack of or incomplete knowledge). This paper proposes a framework based on Bayes networks that uses the available data at multiple levels of complexity (i.e. components, subsystem, etc) and allows quantification and propagation of both types of uncertainty in a system model prediction. A method to incorporate epistemic uncertainty given in terms of intervals on a model parameter is presented and a numerical example demonstrating the approach is shown.
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_47, © The Society for Experimental Mechanics, Inc. 2011
519
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520 Introduction The reliability of high consequence systems, such as aerospace components, has been traditionally established by testing individual systems and verifying their performance is within some acceptable limits. Although full scale testing is currently not feasible for some systems under actual use environments, some limited testing is often available for components, subsystems (i.e. groups of components) and a very limited number of tests of the full system in other use environments. Modeling and simulation attempts to fill the gap left by the lack of full scale testing at the system level for actual use environments. Because component level data are usually cheaper and easier to obtain relative to the system data, it is advantageous to have the ability to build individual models of the component and/or subsystems using available data and incorporate them into a system level model. This leads to a hierarchical approach to building system level models and consequently the uncertainty in the system model is a function of the component level data and of the knowledge not captured in the component or subsystem level data. Furthermore, because tests cannot be performed for many actual use environments, the model is required to extrapolate beyond the data it was developed from. To establish confidence in an extrapolated model prediction, sources of uncertainty must be identified, quantified and propagated to the response quantity of interest at the system model. Bayesian updating techniques and Bayes networks have been used to incorporate the available data, update the model parameters and consequently the model predictions, to reflect the new information that was previously not available for any individual level of complexity. The use of Bayes networks to incorporate various levels of data was demonstrated in [1] and subsequently applied to the example problem used in this paper in [2]. In these previous works, only aleatoric uncertainty (or variability) was considered. Before any ideas can be put forth to address the propagation of uncertainties in a hierarchical model when both aleatoric and epistemic uncertainties are present, it is worthwhile to define each type, see how they come about and how they have been treated. Aleatoric Uncertainty This type of uncertainty is also referred to as variability, irreducible uncertainty, inherent uncertainty or stochastic uncertainty. This is the type most commonly associated with variability due to hardware-to-hardware and experimental setup-to-setup variability of nominally identical systems. It is also associated with material properties data and loading data. In general, a statistically significant database, fully relevant to the application is available. For the most part, a probabilistic interpretation can be assigned to input and output variables and all the machinery associated with probability theory can be used to propagate and analyze this type of uncertainty. Techniques for quantification and propagation of this type of uncertainty have been well established for many years and therefore, will not be further examined in this research. Epistemic Uncertainty This type of uncertainty is also known as reducible uncertainty, subjective uncertainty or lack of knowledge. Some areas where this arises are when alternate plausible models are available, cases where there is non-existent, sparse, incomplete, or inconsistent experimental data, model approximations, expert elicitation that expresses subjective rather than data based on observations and, in general, where there is lack of information about the behavior of a system. It is found in the literature ([3, 4, 5]) that this type of uncertainty is treated by two types of methods: Non-Probabilistic Methods x Evidence (Dempster-Shafer) theory x Possibility theory x Fuzzy set theory x Interval analysis Probabilistic Methods x Bayesian approach or classical probability approach using transformations of bounds to probability density functions It is important to note that both types of approaches have their pros and cons and this paper will not make an attempt to demonstrate why one approach is better than another but simply use an approach that is more suitable to be implemented within the overall analysis framework. For a comprehensive review of how aleatoric and epistemic uncertainty can be treated, the proceedings from Sandia National Laboratories’ workshop on alternative representation of epistemic uncertainty [5] provide a comprehensive exposition of techniques to treat problems
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521 with two types of variables, one type described by probability distributions and the other type described by interval data. This forum presented various approaches, both probabilistic and non-probabilistic. Historically, probability theory has been used to represent epistemic uncertainty [6, 7] from which a separation of aleatoric and epistemic uncertainty involves two probability spaces, one for each type of uncertainty. However, many have expressed concern about modeling epistemic uncertainty via probability density functions with the main issue being the implication of a higher resolution of knowledge than what is really present [8]. Using probability theory to model epistemic uncertainty has its drawbacks but in the context of this work, it provides a feasible approach to incorporating this type of uncertainty into our Bayes network approach. In addition to this, by treating epistemic uncertainty in a probabilistic way, it allows for a single estimate of the probability of the system response quantity of interest (or probability of failure if required) which could facilitate the formulation of a risk informed decision making methodology. The current paper presents a methodology that model both aleatoric and epistemic uncertainty using a probabilistic framework and incorporates it into a hierarchically developed system model. In this paper, epistemic uncertainty due to data uncertainty is the main source of this type of uncertainty. Data uncertainty is treated in a probabilistic manner via a family of flexible distributions known as the Johnson distribution. Model error, a source of uncertainty, is quantified and discussed but its sources are not investigated. Example Problem Description The example problem depicted in Figure 1 was chosen to examine the effect of sources of aleatoric and epistemic uncertainty and the data and models from this will be used to develop a Bayes network representation of the full system.
Figure 1. Pictorial of the example problem This problem framework was developed at Sandia for the purpose of implementing the rigorous uncertainty quantification and model validation methodology developed there. The problem has the following features: x It is a multi-component problem where one branch of the problem involves a mechanical joint and the other is a type of encapsulating foam. Both are energy dissipating mechanism. x It is a multi-level problem where the phenomena observed at the lowest level is assumed to be present at subsequent levels, i.e. damping in the joints and foam is assumed similar at all levels. This might turn out to be an incorrect assumption.
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522 x x x
The individual component branches converge to a system level hardware where the two couple together. The interaction of the two physics is a potential source of epistemic uncertainty since no information of this coupled behavior is available. Experimental data consists of repeated tests on several, nominally identical hardware systems. These are intended to quantify the variability (aleatoric uncertainty) inherent in a physical system. Finite element models are built and calibrated to simulate a particular behavior of the physical hardware. The model parameters have been calibrated from simple, discovery experiments (shown as level 0) aimed at isolating the particular physical phenomenon that the model is trying to represent.
Several observations regarding the example problem and their potential implication are: x In general, there is no correspondence between the hardware that was tested at the various levels. For example, the single joint tested in level 1 is not part of the 3 leg system (at level 2). Similarly for the foam, the piece of foam in level 1 is not the same one (nor does it come from the same batch) as the foam in level 2. This issue could make relating data from one level to another difficult. x The type of data collected for the joints and foams are different. In general for the joints, time domain data is available and for the foam, frequency domain data is available. x No additional experiments can be performed nor can additional hardware be build. x Model runs can be made if necessary. Surrogate models, particularly of the system level, might be necessary to expedite the calculations The data and models available for this example are described in [2, 9]. Just to recall, the available data consisted of the following: Level 0: Material Parameter Characterization Foam: x Many encapsulating foam samples with a nominal density of 20 pcf were fabricated. x Physical measurements, tension/compression and torsional experiments were performed on the samples. x Material density, elastic and shear moduli were obtained. Joints: x A total of 45 experiments were conducted which consisted of 5 repetitions of the experiment for each of the nine leg configurations (The connection is shown in Figure 1-Level 0/Joints while mounted on an electro-magnetic shaker). x Experiments consisting of sine sweeps at various low-, medium-, and high-load levels were performed. x Data in the form of force versus energy dissipation were used to calibrate a constitutive model that seeks to represent the behavior of the physical joint. x Variability from sample to sample and test to test was quantified with these samples and tests. Levels 1 and 2: Sub-component and component level Foam: x 6 test samples and modal tests were conducted at level 1. x 3 test samples and modal testing was performed at level 2. x Material density is measured and modulus of elasticity is calculated for each test case at both levels. x Natural frequencies of the deformation in the axial direction were obtained for each experiment at each level. Joints: x 45 joint samples tested with an impulse type excitation were conducted at level 1. x 27 joint samples were tested using a wavelet type input excitation at level 2. x In both levels, acceleration time histories were recorded and energy dissipation was calculated for each experiment
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523 Uncertainty Quantification and Propagation We now look at quantifying and propagating both aleatoric and epistemic uncertainty. As described in the introduction, aleatoric uncertainty refers to variability in the response of a component arising from differences in the manufacturing process, assembly of parts, test conditions, etc. This is quantified by the use of multiple pieces of hardware and multiple test repetitions. Epistemic uncertainty arises from not knowing precisely where a parameter lies or having data to establish this. This is also referred to as data uncertainty. Data uncertainty is introduced with respect to the model parameters and arises from the fact that some materials used in engineering systems may not be fully characterized. Usually their parametric description is given in terms of intervals defined by subject matter experts and/or by very limited information but not by full probability distributions. Formally, References 10 through 12 list eight sources from which information is best represented by intervals, including plus-or-minus reports, significant digits, intermittent measurement, non-detects, missing data and gross ignorance. Intervals are obtained by either examination of the limited information available (enough to establish bounding information but not a full probabilistic representation) or by eliciting information from subject matter experts. For the case of expert opinion, one should also consider the relative weight that each expert’s opinion carries and it should be incorporated into the uncertainty analysis. This assessment of weight is, for the most part, a subjective endeavor and a future work topic. In this paper, the parameter describing the foam behavior will be treated as a source of epistemic uncertainty and is assumed to be given as intervals by subject-matter experts. (For demonstration purposes, it is assumed that no data is available, even though data has been collected). The approach taken in this paper uses a transformation of the data uncertainty into a probabilistic description and it is detailed in [13]. This approach starts by estimating the bounds on the first statistical moment of a distribution given the intervals on the parameter of interest. These moments are used to estimate the parameters of the Johnson family of distribution by one of several methods. One of these methods will be described in a subsequent section. Using the hardware and test data collected along with their corresponding finite element models and/or surrogate models, a methodology to quantify and propagate sources of uncertainty using Bayes networks has been developed and details of that work can be found in [2]. From [2], the resulting Bayes network for the example problem is shown in Figure 1.
Figure 2. Bayes network for problem shown in Figure 1
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524 Note that the error nodes, H shown in Figure 2 are associated with the discrepancy in the model predictions relative to the observed data. This error term has a normal distribution with zero mean and some standard deviation, W which will be updated. In the context of this paper, these error terms will be interpreted as sources of uncertainty. The main reason behind this is in the way the network is set up. There is a conditional probability that relates the model at each level back to a set of model parameters denoted by T . These model parameters are updated based on the observed data of the various levels. It is accepted that the models and its parameters can only account for some of the behavior in observed data (i.e. no model is a perfect representation of the underlying phenomena). As a matter of fact, it should be imposed that not all the observed behavior be contained within these parameters. It is generally agreed that sources of difference can come from various sources, such as the model representation of the actual hardware (i.e. dimensions, boundary conditions, contact surfaces, etc), the inability to exactly represent mathematically the dominant physical phenomenon, and model convergence issues. To account for these differences, an error term is included in the formulations. This formulation was proposed by Kennedy and O’Hagan [19]. In the intended use of a Bayes network, one would like to calculate the probability density function (PDF) associated with some nodes of interest in the network. To do this, it is known that each parent node has a PDF associated with it and each child node has a conditional probability density function, given the value of the parent node. The entire network can be represented using a joint probability density function which is given by the general expression:
ܲ(ܷ) = ෑ ܲ൫ܺ݅ ห) ݅ܺ(ݐ݊݁ݎܽ൯
(1)
݅ th
where P(U) is the joint probability of the Bayes network and Xi are the i nodes of the Bayes network [14]. From Figure 1, one formulates the joint probability density function of the entire network, U as: ݂
݂
݂
݂
݂
݂(ܷ) = ݂൫ߠ ݂ ൯ ݂ כ൫ܺ1 หߠ ݂ ൯ ݂ כ൫ܻ1 หܺ1 ൯ ݂ כ൫ܺ1 ห߳1 ൯ כ ݂
݂
݂
݂
݂
݂൫ܺ2 หߠ ݂ ൯ ݂ כ൫ܻ2 หܺ2 ൯ ݂ כ൫ܺ2 ห߳2 ൯ כ ݆
݆
݆
݆
݆
݂൫ߠ ݆ ൯ ݂ כ൫ܺ1 หߠ ݆ ൯ ݂ כ൫ܻ1 หܺ1 ൯ ݂ כ൫ܺ1 ห߳1 ൯ כ ݆
݆
݆
݆
(2)
݆
݂൫ܺ2 หߠ ݆ ൯ ݂ כ൫ܻ2 หܺ2 ൯ ݂ כ൫ܺ2 ห߳2 ൯ ݂ כ൫ܺ ݏหߠ ݂ , ߠ ݆ ൯ The marginal PDF of any node in the Bayes network can be obtained by the integration of the joint PDF (shown in Equation (2)) over all the values of the remaining variables. Thus the Bayes network approach offers a rational and effective methodology to extrapolate inferences from component level information to the system level, as long as the two levels have common, linking nodes. The network also facilitates the inclusion of new nodes that represent the observed data and thus the updated densities can be obtained for all the nodes. The joint probability density function for the network can be updated using Bayes theorem when data is available. The expression for Bayes theorem is:
݂ߠ (ߠ|ܻ) =
݂ߠ (ߠ)݂ߠ {ܻ|ܺ(ߠ)} ߠ݀ })ߠ(ܺ|ܻ{ ߠ݂)ߠ( ߠ݂
(3)
In this expression, the ݂ߠ (ߠ) represents the prior distribution of a parameter(s) of interest given by ߠ which will be updated with the available data. The parameters to be updated are: Joints parameters, ߠ ݆ (from model developed by Smallwood [18]), : x Klin - Linear stiffness, x Knon - Non linear stiffness, x npow - Degree of non linearity of energy dissipation vs force relationship. Foam parameter, ߠ ݂ :
BookID 214574_ChapID 47_Proof# 1 - 23/04/2011
525 x
E - Modulus of elasticity.
Error terms, ߳Ԣݏ: x W - Standard deviation of the error term(s), where the error term is assumed normally distributed with a mean of zero and standard deviation, W The prior distributions of these parameters represent the current knowledge about these parameters expressed as probability distributions with statistics obtained from the level 0 data, in the case of the joint parameters. How to obtain the prior distributions for the foam parameter will be discussed in the following section. The second term in the numerator in Equation (3) is the likelihood function which relates available data back to the parameters of the model. In this work, we use a Normal distribution for the likelihood function. The integration in the denominator of Equation (3) can be conveniently done using Markov Chain Monte Carlo (MCMC) techniques [15]. An MCMC solution to the Bayes network shown in Figure 2 was found using the software WinBUGS [16], a free Bayesian inference software. To simplify and expedite the implementation of the Bayes network for the sample problem, Gaussian process (GP) models were developed and used in lieu of the full 3 dimensional finite element models that represent the hardware at each of the levels. The GP software used in this study was coded by McFarland and Bichon [17]. One of the main tasks to implement this was to make code written in Matlab and Fortran available to WinBUGS which does not allow external calls to other programs; therefore, the GP code was implemented as a function written in Component Pascal and compiled directly into the WinBUGS software. This allowed the software to make the necessary function evaluations which related the model prediction to the calibration parameters and allowed for the updating of the parameters within a Bayesian framework. Treatment of interval data: Methodology and Implementation This section considers a methodology to handle interval data in the parameters of a model. Since the objective is to treat the entire problem in a probabilistic way, the objective will be to use the available interval information and assign a reasonable probability distribution that incorporates the given information (i.e. bounds) and also does not add too much subjective information. For this paper, the intervals on the modulus of elasticity, E given by six subject matter experts are given as: Table 1. Lower and upper limits for modulus of elasticity, E, from six experts Expert
Lower Limit (ksi)
Upper Limit (ksi)
1
32
60
2
35
68
3
40
72
4
42
78
5
48
82
6
50
94
It is assumed that the foam characterization data (collected at level 0) is not available and it is replaced by the experts’ interval data shown in the table above. For comparison purposes, the range of E obtained from the experimental data is between 20 ksi and 70 ksi whereas the experts’ range from 32 ksi to 94 ksi. The range of values given by the experts are their best estimate of the parameter E at the system level and does not necessarily need to coincide with the available experimental data. For the joints, the level 0 data is still available and thus can be used to construct probability density functions of the parameters of the joint model.
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526 Johnson Family of Distributions As a basis to construct a probabilistic representation of the foam modulus of elasticity, the Johnson family of distribution [20] is used in this research. The reason for choosing Johnson family of distributions is that it is a flexible set of probability distributions capable of representing a wide array of conventional probability distributions [13]. If X is a continuous random variable with distribution function F ( x) P ( X d x) , Johnson [20] proposed four normalizing translations of the general form:
ܺെߦ ܼ = ߛ +ߜ ݃כ൬ ൰ ߣ
(4)
where Z is a standard normal random variable, J and G are shape parameters, O is a scale parameter, location parameter and g(.) is a function that defines the four distribution families as:
ln()ݕ ۓ ۖln ቀ ݕ+ ඥ( ݕ2 + 1)ቁ ݃(= )ݕ ۔ln(ݕΤ(1 െ ))ݕ ۖ ݕە
[
is a
for SL (lognormal) family for SU (unbounded) family for SB (bounded) family for SN (normal) family
(5)
If sample data is available, choosing which family of distributions to use is accomplished by the following procedure [21]: 1. Estimate the first four central moments, m1, m2, m3, and m4.of the sample data, X as:
݉1 )ܺ(ܧ ؠ ݉݇ ܺ(ܧ ؠെ ݉1 )݇
(6)
݇ = 2,3,4
(7)
where E(.) is the expected value of the quantity inside the parenthesis. 2. Calculate the skewness and kurtosis:
ߚ1 ݉ ؠ32 Τ݉23
(8)
ߚ2 ݉ ؠ4 Τ݉22
(9)
3. Use the identification chart in Figure 3 to determine the appropriate distribution family to use.
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527
Figure 3. Johnson distribution family identification chart 4. To fit the distribution parameter O , [ , J and G in Equation (4) to one of the family of distributions shown in Equation (5), the following methods could be used: x method of matching moments (using the first four moments of the data), x percentile matching (where a desired value is specified at a given percentile point(s)) x least squares estimation and x minimization of the error norm of Johnson distribution when compared with empirical cumulative distribution function (CDF). In the chart, SU represents an unbounded distribution (support is ±λ) and SB is a bounded distribution. The bounded family of distributions will be used for the problem examined in this paper for the simple reason that the physics of the problem establishes natural bounds on the variable being modeled. That is, the modulus of elasticity has to be greater than zero and it has a physical upper limit. To use the Johnson family of distributions when interval data is available and when a moment based approach is desired, the process outlined above can still be applied except that instead of calculating moments from sample data, the moments would come from sampling within the bounds on the moments using the formulas presented in [13]. One important issue to keep in mind when evaluating the various techniques to describe the epistemic variable is that each of these methods only produces a prior distribution for a given parameter. This prior distribution will be updated with any available data through the use of the Bayes network construct which was described above. When sufficient data is incorporated into the analysis during the Bayesian updating process, the posterior probability distribution tends to have diminishing dependence on the prior distribution, regardless of the form of the prior. It is important to note that the priors should be defined over a range that includes values of the likelihood function (i.e. where data is available); otherwise, the resulting posterior distribution will resemble a delta function. Method of percentile matching and mean bounding This method starts with the bounding information given by subject experts and computes the empirical cumulative distribution function (ECDF) of the experts’ lower limit specifications and the ECDF of the experts’ upper limit specifications. The ECDF is a cumulative distribution that concentrates probability 1/n at each of the n numbers in
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528 a sample [22, 23]. Let x1 … xn be independent and identically distributed (iid) random variables with the cumulative distribution function (CDF), F(x). The empirical distribution function Fn(x) based on sample x1 … xn is the step function defined by:
= )ݔ( ݊ܨ
number of elements in the sample ݔ ݊ (10)
݊
1 = ݅ݔ(ܫ )ݔ ݊ ݅=1
where I(xi [) is the indicator of the event in parenthesis. Once the two bounding ECDFs are calculated, the 10 th and 90 percentile points are obtained from each. These are shown in Figure 4 by red asterisks.
th
Figure 4. ECDFs for lower and upper bounds on E and 10th and 90th percentile points A procedure to generate realizations of the parameters of a bounded Johnson distribution that is based on the experts’-given bounds shown in Table 1 and enforces the ECDFs shown in Figure 4 is described below: 1. Start by constructing the bounding ECDFs from the experts’ bounds and identifying their corresponding th th 10 and 90 percentile points. This is shown in Figure 4. The percentile points are tabulated below. th
th
Table 2. Lower and upper bounds on the 10 and 90 percentile points based on ECDFs of experts' given intervals Lower Bound
Upper Bound
th
32
60
th
50
94
10 Percentile point 90 Percentile point
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529 th
th
2. Assume that the 10 and 90 percentile points follow a uniform distribution with limits given by the lower th th and upper bounds on the 10 and 90 percentile points (shown in Table 2). These distributions are shown in Figure 5.
th
th
Figure 5. Uniformly distributed 10 percentile (upper graph) and 90 percentile (lower graph) with limits from ECDFs from experts’ given bounds th
th
3. From the distributions shown in Figure 5, sample n realizations of the 10 and 90 percentile values and denote these as:
(10th percentile)݅ ࢻ=ቌ ڭ (10th percentile)݊
(90th percentile)݅ ቍ ڭ (90th percentile)݊
݅ = 1…݊
(11)
4. Now, from the form of the bounded Johnson distribution given in Equations (4) and (5) and using the given interval data, the parameters O and [ can be established.
[ 32 O 62 st
5. Now use the method of percentile matching and the bounds on the 1 moment to estimate the remaining parameters, G and J . This is done using an optimization formulation as shown below: Find ߜ, ߛ with
ܺ = ߦ + ߣ ݂ כെ1 ൬ where
ܼെߛ ൰, ߜ
ߦ < ܺ <ߦ+ߣ
(12)
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530
݂ െ1 (= )ݖ
1 (1 + ݁ െ) ݖ
(13)
subject to:
ߙ(݅, 1) = ߙො1 ߙ(݅, 2) = ߙො2
(14)
ܯ1 ܮ ݉ ෝ 1 ܯ1ܷ where
݉ ෝ 1 = ݔ(ܧെ ߤ) ݔ where
D i,1 : 2
are the rows of Į in Equation (11), Dˆ j , j
(15)
1, 2 are 10th, 90th percentile values and mˆ 1 is the
st
1 moment found from sampled data taken from a bounded Johnson distribution with parameters and M 1L
O , [ , J and G
st
and M 1U are the lower and upper bounds on the 1 moment respectively. The bounds on the 1st moment
are obtained with the following formulation [10, 11] and from the given intervals, the values can be calculated: ܰ
ܰ
݅=1
݅=1
1 1 ൣܯ1 , ܯ1 ൧ = ܽ݅ , ܾ݅ ൩ ܰ ܰ
(16)
ൣܯ1 , ܯ1 ൧ = [41.2, 75.7] The procedure was implemented in Matlab and a probabilistic description of the modulus of elasticity of foam is obtained. 30 realizations of the PDF of E are show in Figure 6. The PDFs shown in Figure 6 show good coverage of the entire range of E and seem plausible realizations of the probability density for this variable.
Figure 6. PDFs of foam modulus of elasticity using percentile matching method and level 0 data – These are the prior distributions of E
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531 To verify that the algorithm is working correctly, a plot of the CDFs for each of the PDFs shown in Figure 6 are th th th shown with their 10 and 90 percentile values along with the ECDFs and the corresponding bounding 10 and th 90 percentile values. These quantities are shown in Figure 7. As expected, the ECDFs bound the realizations of the CDFs from the bounded Johnson distributions.
Figure 7. Estimated 10th and 90th percentile points from 30 generated CDFs The final check on this methodology is to ensure that the first moments of the resulting PDFs fall within the calculated bounds on the first moments based on the expert’s intervals and shown in Equation (16). This is st confirmed in Figure 8. The 1 moment of the distributions obtained with the percentile matching method span the st range defined by the bounds on the 1 moment obtained from the interval data. The realizations of PDFs for the st variable E appear plausible and fall within the prescribed criteria for this method (percentiles matched and 1 moments within the bounds).
st
st
Figure 8. 1 moment of PDFs shown in Figure 6 and bounds on 1 moment from experts’ given bounds
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532 The priors shown in Figure 6 are now propagated through the Bayes network, updated via Markov chain Monte Carlo simulation, and used to obtain samples of the posterior distribution of E. These are then used in the forward propagation to a prediction of the system level response. This sequence is plotted in Figure 9 through Figure 11. To visualize the resulting distributions, a kernel density estimator (KDE) [24] of the computed samples is plotted. A KDE is an approximation to the probability density function (PDF) of a source of values of s and it is computed from n data realizations, s (.) j j 1...n . The form of the KDE used here is: ݊
݂መ= )ߙ( ݏ
1 1 1 2 ݁ ݔെ 2 ൫ߙ െ ( ݆)ݏ൯ ൨ 2߳ ݊ ξ2ߨ߳
െλ<ߙ <λ
(17)
݆ =1
where ߳ is the “width” of a Gaussian kernel.
Figure 9. KDEs of posteriors of foam modulus of elasticity, E, updated with level 1 and level 2 data (Note: Prior PDFs of E, based on level 0 data, are shown in Figure 6) For completenss, the prior and posterior distributions of two of the joint parameters, Klin and npow are shown in Figure 10. The posterior distributions show very little difference from the prior distribution which means that the data used for updating does not add significant amounts of information to the original knowledge (i.e. the prior distribution obtained from level 0 data only).
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533 -5
f(Klin)
1
x 10
0.5 0
1.8
1.9
2
2.1 Klin
2.2
2.3
2.4
2.5 6
x 10
npow
6 4 2 0 0.8
1
1.2
1.4
1.6
1.8
npow Figure 10. KDEs of prior (red line) and posterior (cyan lines) distributions of two of the joints parameters The system level prediction shown in Figure 11 demonstrates the effect of the varying posterior PDFs of the foam modulus of elasticity, E and the propagation of the uncertainty due to the joints parameters. For comparison purposes, the forward prediction obtained using the prior distributions for the foam and joint parameters is also shown in Figure 11. It is clear that a decrease in the level of uncertainty in the system level response denoted by a narrower probability density function is accomplished by introducing data to update the prior knowledge.
Figure 11. KDEs of peak acceleration from predicted responses of system level structures
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534 Summary In this paper, the quantification and propagation of aleatoric and epistemic uncertainty through a hierarchically build model of an aerospace component was presented. Epistemic uncertainty is given in terms of intervals on a model parameter and it is treated in a probabilistic manner. This treatment of interval data enables the use of Bayesian techniques to quantify and propagate uncertainty. In the context of this paper, the modulus of elasticity of the foam material was given in terms of intervals (presumably given by subject matter experts) which in essence bound the “true” value of the modulus of elasticity. To model the interval data, the Johnson family of distributions is used to construct a probability model of the given intervals. For a bounded Johnson distribution, two parameters need to be estimated and this is accomplished by solving an optimization problem. The constraints on this problem are given by: th th 1. matching a set of 10 and 90 percentile values chosen from within the bounds defined by the given intervals and st 2. matching a set of 1 central moments of the distribution chosen from within the bounds on this central moment as calculated from the given intervals. The methodology shown in this paper allows the construction of plausible prior distributions for the modulus of elasticity of foam given the interval information. Prior distributions for the joint model parameters were obtained from a collection of simple experiments and thus this is treated as an aleatoric uncertainty. A Bayes network was used to incorporate experimental data from different levels of complexity and to propagate uncertainty to the system level model. Results obtained from propagating both epistemic and aleatoric uncertainty using a Bayes network were shown. Surrogate models were used to expedite the updating of the Bayes network without loosing much accuracy in the results. From the results, the effect of epistemic uncertainty is reflected as a scatter in the posterior distributions obtained for the system level response of interest. The effect of introducing data into the analysis is shown as a reduction in the variability in the system response’s PDF relative to the ones obtained with just the priors. The effect of the aleatoric part is also reflected in the individual posterior distributions at the system level. It is interesting to note that unlike a traditional analysis when only aleatoric uncertainty is considered, an analysis with both epistemic and aleatoric uncertainty leads to a collection of plausible results. This is a complicated factor and it is the subject of current research. Acknowledgments Sandia is a multi-program laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under Contract DE-AC0494AL85000. References 1. Rebba, R., Model Validation And Design Under Uncertainty, Vanderbilt University, Nashville, TN, 2005. 2. Urbina, A. and Mahadevan, S., “Uncertainty Quantification in Hierarchical Development of Computational th Models”, Proceedings of the 50 AIAA Structures, Structural Dynamics, and Materials Conference, 2009. 3. Oberkampf, W.L., DeLand, S.M., Rutherford, B.M., Diegert, K.V., and Alvin, K.F.,“Estimation of total uncertainty in modeling and simulation”, SAND 2000-0824, Sandia National Laboratories, Albuquerque, NM, 2000. 4. Sentz, K and Ferson, S., “Combination of evidence in Dempster-Shafer theory, SAND 2002-0835, Sandia National Laboratories, Albuquerque, NM, 2002. 5. Reliability Engineering and System Safety (RESS), Alternative Representation of Epistemic Uncertainty, J.C. Helton and W.L. Oberkampf, guest editors, Vol. 85. Nos. 1-3, July-September, 2004. 6. Apostolakis, G., “The Concept of Probability in Safety Assessments of Technological System”, Science, Vol. 250, No. 4986, pp. 1359-1364, 1990. 7. Parry, G.W., and Winter, P.W., “Characterization and Evaluation of Uncertainty in Probabilistic Risk Analysis”, Nuclear Safety, Vol. 22, No. 1, pp. 28-42, 1981.
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535 8. Helton, J.C., Johnson, J.D., Oberkampf, W.L. and Storlie, C.B., “A Sampling-Based Computational Strategy for the Representation of Epistemic Uncertainty in Model Predictions with Evidence Theory”, SAND20065557, Sandia National Laboratories, Albuquerque, NM, 2006. 9. Urbina, A., Paez, T.L., Gregory, D., Resor, B., Hinnerichs, T.D. and O’Gorman, C.C, “Validation of a Combined Non-Linear Joint and Viscoelastic Encapsulating Foam”, Proceedings of the 2006 Society for Experimental Mechanics, St. Louis, MO, 2006. 10. Ferson, S., Kreinovich, V., Hajagos, J., Oberkampf, W., and Ginzburg, L., Experimental Uncertainty Estimation and Statistics for Data Having Interval Uncertainty, SAND2007-0939, Sandia National Laboratories, Albuquerque, NM, 2007. 11. Ferson, S., R.B. Nelsen, J. Hajagos, D.J. Berleant, J. Zhang, W.T. Tucker, L.R. Ginzburg and W.L. Oberkampf, Dependence in Probabilistic Modeling, Dempster-Shafer Theory, and Probability Bounds Analysis. SAND2004-3072, Sandia National Laboratories, Albuquerque, New Mexico, 2004. 12. Osegueda, R., V. Kreinovich, L. Potluri, R. Aló, Non-destructive testing of aerospace structures: granularity and data mining approach. Pages 685-689 in Proceedings of FUZZ-IEEE 2002, Vol. 1, Honolulu, Hawaii, 2002. 13. McDonald, M., Zaman, K., Rangavajhala, S., Mahadevan, S., “A probabilistic approach for representation of interval uncertainty”, under review, Reliability Engineering and Systems Safety, 2009. 14. Jensen, Finn V., Bayesian Networks and decision graphs, Springer-Verlag, New York, 2001. 15. Gilks, G. R., Richardson, S., and Spiegelhalter, D. J., Markov Chain Monte Carlo in Practice, Interdisciplinary Statistics, Chapman & Hall/CRC, London, 1996. 16. Spiegelhalter, D. J., Thomas, A., Best, N. G. and Lunn, D., WinBUGS User Manual Version 1.4. Cambridge, U.K.: MRC Biostatistics Unit, [Online], Available: http://www.mrc-bsu.cam.ac.uk/bugs 2003. 17. McFarland, J.M., Uncertainty analysis for computer simulations through validation and calibration, Vanderbilt University, Nashville, TN, 2008. 18. Smallwood, D., Gregory, D., Coleman, R., “Damping Investigations of a Simplified Frictional Shear Joint,” Proceedings of the 71st Shock and Vibration Symposium, SAVIAC, The Shock and Vibration Information Analysis Center, 2000. 19. Kennedy, M.C. and O’Hagan, A., “Predicting the output from a complex computer code when fast approximation are available”, Biometrika, Vol. 87, 1, pp. 1-13, 2000. 20. Johnson, N.L., “Systems of frequency curves generated by methods of translation”, Biometrika, 36:149-176, 1949. 21. DeBrota, David J., Swain, James J., Roberts, Stephen D., Venkataraman, Sekhar., “Input modeling with the Johnson System of distributions,” Proceedings of the 1988 Winter Simulation Conference, 1998. 22. Cox, D.R. and D. Oakes, Analysis of Survival Data, Chapman & Hall, London, 1984. 23. The Mathworks, Inc, “Matlab: On-line documentation”, Natick, Massachusetts, 2009. 24. Silverman, B. W., Density Estimation for Statistics and Data Analysis, Chapman and Hall, London, 1986.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
The Dangers of Sparse Sampling for Uncertainty Propagation and Model Calibration François M. Hemez 1
H. Sezer Atamturktur2
Los Alamos National Laboratory, X-Division (X-3), Los Alamos, New Mexico 87545
Clemson University, Civil Engineering Department, Clemson, South Carolina 29634
ABSTRACT: Activities such as sensitivity analysis, statistical effect screening, uncertainty propagation, or model calibration have become integral to the Verification and Validation (V&V) of numerical models and computer simulations. Because these analyses involve performing multiple runs of a computer code, they can rapidly become computationally expensive. For example, propagating uncertainty with a 1,000 Monte Carlo samples wrapped around a finite element calculation that takes only 10 minutes to run requires seven days of single-processor time. An alternative is to combine a design of computer experiments to meta-modeling, and replace the potentially expensive computer simulation by a fast-running surrogate. The surrogate can then be used to estimate sensitivities, propagate uncertainty, and calibrate model parameters at a fraction of the cost it would take to wrap a sampling algorithm or optimization solver around the analysis code. In this publication, we focus on the dangers of using too sparsely populated design-of-experiments to propagate uncertainty or train a fast-running surrogate model. One danger for sensitivity analysis or calibration is to develop meta-models that include erroneous sensitivities. This is illustrated with a high-dimensional, non-linear mathematical function in which several parameter effects are statistically insignificant, therefore, mimicking a situation that is often encountered in practice. It is shown that using a sparse design of computer experiments leads to an incorrect approximation of the function. (Publication approved for unlimited, public release on November 4, 2009, LA-UR-09-7227, Unclassified.)
1. INTRODUCTION Design-Of-Experiments (DOE) have become common to support activities of modeling and simulation such as sensitivity analysis, statistical effect screening, uncertainty propagation, and model calibration. With a DOE, one selects a sequence of computer runs defined by varying the input variables of a model such that specific input-output effects can be identified [1-2]. Once the model evaluations have been completed, the output predictions are often used to best-fit a fast-running surrogate model that replaces the computationally expensive simulation. Studies performed at the Los Alamos National Laboratory where this technology is used include the validation of a finite element model developed to simulate the structural response of a threaded assembly to impulse loading [3-5]; quantification of prediction uncertainty for a multiphysics code that simulates casting processes [6]; and other multi-physics applications [7]. More recently, DOE and surrogate modeling have been used to quantify the statistical consistency between populations of measurements and predictions and to facilitate the calibration of model 1
Technical Staff Member, X-Division (X-3). Mailing: Los Alamos National Laboratory, X-3, Mail Stop F644, Los Alamos, New Mexico 87545, U.S.A. Phone: 505-667-4631. Fax: 505-667-3726. E-mail:
[email protected]. 2 Assistant Professor, Civil Engineering Department. Mailing: Clemson University, Civil Engineering Department, 110 Lowry Hall, Box 340911, Clemson, South Carolina 29634-0911, U.S.A. Phone: 864-656-3000. Fax: 864-656-2670. Email:
[email protected]. Currently, long-term visiting faculty staff member at the Los Alamos National Laboratory, Los Alamos, New Mexico 87545, U.S.A. T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_48, © The Society for Experimental Mechanics, Inc. 2011
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parameters. Reference [8] presents an application to a non-linear model of material strength using Hopkinson bar experiments. Reference [9] applies the same methodology to a multi-scale, crystallography-based material model capable of predicting the thermal creep of metals. The success of implementing a DOE to appraise the important input-output sensitivity, to quantify uncertainty and to calibrate model parameters has prompted the application of DOE to problems of increasing size. While the applications to material models discussed in References [8] and [9], for example, involve 7 and 3 parameters, respectively, the simulations of References [3] or [7] are parameterized with 20+ inputs. In other studies, even higher-dimensional problems are being considered. Unfortunately the cost of a two-level, full-factorial design, where each variable is set to either its lower or higher bound, grows as 2N with N input parameters or dimensions. Increasing the dimensionality from, say, N = 5 to 20 parameters grows the number of runs from 32 to over a million if one selects to perform a two-level, full-factorial design. Such computational cost rapidly becomes prohibitively expensive. It is pushing the technology towards sparsely-populated designs. A commonly-encountered rule-of-thumb is that a sparse DOE can be constructed and analyzed based on a total number of computer runs equal to 10 times the number of active variables. For a problem dominated by 20 parameters, it means performing about 200 runs selected using strategies such as an orthogonal array or space-filling Latin hyper-cube design. The main drawback of this approach is that the statistically significant, or active, variables of the model or numerical simulation may not be a priori known. Hence, we are witnessing the increasing usage of extremely sparse designs based on the guideline of “10 runs per dimension” to perform statistical effect screening and develop statistical emulators. Our concern is that analyzing a complex, high-dimensional simulation with too few runs may lead to misleading results. This has been witnessed for a number of studies performed at Los Alamos [10]. Our hypothesis is that, when a very sparse design is employed, the functional form of the emulator trained to replace a computationally expensive simulation influences the sensitivities instead of staying “neutral” in the process. Erroneous sensitivities, in turn, lead to poor-quality emulators and incorrect characterization of parameter or prediction uncertainty. If proven correct, then this mechanism may yield adverse consequences. When performing a screening experiment, for example, we believe that we are discovering which model parameters or interactions control the prediction variability when, in fact, we are only learning the incorrect sensitivities of the emulator. Likewise, results of parameter calibration are incorrect because too dominantly influenced by the functional form of the emulator. Our discussion of the dangers of sparse sampling follows a three-pronged approach. In section 2, we first illustrate the rapidity with which the sparsity of a design increases as the number of dimensions grows. The coverage, for example, of a 150-run Latin hyper-cube design does not exceed 10–3% in a 15-dimensional space, where “coverage” is the volume of the convex hull defined by these 150 points relative to total volume. It suggests that the generally accepted ruleof-thumb that recommends constructing designs with “10 samples per active dimension” may be inappropriate when the numerical simulation features a discontinuity or non-linearity that cannot be captured adequately with so few sampling points. In sections 3 and 4, we show an example where sampling a high-dimensional mathematical function with too few points leads to an incorrect identification of significant effects. This is observed whether the sensitivities are estimated from an analysis-of-variance (section 3) or Markov Chain Monte Carlo random walk (section 4). The detrimental effects for parameter calibration are also illustrated in section 4. Finally, an explanation is suggested in section 5 in the particular case where the fast-running surrogate is a Gaussian Process Model (GPM). We prove that the covariance matrix of the statistical emulator is bounded by an upper limit that depends on the maximum distance between samples of the design. Our numerical simulation indicates beyond any statistical
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uncertainty that, as the number of dimensions increases, the upper bound rapidly decreases. It means that the structure of the covariance matrix converges asymptotically towards a constant, leading to fast-running surrogates that model fluctuations of the numerical simulation no better than random noise. We conclude that the rule-of-thumb of “10 samples per active dimension,” while appropriate for low-dimensionality problems, is not recommended when dealing with numerical simulations that combine a large number of variables, significant interactions, and non-linearity or discontinuity. Based on the findings of this study, “low-dimensionality” means not exceeding N § 10 variables. Our conclusion is especially relevant when relying on Gaussian process modeling to learn the important sensitivity, infer parameter uncertainty, or calibrate a computer code (see section 5).
2. THE ISSUE OF COVERAGE OF THE SAMPLING SPACE Before starting to illustrate the dangers of generating too-sparse a sample to quantify model sensitivities or prediction uncertainty, the concept of coverage of the design space is briefly discussed. A metric of coverage is needed to measure the region of the design space that is “filled” with points of a given sample, as discussed in Reference [8]. For simplicity, it is assumed in the manuscript that all variables are unit-less and scaled in the interval [-1; 1]. This is achieved without loss of generality using a transform such as:
2
Xk
Z k Z k, Min 1, Z k, Max Z k, Min
(1)
that converts the kth physical variable Zk of the problem to the dimensionless Xk. In equation (1), Zk,Min and Zk,Max denote the minimum and maximum bounds of variable Zk, respectively. The transform (1) converts the N-dimensional physical domain k=1…N [Zk,Min; Zk,Max] in a hyper-cube ([-1; 1])N. Clearly, the total volume of the hyper-cube is equal to 2N. Variable X2 +1
N-Dimensional Space
A B
Latin Hyper-cube Samples
C
Convex Hull
D 2N Samples
-1 1 -1
2
3
4 +1
Variable X1
Figure 1. Illustration of a 2D space and its coverage with a 4-run sample. Coverage of a DOE is, here, simply defined as the volume occupied by the design relative to the total volume of hyper-cube ([-1; 1])N. It can be measured in percentage as:
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Ș 100.
Volume DOE %, 2N
(2)
where Volume(Ɣ) is a function that estimates the N-dimensional volume of the design. Figure 1 gives an illustration in the case of a two-dimensional space (X1; X2) ([-1; 1])2. The total volume is shown as a square domain that extends from -1 to +1 value on each axis. It is assumed that four samples are available, represented by red dots. The dashed, red line that encloses the four sample points symbolizes the convex hull of the design, that is, the smallest possible convex volume that includes all points. The convex hull and its volume are estimated, for example, with the MATLABTM function “qhull.m.” PDF
PDF
PDF
X1
Equal-probability Intervals of X1
Xp
XN
Equal-probability Intervals of Xp
Initialize: “k = 1” Test: “k d N”?
Equal-probability Intervals of XN
No
LHS Design is Complete
Yes
1) Form the kth Sample by Randomly Selecting an Interval for Xp in the List of Remaining Intervals, for p = 1 … N. 2) Delete Intervals Selected for the kth Sample in the N Lists of Remaining Intervals.
Increment: “k Å k+1” Figure 2. Main steps of an algorithm for Latin Hyper-cube Sampling (LHS). Two designs of experiments are illustrated in Figure 1. The first one is a two-level, full-factorial design represented by the green dots. A full-factorial design consists of assembling all possible combinations of “low” and “high” values for each variable, that is, Xk = -1 and Xk = +1. Full-
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factorial designs offer desirable properties for sensitivity analysis and effect screening. They can be used to model the main effects and linear interactions between input variables without any statistical aliasing that could adversely bias the results [11-12]. In this work, we focus on another class of designs known as stratified sampling and, specifically, the Latin Hyper-cube Sampling (LHS) technique [13-14]. The red dots of Figure 1 show a 4-run LHS. This algorithm is designed to “spread” sample points within the design space and provide good coverage while keeping the element of chance necessary to arrive at a random sample. Full-factorial designs, however, are expensive and can rarely be deployed in situations that combine “large” numbers of variables, say, N = 15 or more, and computationally expensive models. Consider, for example, a finite element calculation that takes only 10 minutes to run on a single processor machine. Sampling 15 variables with a full-factorial design requires a total of 215 runs which means 215 x 10 minutes = 5,461 hours, or over 7½ months of total computing time. Without access to parallel computing, analyzing such a design may be out-of-reach.
100%
10%
p=½
1%
Figure 3. Statistics of coverage for a 150-run LHS design as a function of dimension N. (This is for 4 N 9, with statistics based on 1,000 independent trials.) To build a LHS, one starts by choosing the total number of points in the sample. In the case of Figure 1, the number of model evaluations is equal to N = 4. The next step is to sub-divide each variable -1 d Xk d +1 into N equal-probability intervals. Figure 1 shows a simplification where the probability distributions of variables X1 and X2 are independent, uncorrelated, and uniform. The four equal-probability intervals become four equal-length intervals labeled {1; 2; 3; 4} for X1 in the horizontal direction and {A; B; C; D} for X2 in the vertical direction. Sampling consists of randomly selecting and combining these equal-probability intervals for all variables Xk. After an interval has been selected for one of the variables, it is deleted from the list such that the same interval can never be selected twice. In Figure 1, for example, the first combination selected at random may be 4-A. Once selected, column 4 of X1 and row A of X2 are deleted to guarantee that neither of them is chosen a second time. Repeating the procedure N times, that is, until exhaustion of all intervals, completes the LHS algorithm. The LHS algorithm is illustrated in
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Figure 2 and discussions of its implementation, properties, and applications can be obtained from References [13-14]. It can be verified in Figure 1 that the same row or column is never selected twice, which is the trademark of a LHS. The first issue discussed is the coverage of the design space. When computing resources are scarce, or answers must be provided rapidly, it is common practice to limit the number of model evaluations of a sparsely-populated design to about 10 times the number of variables Xk. For example, a problem with N = 15 variables would be analyzed with a LHS design populated with 150 runs. To illustrate how sparsely-populated this is, Figure 3 plots on a semi-log scale the mean coverage KMean of a 150-run LHS design for dimensions 4 N 9.
Ș | 10–3%
N = 15 Figure 4. Extrapolation of the average coverage KMean of a 150-run LHS design. To arrive at the statistics pictured in Figure 3, a 150-run LHS design is selected and its degree of coverage is calculated based on equation (2). Because the combinations (X1; X2; …; XN) that define the LHS runs are randomly selected, this procedure is repeated a 1,000 times to estimate the statistics of Figure 3. The solid, red line shows the mean coverage (KMean) and the dotted, black lines follow the 25% and 75% quartiles of the statistical distributions. The height of each box-plot, shown with a solid, blue line, extends to +/- one standard deviation (ıȝ) away from the mean coverage. Because calculating the convex hull of a set of 150 points in an N-dimensional space rapidly overwhelms MATLABTM’s algorithmic capability, the analysis is restricted to dimensions 4 N 9 in Figure 3. To go to the higher dimensions that are representative of commonly-encountered problems, an extrapolation of the mean coverage is performed. Best-fitting a simple polynomial model suggests:
log10 Ș Mean 2.1 0.0076 N 0.024 N 2 .
(3)
Figure 4 illustrates the extrapolation law of equation (3). It is observed that the average level of coverage KMean of a 150-run LHS design in N = 15 dimensions is not expected to exceed 10–3% which is, to say the least, extremely sparse.
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In light of this simple analysis that could be repeated with other types of sparse designs, one has to ask whether such an extreme sparsity can provide enough information to appropriately “learn” the sensitivities of a computational black-box model or develop a meaningful fast-running surrogate. If sampling in a 15-dimensional space is limited to 150 model evaluations, then the mechanical or physical processes being modeled by the code (that one is attempting to understand through statistical analysis) would have to offer an extreme level of regularity and “smoothness” throughout the design space. If the behavior of the model output Y, on the other hand, exhibits a non-linearity or sudden discontinuity as input parameters (X1; X2; …; XN) vary, then an average level of coverage of KMean = 10–3% may not suffice to capture the phenomena. Our first conclusion is that these extremely sparse designs may yield misleading or incorrect findings. This is demonstrated in sections 3 and 4 using sensitivity analysis and calibration.
3. THE ISSUE OF SENSITIVITY ANALYSIS AND EFFECT SCREENING The second issue addressed is sensitivity analysis, also known as effect screening [12, 15]. It refers in a broad sense to the identification of input parameters Xk, or combinations of input parameters such as (Xp·Xq), that are most responsible for explaining and controlling how the model prediction Y varies. Using a simple illustration, we suggest that sensitivity analysis may yield erroneous results when it is based on a too sparsely-populated design. For illustration, we analyze the Rosenbrock function [16]. It is a test function commonly used in the discipline of numerical optimization because it varies rapidly and defines multiple “valleys.” This makes it difficult for gradient-based optimization algorithms to reach for a global minimum.
(5-a) Rosenbrock function in 2D. (5-b) Contour of the 2D function. Figure 5. Illustration of the Rosenbrock function in 2D space (X1; X2). The Rosenbrock function in the N-dimensional space (X1; X2; …; XN) is defined as:
¦ 1 X
2
Y
k 1(N 1)
k
100 X k 1 X 2k , 2
(4)
where variables Xk are, as before, scaled in [-1; +1]. In two dimensions, the function becomes:
Y
1 X1 2 100 X 2 X12 2 ,
(5)
and an illustration of what it looks like is given in Figure 5. The formation of a “valley” that makes optimization difficult can be observed on the three-dimensional view of Figure 5 (left), and its contour plot (right). The “valley” eventually bifurcates into two branches, one progressing towards the point (X1 = +1; X2 = -1) and the other one going towards point (X1 = +1; X2 = +1). It
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can be easily verified that the global minimum of the Rosenbrock function is reached at Xk = +1, for k = 1 … N, which is the only point where the function is valued Y = 0 exactly. To study the adverse effect that sparse sampling may exercise when attempting to understand the statistical influence of input variables Xk on the prediction Y, the Rosenbrock function is first modified. Its non-linear nature is preserved but the influence of its input variables is modulated somewhat such that some inputs exercise a strong influence while others do not. This change is achieved by replacing the original function of equation (4) by the following variant:
Y
¦ 1 X
2
k 1(N 1)
k
C k 1 X k 1 X 2k , 2
(6)
with dimensionless coefficients Ck defined in Table 1 for a 17-dimensional space. Table 1. Coefficients Ck for the Rosenbrock variant of equation (6). Variable
Coefficient
Variable
Coefficient
2 3 4 5 6
C2 = 5 C3 = 3 C4 = 2 C5 = 10 C6 = 5
7 8 9 10 11-to-17
C7 = 3 C8 = 2 C9 = 10 C10 = 1 C11-to-C17 = 1
Coefficients Ck are defined in Table 1 such that only two variables have a dominant effect: X5 and X9. In addition, four variables moderately control the function variation: they are X2, X3, X6, and X7 whose coefficients are equal to either 3 or 5. Finally, variables X10 to X17 exercise the least influence over the variability of equation (6) because their coefficients Ck are equal to 1. It is expected that a statistical analysis of the influence that input variables exercise on the function prediction Y identify the main effects X5 and X9 as dominant. The other expected result is that the main effects X10 to X17 are not statistically significant because, everything else being equal, their coefficients C10 to C17 are ten times smaller than C5 or C9. To verify this assertion, a two-level, full-factorial design is first analyzed. In a 17-dimensional space, this means performing a total of 217 = 131,072 model evaluations for every combination of “low” levels (Xk = -1) and “high” levels (Xk = +1). The large number of runs is, here, feasible because the analytical function (6) can be evaluated in a fraction of second. In general, this would not be possible on a single processor if the calculation takes more than a few minutes, hence, justifying the commonly-encountered interest for sparse designs. Predictions generated by the full-factorial DOE are analyzed with an analysis-of-variance, or ANOVA, that screens for statistically significant main effects [17]. Statistical screening is also referred to as sensitivity analysis, but it is emphasized that these “sensitivities” are not local derivatives or gradients. Instead, they are statistics that estimate the overall influence that a given variable, such as a main effect Xk, or group of variables, such as a linear interaction Xp·Xq or quadratic effect Xk2, exercise on the output prediction Y. Screening is briefly explained next. The basic principle of ANOVA is to decompose a variance into the summation of two terms:
>
@
ı 2 Y ı 2 E>Y | X k @ E ı 2 Y | X k ,
(7)
where the left-hand side ı2(Y) is the observed variance of predictions Y estimated using runs of the DOE. (It is simply a calculation of variance or standard deviation that uses, for example, the function “std.m” of MATLABTM.) Two terms appear in the right-hand side decomposition (7). The first term ı2(E[Y|Xk]) is the variance of the conditional expectation of Y, given that the kth
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variable Xk is known. It is, again, a simple calculation of variance except that it is applied to runs where Xk is kept constant. With a full-factorial design, one simply identifies all runs where Xk is constant and equal to one of its levels. The calculation of variance is restricted to these runs. The procedure is then repeated for the other levels of the design where “Xk = constant.” The second term of the decomposition E[ı2(Y|Xk)] is a coefficient of influence that is estimated knowing the other two terms of equation (7). It measures the overall influence that variable Xk exercises on the variance of predictions Y. Assume that the value of Xk becomes known through some other means and can be kept constant in the design. One can then ask the question: how does knowing Xk reduce the overall prediction variability? Obviously, if predictions Y and E[Y|Xk] exhibit the same variance, then knowing variable Xk does not reduce the overall variability. The coefficient of influence in the right-hand side of equation (7) would, then, be close to zero. The other extreme is a situation where knowing variable Xk significantly reduces the variability of predictions Y. In this case, the variance of the conditional expectation E[Y|Xk] differs from the variance of all predictions Y and the coefficient of influence becomes “large.” The importance of main effect Xk in equation (7) is therefore indicated by the difference between variances ı2(Y) and E[ı2(Y|Xk)] or by the correlation ratio Ș2 = E[ı2(Y|Xk)]/ı2(Y). We choose, here, to estimate the correlation ratio Ș2 with the R2 statistic:
R2
1
¦ Y ȝ ¦ Y ȝ
¦
l 1... N Level p 1... N(l)
(l) 2 Y
(l) p
2
p
,
(8)
Y
p 1... N Runs
where it can be easily verified that equation (8) derives directly from the decomposition (7). In equation (8), the predictions Yp are indexed with subscript (Ɣ)p that denotes the run number, for p = 1 …NRuns, and ȝY is the overall mean value of all runs in the DOE:
ȝY
E>Y @ |
1 N Runs
¦Y . p
(9)
p 1... N Runs
Likewise the predictions Yp(l) are indexed with superscript (Ɣ)(l) that denotes a run performed when the kth variable Xk is kept constant at its lth level (Xk = constant = Xk(l)), and ȝY(l) is the conditional expectation, or mean value, that averages over these runs only:
ȝ (l) Y
>
E Y | Xk
@
X (l) k |
1 ¦ Yp(l) , N (l) p 1... N(l)
(10)
where N(l) represents the number of runs available to estimate the conditional statistics for the lth level of variable Xk. To complete the description of equation (8), NLevel denotes the number of independent levels available to analyze variable Xk. With a two-level, full-factorial design, for example, it would be equal to NLevel = 2 for all main effects. In the following, the R2 statistics are estimated for all main effects, one at a time. When a large R2 value is found, where “large” is relative to the other values, it indicates that the corresponding variable has a significant main effect that controls how Y changes when variables (X1; X2; …; XN) are varied simultaneously. Figure 6 illustrates the results of ANOVA with the full-factorial (2N) design. It can be observed that variables X5 and X9, because they feature higher R2 statistics that those of the other effects, are identified as the most statistically significant. The figure also indicates unambiguously that variables X10 to X17 are not significant. Between these two extremes, the second-most influential variables X2 and X6, whose coefficients in equation (6) are equal to C2 = C6 = 5, are found. They are closely followed by variables X3 and X7, whose coefficients are C3 = C7 = 3 in Table 1. This
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simple example verifies that, because it features no statistical aliasing, the identification of main effects using a full-factorial design is essentially “perfect.”
Figure 6. R2 statistics for main-effect analysis with a full-factorial design.
Figure 7. R2 statistics for main-effect analysis with a Box-Bhenken design. Figure 7 illustrates the results of ANOVA with a Box-Bhenken Design (BBD) [18]. This design is commonly encountered in physics and engineering because it reaches a compromise between coverage of the design space and aliasing structure for problems that are dominated by main effects and linear interactions. The BBD is suitable to analyze phenomena or black-box models whose input-output functions can be reasonably modeled as linear polynomials:
Y
ȝY
¦ȕ
k 1 N
k
Xk
¦
¦ȕ
p,q p 1... N q (p 1) N
Xp Xq İ ,
(11)
where İ represents the “white,” Gaussian noise of the un-modeled input-output dynamics. The BBD used here features a small number of runs relative to the full-factorial 2N combination, with only 545 evaluations of the Rosenbrock function in our 17-dimensional space.
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It is clear from Figure 7 that an ANOVA performed from the BBD fails to identify the significant variables X5 and X9. Variable X5 is identified correctly, but one also observes that variable X4 is found to be significant. Variable X4 erroneously shows up as important, not because it actually is significant, but through its interaction with X5 in equation (6). Not enough runs are available to distinguish the main effect from the linear interaction, which causes this statistical aliasing. Likewise variables X8 and X9 are aliased when the correct answer is that only X9 is statistically significant to explain the variability of predictions Y. Figure 7 also illustrates that, as expected, the influence of variables X10 to X16 is relatively low. Variable X17, whose influence should also be low, appears more significant than its neighbors. It is another adverse effect of aliasing due to employing this sparse design, relative to the complexity of the Rosenbrock function. Sensitivity analysis results obtained with LHS designs are discussed in section 4, jointly with the analysis of parameter calibration. Whether a LHS or other sparse design is employed, a similar conclusion applies: the analysis rapidly starts to yield erroneous sensitivities if too few runs are available to characterize the main effects and linear interactions of a complicated function.
4. THE ISSUE OF SURROGATE MODELING AND PARAMETER CALIBRATION The third and last issue that we address is parameter calibration. Calibration does not necessarily refer to solving an optimization problem with the goal of searching for parameters that define the model whose predictions best-match the measurements. Calibration refers, here, to inference uncertainty quantification. The goal is to find the probability distribution of input parameters such that, when sampled, the resulting prediction uncertainty matches the variability of physical measurements. Our discussion, however, applies to both cases since deterministic calibration can be viewed as a “subset” of inference uncertainty quantification. To solve the inference uncertainty problem and propagate uncertainty “backwards” through the model, one must first formulate a mathematical equation that connects physical experiments to predictions of the numerical model. We adopt, in this work, the formalism of Kennedy and O’Hagan who propose a statistical model to relate physical measurements to “reality” and model predictions [19]. The starting point is that physical observations never perfectly measure reality:
Y Test X
Y Reality İ Test ,
(12)
where İTest is a zero-mean, stochastic process that represents experimental variability. Equation (12) postulates that the physical observations YTest(X) only differ from reality by the experimental variability (no systematic bias). Second, it is assumed that the computer code or numerical model is not always capable of perfectly representing reality. The difference between the two is another stochastic process that includes systematic bias between reality and model predictions together with random fluctuations. This equation is simply:
Y Reality
Ș X; ș į X ,
(13)
where the symbol į(X) represents the statistical discrepancy between reality and predictions. In equation (13), K(X;ș) denotes the prediction of a model that depends on control parameters Xk, for k = 1 … N, and calibration variables șk, for k = 1 … Nș. Collecting equations (12-13) leads to:
Y Test X
Ș X; ș į X İ Test .
(14)
Simply speaking, the inference uncertainty problem searches for the probability distribution (ș) of calibration variables such that equation (14) is verified, not point-wise, but in a statistical ensemble sense. We rely, here, on a Bayesian inference method implemented in the Gaussian Process Model for Simulation Analysis (GPM/SA) code developed at Los Alamos [20-21].
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Assume a Prior Distribution Prior(ș) for Calibration Variables
First Trial, p = 1
Start Trial Number “p”
Sample Prior(ș)
Make Predictions K(X;ș(p)) for all Control Parameters X
Compare Predictions K(X;ș(p)) to Observations YTest(X) Thought the Likelihood Function L(X;ș) Increment “p m p+1” and Take Another Random Walk From Current Sample ș(p)
Acceptance Test: “Is L(X;ș) Good Enough?”
Take Another Random Walk From the Last-accepted Sample ș(p)
No: Reject Trial
Yes: Accept Trial
Store the Sample ș(p) for Future Estimation of the Posterior Distribution Posterior(ș) Estimate and Store the Sample į(p) of Discrepancy Term
Yes
Iteration Test: “p d pMax?” No
Estimate the Posterior Distribution Posterior(ș) Based on the Population of Samples ș(p) Stored
Estimate the Discrepancy į(X) Based on the Population of Samples į(p) Stored Figure 8. Simplified illustration of the Bayesian method for inference uncertainty. The procedure follows the following steps. The simulation code is first exercised with a userspecified DOE where calibration variables ș are sampled according to their prior probability distribution Prior(ș). The GPM/SA code develops a Gaussian process emulator for predictions
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K(X;ș) based on results of the DOE. A separate Gaussian process emulator is developed for the discrepancy į(X). The next step is to start the Markov Chain Monte Carlo (MCMC) algorithm that explores the (unknown) posterior distribution Posterior(ș) of variables ș. Samples of calibration variables ș are drawn from a MCMC random walk such that the joint probability distribution of “prediction plus discrepancy,” or K(X;ș) + į(X) in equation (14), matches the probability distribution of experimental observations. A new sample is accepted if the likelihood function L(X;ș), defined as a metric of test-analysis correlation:
2 log L X; ș log detȈ D Y Test Ș(X; ș) Ȉ D1 Y Test Ș(X; ș) , T
(15)
passes an acceptance test. (For the exact definition of the likelihood function and covariance matrix ȈD in equation (15), the reader is referred to References [20-21].) The meaning of the acceptance test is that variables ș must yield predictions that are statistically consistent with the measurements YTest(X), modulo experimental variability İTest. The discrepancy term į(X) is also represented by a Gaussian process emulator; it accounts for the lack-of-correlation between predictions and measurements that cannot be compensated for by calibrating variables ș. The hyper-parameters of probability distributions for the calibration variables ș and discrepancy term į(X) are explored simultaneously during the MCMC random walk. Figure 8 gives a conceptual illustration of these successive steps, leaving out many important details of the implementation. The reader is referred to References [22-23] to learn more about the Metropolis-Hastings random walk algorithm. The estimation of sensitivity coefficients is dealt with in Reference [24]. The theoretical framework of this method is described in great details in Reference [19] while an implementation manual is available from Reference [25].
Figure 9. Main-effect sensitivities obtained with a 150-run LHS design. The GPM/SA code is used to estimate the probability distribution of calibration variables such that equation (14) is verified. The analysis is applied to the Rosenbrock function (6) defined in the same 17-dimensional space as before (see section 3). The emulator K(X;ș) is developed from a 150-run, space-filling LHS design. Fiducial “test data” YTest(X) are simulated from the Rosenbrock function to which a 1% zero-mean, Gaussian “white” noise İTest is added. Note that,
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for consistency of notation, the 17 dimensions Xk of section 3 are the calibration variables denoted, here, by șk. By the same token, there is no control parameter in definition (6) of the Rosenbrock function. The prior of calibration variables is an uncorrelated, uniform probability distribution in the hyper-cube ([-1; 1])17. A total of 10,000 MCMC iterations are performed to estimate the posterior distribution and discrepancy term. Finally, recall that at most 6 of the 17 variables are statistically significant, as defined in Table 1. (They are variables ș2, ș3, ș5, ș6, ș7, and ș9 whose coefficients are within 3 d Ck d 10, with ș5 and ș9 being the most dominant ones.) The analysis discussed next asks two questions. The first one is: can the probabilistic sensitivity analysis correctly identify the significant effects? The second question is: can the random walk correctly infer the posterior distribution of calibration variables? Figure 9 answers the first question while Figure 10 answers the second one. During the MCMC iterations, a sensitivity coefficient is estimated each time a new prediction of the emulator K(X;ș) is calculated. Figure 9 depicts graphically the distributions of sensitivity coefficients estimated when one calibration variable șk is varied at a time. This is equivalent to a main effect analysis, as discussed in section 3. Values close to one in Figure 9 indicate the least level of influence while values that deviate significantly from one point to statistically significant main effects. It is observed that the 5th and 9th variables are significant, as expected. This positive result, however, is degraded by the occurrence of false-positives involving the 4th and 8th variables. The reason why ș4 and ș8 are significant has nothing to do with their “true” influence. It is caused, again, by statistical aliasing due to employing a sparse DOE populated with only 150 model evaluations. The conclusion is that the sensitivity analysis obtained with a Gaussian process model does not outperform the ANOVA-based analyses obtained in section 3 with other sparse designs.
Figure 10. Posterior probability distribution of calibration variables. Figure 10 answers our second question. The main diagonal of the figure depicts the marginal probability distributions of calibration variables ș1-to-ș17. The off-diagonal boxes show contour plots of the joint distributions between pairs of variables (șp; șq) for p q. Ideally the marginal distributions should be narrow, hence, indicating little inference uncertainty, and centered about the values used to simulate the test data YTest(X). Clearly this is far from being the case.
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Another concern is that the marginal distributions of the not-significant variables ș10-to-ș17 seem to indicate that some values are more likely than others. This can be observed, for example, for ș14 and ș16. It is dangerous because it could lead an analyst to erroneously “pick” these falsely more likely values and use them in future studies. What happens is, again, a consequence of the incorrect sensitivities built by the statistical emulator when too few runs are available to “sample” the function in a high-dimensional space. We conclude from this simple illustration that an erroneous description of the inference uncertainty is obtained which, in turn, yields incorrect parameter calibration and incorrect quantification of prediction uncertainty.
5. PROPOSING AN EXPLANATION OF THE BEHAVIOR OBSERVED In this section, we attempt to explain the poor performance observed in section 4 when statistical emulators are constructed from runs of a sparse DOE to estimate global sensitivities and estimate inference uncertainty. Our hypothesis centers on the structure of the covariance matrix that is central to any Gaussian process model. This choice is motivated by recent work of Professor Derek Bingham of Simon-Frasier University, Vancouver, Canada [26]. Because the Gaussian process emulators built as surrogates to the test data YTest(X), model predictions K(X;ș), and discrepancy term į(X) play such an important role in the overall process of inference uncertainty, we postulate that the source of difficulties encountered previously must come from the structure of their covariance matrices. This is because a Gaussian process is uniquely specified by its covariance structure. (There can also be a mean or “average” trend that takes the form of a constant value or low-order polynomial of variables (X; ș). The simplicity of this additional term, however, cannot possibly justify the difficulties encountered.) The GPM/SA code used in section 4 assumes that the covariance between two points X and X* of the multi-dimensional space, that includes control parameters Xk and calibration variables șk, takes the following functional form: 2 4 (X X* ) 1 k k Cov X; X* | ȡ , Ȝ 1d k d N k
(16)
where symbol Ȝ is the precision parameter of the covariance matrix; ȡk denotes the correlation between outputs of the code when they are evaluated at inputs (X; X*) that vary in only the kth dimension; and subscript (Ɣ)k represents the coordinate of a point X or X* in the kth dimension. As noted in Reference [21], the quantities ȡk control the dependence strength of the covariance in each of the directions of variables X. Not shown in equation (16) is the white-noise component often added to the covariance for numerical stability, in the form of an identity matrix multiplied by a constant (1/ȜN). This noise is kept “small” relative to the other contributions by defining its precision parameter ȜN such that (1/ȜN) << (1/Ȝ). Obviously the covariance model of equation (16) is not unique and other functional forms can be defined. Most models, however, share the characteristic that the influence of fluctuations that cause the variance should decrease with distance between samples. This is true whether “distance” is projected in a single direction, such as |Xk – X*k| in equation (16), or defined as the Euclidean distance ||X – X*||2. This property makes sense because the variability of predictions obtained from samples located far away should not influence the covariance matrix as strongly as fluctuations coming from neighboring samples. Our hypothesis is that, when too few runs are available for training the covariance model in a high-dimensional space, the average distance between sample points rapidly increases. Large distances attenuate the covariance structure, therefore, rendering the final model no better than random noise, that is, a value that remains more-or-less constant across the space. In the
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particular case of the GPM/SA code, for example, equation (16) shows that all it takes to “kill” the covariance is to find one projected distance |Xk – X*k| that is larger than the others. Because these individual contributions are multiplied together, a single one suffices to severely attenuate the overall strength of the covariance structure. To test our hypothesis, an upper bound of the covariance model is sought. The upper bound will indicate how the covariance behaves as dimensionality of the problem increases, just like we did to investigate coverage in section 2. The L2 norm of the covariance matrix can be bounded: 4 (X X* ) k k ȡ
1d k d N
2
d
k
ȡk 1d k d N
2
4 (X X* ) k k 2
2
d
1d k d N
4 (X X* ) k k ȡ
2
Max
,
(17)
where ȡMax denotes an upper bound of correlation matrices ȡk. Quantity ȡMax can be defined, for example, with the maximum spectral radius of these matrices. In the univariate case, it is simply the maximum correlation. For our purpose of deriving an upper bound, it suffices to state that, because it represents an upper-limit correlation, the value of ȡMax is bounded within 0 d ȡMax d 1. The next step is to apply simple linear algebra to convert the upper bound (17) to:
1d k d N
4 (X X* ) k k ȡ
2
Max
ȡ Max 1dk¦dN 4 (Xk X*k )
2
2
ȡ Max 4 ||X X*||2 .
(18)
Collecting equations (16-18), and omitting the potential contribution of a white noise component (1/ȜN) because it only introduces a constant “bias,” the upper bound is found to be:
Cov X; X* 2 d
1 4 D2 ȡ Max DOE , Ȝ
(19)
where the upper-limit correlation ȡMax is elevated to an exponent proportional to the maximum distance between all sample points of the DOE, or “diameter” DDOE of the design:
D DOE
max
(X;X *) DOE
X X* 2 .
(20)
The overall influence of the covariance matrix of a Gaussian process emulator cannot exceed the upper bound found in equation (19). Evaluating this asymptotic limit numerically, therefore, can be useful to indicate how the covariance matrix behaves as the number of dimensions of the hyper-space, or number of sampling points available from the design, increases. A numerical illustration is summarized in Figures 11 and 12. The difficulty is to calculate the diameter defined in equation (20) because DDOE depends on points of the design which, in the case of a LHS, are drawn randomly. To address this difficulty, a thousand trials are performed and statistics are estimated from these trials. For each trial, a 150-run LHS design is generated and all distances between its N·(N–1)/2 = 11,175 pairs of sample points (X; X*) are calculated. The value of DDOE is the maximum Euclidean distance ||X – X*||2 found in this manner. It can be observed from Figure 11 that both average and maximum distances increase as the number of dimensions vary from N = 4 to N = 20. The vertical, solid lines (shown in color red) represent the ± 3ı-bounds of standard deviation added to verify that the trends are statistically significant. Note that the ± 3ı-bounds are plotted but barely visible for the average distances (shown in color green). This is because averaging is equivalent to a L1 norm while the maximum distance is a L norm. The L1 norm converges faster than the L norm because of the wellknown inclusion property of Hilbert spaces, that is, Lp+1 Lp for 1 d p d +. It explains why the
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variability of average distances, found to be less than 0.05%, is much smaller than the variability of maximum distances, found to be § 3%, based on the same number of random trials.
Figure 11. Average and maximum distances for a population of 150-run LHS designs.
Figure 12. Upper bound of the L2 norm of the covariance model for 0.5 d ȡMax d 0.9. The moderate increase of the diameter DDOE in Figure 11 suffices to significantly attenuate the “strength” of the covariance matrix of a Gaussian process model. This is illustrated in Figure 12 that plots the upper bound of equation (19) for several values of ȡMax ranging from 50% to 90% correlation. Even in the least severe case where ȡMax = 90%, analyzing the Rosenbrock function of section 4 in a 17-dimensional space instead of restricting it to N = 6 dimensions reduces the overall covariance strength by over a factor 10. Recalling that only six variables are statistically significant in Table 1, it implies that the covariance model is erroneous by construction, simply
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because restricting the analysis to 150 model evaluations in a 17-dimensional space artificially increases the diameter DDOE of the design. This simple example illustrates our hypothesis: training a Gaussian process emulator with too few samples in a high-dimensional space deteriorates the quality of the model’s covariance matrix. An incorrect covariance structure, in turn, leads to erroneous sensitivities and calibration, as demonstrated in section 4. It is emphasized that this conclusion is based on an upper-bound analysis and what may actually be observed in practice can only be worse.
6. CONCLUSION Our objective is to raise awareness of the potential dangers of relying on sparse designs to support simulation activities such as statistical effect screening, uncertainty propagation, and model calibration. Combining a design of computer experiments to the development of a fastrunning surrogate model is attractive to replace a computationally expensive simulation, but it can also lead to erroneous results. First, we illustrate the rapidity with which the sparsity of a design increases as the number of dimensions grows. The coverage, for example, of a 150-run Latin hyper-cube design does not exceed 10–3% in a 15-dimensional space, where “coverage” is the volume of the convex hull defined by these 150 points relative to total volume. It suggests that the generally accepted rule-of-thumb that recommends using designs with “10 samples per active dimension” may be inappropriate when the numerical simulation features a discontinuity or non-linearity that cannot be captured adequately with so few points. Next, we show an example where sampling a high-dimensional mathematical function with too few points leads to an incorrect identification of significant effects. This is observed whether the sensitivities are estimated from an analysis-of-variance or Markov Chain Monte Carlo random walk. It is a concern because the sensitivities become those of the, possibly incorrect, statistical emulator and not those of the numerical simulation. Finally, we suggest an explanation of this phenomenon in the particular case where the fast-running surrogate is a Gaussian process model. We prove that the covariance matrix of the statistical emulator is bounded by an upper limit that depends on the maximum distance between samples of the design. Our numerical simulation indicates beyond any statistical uncertainty that, as the number of dimensions increases, the upper bound rapidly decreases. It means that the structure of the covariance matrix converges asymptotically towards a constant, leading to fast-running surrogates that model fluctuations of the numerical simulation no better than random noise. We conclude that the rule-of-thumb of “10 samples per active dimension” should be revisited to devote, at least, 100 runs per dimension when the numerical simulation explored is believed to feature discontinuity and/or non-linearity. This may not be as expensive as it sounds, given the ever-increasing availability of computing resources. The alternative is to proceed in two stages, first, with a screening experiment to identify the statistically significant variables; then, with a second design to develop an emulator for uncertainty quantification and parameter calibration. Screening methods, such as the Morris one-at-a-time design [27-28], are available that may be applicable to large-dimensional problems while capable of correctly identifying the main effects and interactions of a non-linear function. Future work will investigate these screening methods.
ACKNOWLEDGEMENTS This work is performed under the auspices of the Verification and Validation (V&V) program for Advanced Scientific Computing (ASC) at Los Alamos National Laboratory. The first author is grateful to Mark Anderson, V&V program manager at LANL, for his continuing support. The authors also express their gratitude to Professor Derek Bingham, Simon-Frasier University,
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Vancouver, Canada, for his kind willingness to share his insight with them. Los Alamos National Laboratory is operated by the Los Alamos National Security, LLC for the National Nuclear Security Administration of the U.S. Department of Energy under contract DE-AC52-06NA25396.
BIBLIOGRAPHICAL REFERENCES [1] Sacks, J., Welch, W.J., Mitchell, T.J., Wynn, H.P., “Design and Analysis of Computer Experiments,” Statistical Science, Vol. 4, 1989, pp. 409-423. [2] Santner, T.J., Williams, B.J., Notz, W.I., The Design and Analysis of Computer Experiments, Springer-Verlag Publishers, 2003. [3] Maupin, R., Hylok, J., Rutherford, A., Anderson, M., “Validation of a Threaded Assembly, Part I: Overview,” 6th European Conference on Structural Dynamics, Paris, France, September 5-7, 2005. [4] Hylok, J., Rutherford, A., Maupin, R., Anderson, M., Groethe, M., “Validation of a Threaded Assembly, Part II: Experiments,” 6th European Conference on Structural Dynamics, Paris, France, September 5-7, 2005. (Los Alamos Technical Report LA-UR-05-0931.) [5] Rutherford, A., Maupin, R., Hylok, J., Anderson, M., “Validation of a Threaded Assembly, Part III: Validation,” 6th European Conference on Structural Dynamics, Paris, France, September 5-7, 2005. (Los Alamos Technical Report LA-UR-05-2500.) [6] Lam, K., Allen, D., Tippetts, T., “Engineering Verification and Validation Assessment of Truchas for Induction Heating,” Technical Report LA-UR-07-7134, Los Alamos National Laboratory, Los Alamos, New Mexico, October 2007. [7] Sigeti, D.E., Buescher, K.L., Vaughan, D.E., Cooley, J.H., Hemez, F.M., “Simulation of Integrated Effects Tests for the Fiscal Year 2008 ASC Primary Verification and Validation Milestone,” Technical Report LA-CP-08-1154, Los Alamos National Laboratory, Los Alamos, New Mexico, September 2008. (Not available for public release.) [8] Hemez, F.M., Atamturktur, S.H., Unal, C., “Prediction with Quantified Uncertainty of Temperature and Rate Dependent Material Behavior,” 11th AIAA Non-Deterministic Approaches Conference, Palm Springs, California, May 4-7, 2009. (Los Alamos Technical Report LA-UR-08-6741.) [9] Atamturktur, H.S., Lebensohn, R., Higdon, D., Williams, B., Hemez, F.M., Unal, C., “Predicative Maturity: A Quantitative Metric to Optimize Complex Simulations via Systematic Experimental Validation,” Technical Report LA-UR-09-7226, Los Alamos National Laboratory, Los Alamos, New Mexico, October 2007. [10] Hemez, F.M., “A Technical Note on the Goodness-of-fit of Statistical Emulators,” Technical Memorandum X-3:10-001-C, Los Alamos National Laboratory, Los Alamos, New Mexico, October 2009. (Not available for public release.) [11] Myers, R.H., Montgomery, D.C., Response Surface Methodology: Process and Product Optimization Using Designed Experiments, Wiley Inter-science Publishers, 1995. [12] Saltelli, A., Chan, K., Scott, M., Sensitivity Analysis, John Wiley & Sons Publishers, 2000. [13] McKay, M.D., Beckman, R.J., Conover, W.J., “A Comparison of Three Methods For Selecting Values of Input Variables in the Analysis of Output From a Computer Code,” Technometrics, Vol. 21, No. 2, 1979, pp. 239-245. [14] Tang, B., “Orthogonal Array-Based Latin Hypercubes,” Journal of the American Statistical Association, Vol. 88, 1993, pp. 1392-1397.
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[15] Saltelli, A., Tarantola, S., Campolongo, F., Ratto, M., “Sensitivity Analysis in Practice: A Guide to Assessing Scientific Models,” Probability and Statistics Series, John Wiley & Sons Publishers, 2004. [16] Rosenbrock, H.H., “An Automatic Method for Finding the Greatest or Least Value of a Function,” Computer Journal, Vol. 3, 1960, pp. 175-184. [17] Lindman, H.R., Analysis of Variance in Complex Experimental Designs, W.H. Freeman & Co. Ltd Publishers, 1975. [18] Box, G.E., Hunter, J.S., Hunter, W.G., Statistics for Experimenters: Design, Innovation, and Discovery,Wiley-Interscience Publishers, 2nd Edition, 2005. [19] Kennedy, M., O’Hagan, A., “Predicting the Output From a Complex Computer Code When Fast Approximations Are Available,” Biometrika, Vol. 87, 2000, pp. 1-13. [20] Williams, B., Higdon, D., Gattiker, J., Moore, L., McKay, M., Keller-McNulty, S., “Combining Experimental Data and Computer Simulations With an Application to Flyer Plate Experiments,” Bayesian Analysis, Vol. 1, 2006, pp. 765-792. [21] Higdon, D., Gattiker, J., Williams, B., Rightley, M., “Computer Model Calibration Using HighDimensional Output,” Journal of the American Statistical Association, Vol. 103, No. 482, June 2008, pp. 570-583. [22] Metropolis, N., Rosenbluth, A., Rosenbluth, M., Teller, A., Teller, E., “Equations of State Calculations by Fast Computing Machines,” Journal of Chemical Physics, Vol. 21, 1953, pp. 1087-1091. [23] Hastings, W.K., “Monte Carlo Sampling Methods Using Markov Chains and Their Applications,” Biometrika, Vol. 57, 1970, pp. 97-109. [24] Oakley, J., O’Hagan, A., “Probabilistic Sensitivity Analysis of Complex Models,” Journal of the Royal Statistical Society, Vol. 66, 2004, pp. 751-769. [25] Williams, B., Gattiker, J., “Using the Gaussian Process Model for Simulation Analysis (GPM/SA) Code,” Technical Report LA-UR-06-5431, Los Alamos National Laboratory, Los Alamos, NM, December 2006. [26] Bingham, D., Higdon, D., Williams, B., “Gaussian Process Models for High Dimensional Computer Experiments,” Working Technical Paper, Simon-Frasier University, Vancouver, Canada, September 2009. [27] Morris, M.D., “Factorial Sampling Plans for Preliminary Computational Experiments,” Technometrics, Vol. 33, 1991. [28] Campolongo, F., Cariboni, J., Saltelli, A., “An Effective Screening Design for Sensitivity Analysis of Large Models,” Environmental Modelling and Software, Elsevier Publishers, Vol. 22, 2007, pp. 1510-1518.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Geological Stress State Calibration and Uncertainty Analysis John McFarland1 , Alan Morris2 , Barron Bichon1 , David Riha3 , David Ferrill4 , Ronald McGinnis5 Southwest Research Institute, 6220 Culebra Road, San Antonio, TX 1
Research Engineer 2
3 4
Staff Scientist
Principal Engineer
Department Director
5
Research Scientist
NOMENCLATURE n p θ y s ε G(·, ·) σ j π(·)
Number of observed data points Number of calibration parameters Vector of calibration parameters Vector of observed data Vector of observable, independent variables Vector of model calibration residuals Functional relationship between model inputs and outputs Magnitude of jth principal stress Bayesian prior distribution function
ABSTRACT The stress state is an important controlling factor on the slip behavior of faults and fractures in the earth’s crust and hence on the productivity of faulted and fractured hydrocarbon reservoirs. Uncertain or poorly constrained estimates of stress states can lead to high risk both in drilling and production costs. Current methods for stress tensor estimation rely on slip vector field data, however, this information is not generally available from datasets that are commonly used in the oil and gas industry. This work presents an approach whereby predicted slip tendency is used as a proxy for fault displacement, which can easily be extracted from datasets routinely used by the oil and gas industry. In doing so, a calibration approach is developed in order to estimate the parameters governing the underlying stress state by calibrating slip tendency R predicted by the 3DStress software to match measured slip displacement. A Bayesian approach is employed, and several uncertainty sources are accounted for in the estimation process, including the impacts of limited data and correlated data taken from geologically similar measurement locations.
1
INTRODUCTION
Faults and fractures provide important pathways for subsurface fluid flow in many geologic settings including aquifers, geothermal reservoirs, and hydrocarbon reservoirs. They act as both conduits for and barriers to flow and are therefore the primary structural determinants of aquifer and reservoir compartmentalization. In situ crustal stresses, because they control the slip behavior of faults and fractures, also exert an important control on the fluid conductivity of faulted and fractured systems. Uncertain or poorly
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_49, © The Society for Experimental Mechanics, Inc. 2011
557
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558 constrained estimates of stress states can lead to high risk both in energy exploration and production, and in the estimation of reserves. Several methods exist for the determination (inversion) of the stress tensor based on observed effects of that stress tensor using data sources such as earthquake focal mechanisms[1] , paleostress indicators[2, 3] , and microseismieity[4] . All these methods rely on knowledge of the slip vector field generated by the stress state being sought (see Ref. 5 for a review). However, slip vector information is not generally available from such data sets as seismic reflection data and microseismic swarms commonly used in the oil and gas industry. In this paper we develop a stress inversion technique that allows the estimation of the stress state based on fault displacement data, which can easily be extracted from data sets routinely used by the oil and gas industry. Our approach is formulated as a R 1 model calibration problem: we make use of the 3DStress software tool that accepts the stress state as an input and has as output a measure of fault slip tendency. Such an approach brings a unique perspective to the problem, where the stress state is typically estimated in an ad-hoc, trial and error procedure that requires an experienced analyst to manually “tune” the stress state based on the observed displacement data. Our approach goes beyond estimating a simple point value for the stress state, though. We employ a Bayesian model calibration procedure that develops a comprehensive representation of the uncertainty associated with the estimated stress state. This approach helps us quantify the degree to which we can perturb the stress state while still maintaining a match to the field data. Further, the resulting uncertainty representation in the stress state can be used via uncertainty propagation to quantify the uncertainty associated with new predictions about slip tendency obtained from the model. Section 2 will provide background on the type of geological data that we are using and how it relates to the computational model for predicting slip tendency. Section 3 presents the Bayesian model calibration approach. The analysis is discussed in Section 4, and we present stress state results both with and without the use of a correlation function to describe data dependencies.
2
FAULT SLIP MEASUREMENT AND PREDICTION
The earths crust is subject to stresses that are the combined result of gravitational loading and heat-transfer-driven tectonic processes. One type of response to these stresses is the formation and propagation of fractures. In geological terminology, shear fractures that accommodate measurable amounts of displacement are called faults. Locally, stress systems are approximately homogeneous, and can be considered as second rank tensors that are most commonly described in terms of three mutually perpendicular principal stresses. Some regions are subject to earthquakes, evidence that present-day in situ stresses are capable of generating fractures and causing them to slip, whereas other regions are seismically inactive. However, even seismically “quiet” regions contain rocks that are faulted and fractured. Whether seismically active or not, faults and fractures provide important pathways for subsurface fluid flow in many geologic settings[6] including aquifers, geothermal reservoirs, and hydrocarbon reservoirs.
2.1
Slip Tendency Analysis
Slip tendency analysis is based on the premise that the resolved shear and normal stresses on a surface are strong predictors of both the likelihood and direction of slip on that surface[7] . The method has been used successfully to characterize fault slip[8] and fault slip directions[9, 10] . Fractures in high slip tendency orientations are, in many cases, better flow conduits than fractures in low slip tendency orientations[6, 7, 11, 12] . The effect of stress anisotropy is greatest when the effective stress conditions on a fault or fracture approach those required for slipthe so-called critical stress[13−15] . Thus, preferential fluid flow through fault and fracture pathways is more pronounced the greater the differential stress and the greater the area of faults and fractures experiencing high slip tendency[7, 12, 16, 17] . Analysis of the effects of in situ stresses on existing faults and fractures is extremely important to optimizing hydrocarbon production from fractured reservoirs, designing and interpreting “hydrofracs” to stimulate hydrocarbon production, designing and operating fractured rock geothermal fields, and predicting the effects of underground CO2 sequestration. In addition to the fault and fracture network, the in situ stresses are key inputs to this analysis. R 1 3DStress is
a commercial software tool developed at the Southwest Research Institute
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559 2.2
Stress Inversion
Several methods exist for the determination (inversion) of the stress tensor from the effects of that stress tensor using data sources such as earthquake focal mechanisms[1] , paleostress indicators[2, 3] , and micro-seismicity[4] . All these methods rely on knowledge of the slip vector field generated by the stress state being sought. However, slip vector information is not generally available from such data sets as seismic reflection data and microseismic swarms commonly used in the oil and gas industry. Our stress inversion technique uses slip tendency as a proxy for fault displacement, a measure that can easily be extracted from data sets routinely used by the oil and gas industry. If it can be assumed that the fault displacements were all generated by the same stress state, then this type of data provides information about that stress state. Different stress states would be more or less likely to have produced the observed pattern of fault displacements. Thus the process of using observed displacement data to estimate the underlying stress state in conjunction with a software tool that predicts slip tendency is an example of an inverse problem, also known as a model calibration problem (model calibration will be discussed in Section 3).
2.3
Geological Field Data
For the purpose of developing an inversion technique, the most useful natural data set is one that incorporates detailed fault slip data from a relatively small volume of rock, and for which the assumption that fault slip was geologically synchronous is appropriate. The Canyon Lake Spillway Gorge in Comal County, Texas contains faults that are amenable to this analysis and can be mapped at the required level of detail. Several sub-horizontal bedding-parallel pavements adjacent to the Hidden Valley fault[18] are cut by networks of small-displacement (≤ 1 m) normal faults (Figure 1). The surfaces of these faults are clearly visible and are commonly decorated with slickenline indicators in the form of grooves, ridges and swales, or fibrous calcite. High resolution stratigraphic mapping[19] and close inspection of faulted strata permit precise measurements of displacement parallel to slickenlines on exposed fault surfaces. Approximately 348 slip surfaces were identified and measured (strike, dip, and rake using the right-hand rule, and displacement), and their locations recorded using a real-time kinematic global positioning system.
Figure 1: Measuring orientation and slip data on small faults at Canyon Lake Gorge. Measurement sites are temporarily marked with tape then surveyed using a real-time kinematic global positioning system (staff with antenna).
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560 3
BAYESIAN MODEL CALIBRATION AND UNCERTAINTY QUANTIFICATION
We use the term model calibration to refer to the process of estimating or tuning unobservable model parameters in order to improve agreement between model output and experimentally observed responses. Note that in most cases of interest, the relationship between the model inputs and outputs is only observable by exercising a computational simulation, meaning that the model can not be manipulated analytically or inverted to directly solve for the unknown parameters. As such, model calibration is a numerical process, much like numerical optimization. We formulate the calibration problem in a general sense by first formalizing the relationship between the model inputs and outputs as follows: y = G(θ, s), (1) where y is the model output (also known as the response or dependent variable), θ is a p-dimensional vector of unknown calibration parameters, s is a vector of observable independent variables (sometimes referred to as covariates), and G(·, ·) represents the functional relationship defined by the computer simulation. Consider that the calibration parameters are to be estimated using n experimental observations y = (y1 , . . . , yn )T of the dependent variable(s) that correspond to the values of the independent variables s1 , . . . , sn . The statistical model that relates the predicted and observed values of the response is then written as yi = G(θ, si ) + εi , i = 1, . . . , n,
(2)
where the εi are the random error terms (the model of Eq. (2) is referred to as a nonlinear regression model). Thus, the objective of the calibration analysis is to obtain estimates of the calibration parameters θ1 , . . . , θp based on the experimental measurements y1 , . . . , yn . Perhaps the most straightforward approach for tackling the problem is to formulate the analysis as a nonlinear least-squares problem (see Ref. 20). However, there is an important feature of model calibration problems that is sometimes overlooked, which is that there may be a wide range of model calibration parameters that provide comparable fits to the observed data. In the literature, this is sometimes referred to as the problem of non-uniqueness. In fact, the degree to which a range of parameters may fit the data is related to the amount of data available, and the range generally decreases as more data become available. In these situations we say that the parameters are subject to uncertainty, meaning that the “true” values of the calibration parameters are not known. When performing a calibration analysis, it is important to be aware that the best fitting (e.g. leastsquares estimator) value of the calibration parameters is not necessarily the only feasible value. Several techniques are available for quantifying the degree of uncertainty (or conversely confidence, as in confidence intervals or confidence regions) that exists in estimated calibration parameters (refer to Ref. 20 for a description of classical approaches). For this work, we adopt a Bayesian approach, which allows us to develop a comprehensive representation of our complete state of knowledge regarding the unknown calibration parameters (Ref. 21 discusses differences between classical and Bayesian uncertainty quantification approaches). Bayesian analysis is based on the single equation known as Bayes’ theorem: f (θ | y) = R
π(θ)f (y | θ) , π(θ)f (y | θ) dθ
(3)
where as above, θ contains the variables being estimated (calibration parameters), y contains observed data, π(θ) is known as the prior distribution for the unknowns, f (y | θ) is the likelihood of observing the data given a particular value of the unknowns, and f (θ | y) represents the posterior state of knowledge about the unknowns. The goal is to compute the posterior distribution for the unknowns, which is actually a probability density function that captures the complete state of knowledge about the unknowns. This posterior density function can be used to quantify the “most likely” values of the unknowns, associated uncertainty, correlations among components, etc. The prior distribution for the unknowns, π(θ), is also a probability density function2 , and it is used to capture all information about the unknowns that is available before taking into account the data y. The prior distribution can be used to incorporate parameter constraints by defining π(θ) = 0 wherever θ is not a feasible parameter vector. 2 In most cases, the prior distribution is allowed to be an improper probability density function, meaning that its integral may not converge.[25] In fact, the reference prior distribution that we employ in Eq. (5) is improper.
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561 The last element of Bayes’ theorem is the likelihood function, f (y | θ), and it serves the purpose of measuring the agreement between the data and predictions (note that the likelihood function is viewed as a function of θ; the observations y hold fixed values). The specific form of the likelihood function is based on the joint probability density function used to model the error random variables, ε1 , . . . , εn . The most common assumption is that the errors are independent and identically distributed and have a normal distribution with zero mean and some unknown variance, σ 2 . In this case, the likelihood function is given by f (y | θ, σ 2 ) =
n Y i=1
» – 1 (yi − G(θ, si ))2 √ exp − . 2σ 2 σ 2π
(4)
Note that independence for the error terms implies independence for the experimental observations y1 , . . . , yn . Whether or not this assumption is appropriate will depend on the application and should be carefully considered. If the measurements are not independent, then different error models may be evoked. Dependent errors are typically modeled using the multivariate normal distribution, in which case an error correlation matrix is also required. The correlation matrix would usually be constructed by first formulating a correlation function in terms of the independent variables, si , . . . , sn , and additional correlation parameters, say φ. Ideally one would treat the correlation parameters as additional objects of Bayesian inference, however when the number of observations n is large, this may not be feasible, in which case an alternative approach is to first derive point estimates (perhaps using maximum likelihood) for the correlation parameters, and then to treat them as knowns for the remainder of the analysis. The process of actually computing the posterior distribution, f (θ | y), is fairly involved. Numerical sampling techniques such as Markov Chain Monte Carlo simulation are widely used; details are given by Ref. 21. However, once the posterior distribution is obtained, it provides a comprehensive, quantitative representation about the state of knowledge of the unknowns. Refs. 21–24 provide detailed case studies of the use of Bayesian inference in this fashion.
4 4.1
ANALYSIS Introduction
The stress state is an important controlling factor on the slip behavior of faults and fractures in the earth’s crust. However, data sets commonly available in the oil and gas industries do not establish a direct connection with the stress state. Instead, these data sets typically capture slip displacements at various locations on one or more faults. Commercial software packages are currently available to predict slip tendency based on a given stress state, but the analyst is required to specify the stress state. In practice, this often gives rise to an ad-hoc “tuning” process, in which the stress state is manually (and subjectively) adjusted in order to obtain some level of agreement between the slip tendency predicted by the simulation and observed slip displacements measured in the field. We will present a rigorous approach in which the tuning process is formulated as a model calibration problem, as discussed in Section 3.
4.2
Approach
The objective of the Bayesian calibration approach is to construct the posterior distribution for the unknowns, based on some observed data. Here the unknowns are the set of parameters that describe the stress state, and the observed data consist of measurements of fault displacement. Referring to Eq. (2), the first step is to formalize definitions for the observed data, y, the calibration parameters, θ, the independent variables, s, and the performance model G(·, ·). As mentioned above, the data y simply consist of n measurements of actual surface displacements at various locations on one or more faults. The independent variables, s, that are observable and associated with each measurement are the strike and dip angles that define the fault orientation. Note that for each displacement measurement there are corresponding measurements of the strike and dip angles for that location. The vector of calibration parameters, θ, contains the full set of parameters necessary to define the stress state. This stress state consists of the orientations and magnitudes of three principal stresses. Section 4.2.1 below provides a discussion of how we choose to represent the stress state for the purpose of calibration, in order to comply with certain conventions and constraints.
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562 In addition to the calibration parameters, the error variance, σ 2 , is also treated as an object of Bayesian inference, so that its posterior distribution is obtained as well. The prior distribution that we use is the non-informative reference prior[25] π(θ, σ 2 ) ∝
1 , σ2
(5)
which is independently uniform in the calibration parameters and log σ 2 . The performance model, G(·, ·), defines a predictive relationship between the calibration parameters, the independent variables, and the response. In this case the performance model accepts as inputs the stress state as well as the fault orientation (defined by the strike and dip angles) and returns a predicted fault displacement. This application presents an additional challenge, though, which is that the actual computational model does not predict real displacement but instead slip tendency. This necessitates the formulation of a linking function that converts slip tendency to something that is comparable with actual displacement. This is discussed in more detail in Section 4.2.2, but we note that the model operator G(·, ·) is actually two functions, the slip tendency analysis engine and the conversion from tendency to displacement, so that the values returned by G(·, ·) have the same units as the measurements y.
4.2.1
STRESS STATE PARAMETRIZATION
As mentioned above, the in situ stress state consists of the orientations and magnitudes of three principal stresses: σ 1 , σ 2 , and σ 3 . As such, it is necessary to chose a set of calibration parameters that completely define the stress state. We first consider the parameters that define the orientation of the stress state. The orientation of the stress state is constrained such that the three principal stresses remain mutually orthogonal. As such, there are three degrees of freedom associated with the stress state. We choose a coordinate system that defines the orientation in terms of an initial state and three global, ordered rotations. The initial stress state has the first principal stress vertical and the second and third in the horizontal plane, with the second directed towards north and the third directed towards east. The first rotation is referred to as the azimuth, and it is a left-handed (clockwise as seen looking down on the horizontal plane) rotation about the vertical axis. The azimuth defines the compass direction of the second principal stress. The second rotation is referred to as the plunge, and it defines a right-handed rotation about the axis of the second principal stress. The third rotation is referred to as the tilt, and it defines a right-handed rotation about the axis of the third principal stress. Next consider the parameters that govern the magnitudes of the principal stresses. The first thing to note is that the simulation that predicts slip tendency is not a function of the absolute stress magnitudes, but only their relative magnitudes. We take this to mean that there are only two “real” degrees of freedom associated with the stress magnitudes, assigning a nominal value of 100 to the first principal stress. In practical applications, the type of stress state may be known beforehand based on knowledge of the predominant fault type, earthquake activity, or other geological information. For the present work, the stress state is what is referred to as a normal faulting environment, meaning that the largest principal stress (here referred to as the first principal stress) is approximately in the vertical direction, and the intermediate (second) and least (third) principal stresses lie approximately in the horizontal plane. In fact, the approximate compass direction (azimuth) of the second principal stress can also be inferred based on the predominant strike angle associated with the measurement data. Because the stress magnitudes are governed by different parameters than the orientations, we want to introduce some type of constraint to ensure the correct ordering of magnitudes between the first, second, and third principal stresses. It should be noted that theoretically, such a constraint could be evoked in terms of a prior distribution that only has support where the constraint is met. This is a valid strategy, but we have found that the problem is better conditioned (that is, the Markov Chain Monte Carlo sampling algorithm is able to explore the parameter space more efficiently) if we instead choose a parametrization that itself takes care of the constraint. As mentioned above, we assume a nominal value for the magnitude of the first principal stress: σ 1 = 100 (6) Given the constraint that σ 1 ≥ σ 2 ≥ σ 3 , we define a parameter (say, θ1 ) governing the second magnitude in terms of a
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563 fraction reduced from the first magnitude. Formally, we have3 : θ1 =
σ 1 − σ 2 σ 2
(7)
The parameter governing σ 3 is defined similarly as a fraction reduced from σ 2 : θ2 =
4.2.2
σ 2 − σ 3 σ 3
(8)
COMPARISON FUNCTION FOR MODEL OUTPUTS AND FIELD MEASUREMENTS
As mentioned above, the computational model has as output slip tendency as opposed to real displacement. Slip tendency is a measure that mostly takes values between zero and one, with zero indicating a low propensity for the surfaces to slip and one indicating a high propensity for slip. Further, we do not necessarily expect a linear relationship between slip tendency and slip magnitude. This is because the physical process of two surfaces slipping past each other is something that happens slowly over time. In fact, different faults or fault locations may have “started” slipping at different points in time. What this means is that a surface with a high slip tendency may have either a large actual displacement (started slipping early) or a small actual displacement (started slipping late). However, a surface with a low slip tendency should have little or no actual displacement. We will let y˜ denote the slip tendency measure that is output from the computational model. To simplify the formulation, instead of directly deriving a transformation from y˜ to G(·, ·), we construct an error measure, y − G(·, ·), which is equal to ε from Eq. (2). The error measure intends to capture the following features:
1. We expect a positive relationship between tendency and displacement 2. Large displacement with small tendency should be penalized more than small displacement with large tendency.
We propose the following error measure (a.k.a. penalty function) based on these considerations: “ ” 8 y y <2 × − yˆ if ymax > yˆ, ymax “ ”2 ε= y :−0.5 × yˆ − otherwise, ymax
(9)
where
y˜ − 0.2 , (10) 0.8 − 0.2 y and ymax is the maximum observed real displacement (the term serves to normalize the observed displacements). Note ymax that the constants 2, 0.5, 0.8, and 0.2 could all be adjusted, but we have found that these are reasonable values for our application.4 yˆ =
y ) equals the The basis for Eq. (9) is that the ideal state is that in which each normalized displacement measurement ( ymax corresponding normalized prediction (ˆ y): in these cases the corresponding penalty or error measure is zero. Deviations from this ideal state are penalized based on whether the actual displacement is more or less than “expected.” If the actual displacement is more than expected (the first case in Eq. (9)), the penalty is more severe, and if the actual displacement is less than expected (the second case) the penalty is not as severe.
As discussed in Section 3, the most common statistical model for the residuals is the normal distribution model with expectation zero. As Eq. (9) is broken into two case statements, we would not expect the residuals to have a symmetric distribution about the origin, and the analysis results confirm that they do not. Nevertheless, we will still apply a normal distribution model for the residuals. From an implementation standpoint such a formulation is preferred, both because of its more intuitive behavior 3 Note that internally to the MCMC algorithm we actually work with log θ , which enables the sampler to explore values on the entire real 1 line, since θ1 ≥ 0. 4 In fact, here 0.2 represents the smallest slip tendency for which we expect any actual measured displacement, and 0.8 is the slip tendency that we expect to correspond with the largest measured displacement.
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564 and because the use of a likelihood function based on a skewed distributions can create convergence problems for the MCMC sampler. From a theoretical standpoint, the principle of maximum entropy provides some justification for the use of the normal distribution5 , even though the residuals are not expected to be symmetric about zero.
4.2.3
ADDRESSING CORRELATIONS AMONG MEASUREMENTS
As discussed in Section 3, the Bayesian inference procedure requires the formulation of a statistical model for the residuals, εi , i = 1, . . . , n. The usual approach is to assume that the residuals are independently and identically distributed, each following a normal distribution with zero mean and some (possibly unknown) variance. However, whether or not the assumption that the residuals are independent is appropriate deserves careful consideration, and such a decision will need to be made on a case by case basis. The important factor to consider is whether or not the individual measurements are related to one another. For example, if the measurements are taken at closely spaced intervals in time (or similarly closely spaced locations in space) from the same experiment, then it is likely that the assumption of independence is not appropriate. This consideration is particularly important, because from an uncertainty standpoint, assuming independence is non-conservative. For the current work, it is not clear beforehand whether or not the assumption of independence will be appropriate, and in fact we will compare results obtained but with and without such an assumption. Recall that the n field measurements are all taken from the same geographical region. It is possible that certain measurements may have been taken at locations that are close enough together (possibly on the same fault) that these measurements are not actually providing independent information about fault displacement (picture a geologist measuring fault displacement at one location and then moving one centimeter along the fault and taking another measurement of displacement). The first challenge is to decide how the correlations among measurements will be characterized. The natural mechanism is through the formulation of a correlation function, defined in terms of some observable coordinates associated with each measurement (perhaps a subset of the independent variables s). Two approaches were considered: (a) a correlation function defined in terms of the northing and easting (coordinates defining the geographic location of each measurement), and (b) a correlation function defined in terms of the strike and dip angles of each measured fault (which are the also the elements of s). The first approach may seem more intuitive, particularly to non-geologists, but it admits the possibility of recognizing two geographically “close” measurements as being correlated, even though they may be on completely separate faults. The second approach makes more sense from a geological perspective, as the strike and dip angles are what define a particular fault, and in conjunction with a stress state, what govern the slip tendency. Taking the second approach, the correlation function is formulated as6 " „ «2 „ « # dipi − dipj 2 strikei − strikej c(yi , yj ) = exp − − , φ1 φ2
(11)
where φ1 and φ2 are the correlation lengths associated with the strike and dip angles, respectively. As discussed in Section 3, the full Bayesian approach would treat the correlation lengths as unknowns. For the current analysis, though, such an approach is not practical because it involves the inversion of the n × n correlation matrix at each step in the Markov Chain Monte Carlo simulation. A more practical approach is to estimate the correlation lengths a priori and then treat them as known constants for the remainder of the analysis. This can be done using the common “two-stage” estimation procedure[20] in which a point estimate for the calibration parameters θ is first obtained under the assumption that the residuals are independent, and then a maximum likelihood procedure is used to obtain point estimates for the correlation lengths. The likelihood function for the unknowns is now based on the multivariate normal probability density function. If we denote the n × n correlation matrix by R, then the likelihood function is given by » – ` ´−1/2 1 f (y | θ, σ 2 ) = (2π)−n/2 σ 2n |R| exp − 2 εT R−1 ε , (12) 2σ where ε is the vector of residuals, ε1 , . . . , εn . 5 The normal distribution is a maximum entropy distribution among all continuous distributions with known mean and standard deviation. As such, its use introduces a certain amount of conservatism into the results. 6 Note that because the model G(·, ·) is deterministic, the correlation among the residuals is the same as the correlation among the field measurements.
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565 4.3
Results
For the first analysis, the observations are assumed to be independent, so that the likelihood function of Eq. (4) is employed. The stress state is estimated based on n = 329 measurements of real slip displacement7 , with corresponding observations of the strike and dip angles (s) for each fault. The posterior distribution is constructed using Markov Chain Monte Carlo sampling with a multivariate Gaussian random walk proposal density to obtain 20,000 samples from the posterior distribution8 . The proposal covariance matrix is automatically adapted based on the history of the chain[26] , and additional re-scaling of the step sizes is also used to achieve an efficient exploration of the parameter space. After convergence, excellent mixing properties were observed, as shown in Figure 2.
Figure 2: Markov Chain for three parameters after convergence.
The posterior mode of the estimated stress state is described in Table 1. Recall that the stress state is derived from the five calibration parameters used in the Bayesian inference process (see Section 4.2.1). The stress state is displayed graphically in the stereograph plot of Figure 3. The space of the stereograph plot is defined over the strike and dip coordinates, so it represents a visualization of the slip tendency as a function of fault orientation.
TABLE 1: Posterior mode for estimated stress state treating observations as independent Principal Stress σ1 σ2 σ3
Magnitude 100 56 31
Azimuth 30◦ 52◦ 140◦
Plunge 70◦ -19◦ 7◦
The most likely (mode) stress state from the posterior can also be visually compared with the observed data. This is done in Figure 4, which plots the orientations of the measured faults on top of the slip tendency stereograph. The measured faults are broken into three categories: high, medium, and low displacement. For good agreement between the predictions and observations, we expect the high measured displacements to correspond to regions of high slip tendency, the medium measured displacements to mostly also correspond with regions of high slip tendency, and the low measured displacements to mostly appear in regions of moderate slip tendency. Visually, the stress state corresponding to the posterior mode achieves excellent 7 Some 8 After
of the 348 measurements mentioned in Section 2.3 were not used in the analysis because they contained incomplete information. convergence, 80,000 samples were generated, and every fourth sample was stored.
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Figure 3: Stress state corresponding to posterior mode assuming independent observations. The colored contours represent slip tendency as a function of the strike and dip (fault orientation).
agreement between predicted and observed slip. Notice that the high-displacement measurements have a fault orientation that corresponds to roughly the highest predicted slip tendency.
(a) High Displacement
(b) Medium Displacement
(c) Low Displacement
Figure 4: Comparison of predicted slip tendency (at the posterior mode stress state, assuming independent observations) with measured fault displacements. Each field measurement corresponds to a dot on the stereograph plot.
The correlations among the stress state parameters are tabulated in Table 2. Note that Table 2 lists σ 2 and σ 3 as opposed to the actual parameters used in the calibration analysis, θ1 and θ2 : while virtually no correlation is seen between the stress magnitudes, a very strong negative correlation of -0.93 occurs between the parameters θ1 and θ2 . Otherwise the correlations among the parameters are mostly minor, with the largest being 0.54 between the azimuth and tilt angles. This relationship is depicted graphically by the joint confidence regions for these parameters in Figure 5. The analysis was also repeated using a more realistic statistical model for the residuals (and hence the observations). For this second analysis, a correlation function was developed for the observations in terms of the fault orientation (see Section 4.2.3). This correlation function is given in Eq. (11) and aims to capture the idea that measured displacements on faults with very similar orientations might not provide statistically independent information.
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TABLE 2: Observed correlation matrix among stress state parameters σ 2 σ 3 Azimuth Plunge Tilt
σ 2 1. 0.1 0.26 0.49 0.01
σ 3 0.1 1. 0.15 0.23 0.01
Azimuth 0.26 0.15 1. 0.1 0.54
Plunge 0.49 0.23 0.1 1. -0.03
Tilt 0.01 0.01 0.54 -0.03 1.
Figure 5: Joint confidence regions for the global azimuth and tilt angles.
Note that the correlation function introduces two additional parameters, φ1 and φ2 , which are the correlation lengths for the strike and dip angles, respectively. As discussed in Section 4.2.3, the large number of field observations makes it computationally prohibitive to treat the correlation lengths as objects of Bayesian inference. Instead, a two-stage estimation process is used in which the residuals from the posterior mode of the preceding analysis are used to estimate the correlation lengths in a maximum likelihood framework. Doing so, the correlation lengths are estimated as 15.2◦ and 0.7◦ for the strike and dip angles, respectively. These correlations reflect a negligible amount of correlation associated with the dip angle and a small amount associated with the strike angle. The issue is somewhat complicated in this application, though, because of the bimodal nature of the fault orientation. The strike angle measures the compass angle of the fault orientation, with the convention that an angle between zero and 180◦ indicates a fault dipping to the west, and an angle between 180◦ and 360◦ denotes a fault dipping to the east. Faults separated by strike angles of 180◦ are known as conjugate sets and are equally likely to occur (this is readily apparent from Figure 4(b)). As such, defining the correlation function in terms of a difference between two strike angles may not be completely appropriate. Nevertheless, we demonstrate such an approach here and leave the possibility of more rigorous approaches for future work. Having estimated the correlation lengths, we treat them as fixed constants for the remainder of the analysis. They are then used to construct the correlation matrix for the observations, R, which is used in the likelihood function of Eq. (12). Because of the relatively short correlation lengths, we do not expect the effect of incorporating correlations to be significant, and the results confirm this expectation. The posterior mode, listed in Table 3, is very similar to the previous result, and in fact the two are not visually distinguishable. Normally, we would expect the introduction of correlations to have a significant impact on the resulting uncertainty in the posterior distributions, because correlated observations provide less information than independent observations. However, in this case, the correlations are small enough that the effect is not large. Figure 6 shows the marginal posterior distribution for the global
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TABLE 3: Posterior mode for estimated stress state using correlation function Principal Stress σ1 σ2 σ3
Magnitude 100 55 30
Azimuth 32◦ 53◦ 140◦
Plunge 68◦ -20◦ 7◦
azimuth angle (which is also the azimuth angle associated with the second principal stress). Here we see only a small difference, and in fact the posterior obtained with the correlation model shows less uncertainty (while this is the opposite of what is expected, the uncertainty in other parameters increased, as expected). Figure 7 plots joint 95% confidence regions for the magnitudes of the second and third principal stresses (recall that the magnitude of the first is arbitrarily set at 100). Again, only a small change is seen, and it is in fact statistically insignificant at the 95% confidence level. The posterior standard deviations are listed in Table 4.
Figure 6: Comparison of marginal posterior distributions for the azimuth angle for independent and correlated observations.
TABLE 4: Posterior standard deviations from analyses treating observations as independent and correlated
Independent Correlated
5
σ 2 2.3 2.4
σ 3 0.49 0.50
Azimuth 2.4 2.3
Plunge 2.2 2.1
Tilt 0.95 0.97
CONCLUSIONS
In industries such as oil and gas, software tools are used to predict slip tendency for faults in the earth’s crust, which aids in energy exploration and production as well as reservoir estimation. The stress state is an important factor in the slip behavior and it is required as an input for such tools. The stress state may be estimated from available geological data, but previous methods for doing so have typically involved subjective tuning of the stress state, requiring expert input and judgement. This paper presents a new approach for stress state estimation from field data (also referred to as stress state inversion) by employing calibration analysis methodology. A Bayesian approach is presented, whereby a best-fitting estimate of the stress state is obtained in addition to a comprehensive representation of the accompanying uncertainty based on the amount of field data available.
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Figure 7: Comparison of 95% joint confidence regions for the magnitudes of the second and third principal stresses for independent and correlated observations.
The stress state estimation process is demonstrated using actual geological field data obtained from the Canyon Lake Spillway Gorge in Comal County, Texas. Two analyses are considered, one which aims to model the correlations present among measurements from similar fault surfaces, but no significant differences are seen between the results. The stress state obtained is shown to produce excellent agreement between the predicted and observed fault slip. Several quantitative and graphical summaries of the uncertainty associated with the estimated stress state are presented as well. However, the uncertainty representation may be most useful when the stress state is used to obtain new predictions, in which case slip tendency confidence bounds could be easily derived via uncertainty propagation.
ACKNOWLEDGEMENTS The research reported in this paper was funded as part of an Internal Research and Development project at the Southwest Research Institute.
REFERENCES [1] Gephart, J. W. and Forsyth, D., An improved method for determining the regional stress tensor using earthquake focal mechanism data: Application to the San Fernando earthquake sequence, Journal of Geophysical Research, Vol. 89, pp. 9305–9320, 1984. [2] Angelier, J., Determination of the mean principal directions of stresses for a given fault population, Tectonophysics, Vol. 56, pp. T17–T26, 1979. [3] Angelier, J., Tectonic analysis of fault slip data sets, Journal of Geophysical Research, Vol. 89, pp. 5835–5848, 1984. [4] Tezuka, K. and Niitsuma, H., Stress estimated using microseismic clusters and its relationship to the fracture system of the Hijiori hot dry rock reservoir, Engineering Geology, Vol. 56, pp. 111–130, 2000. [5] Blenkinsop, T., Lisle, R. and Ferrill, D. A., Introduction to the special issue on new dynamics in palaeostress analysis, Journal of Structural Geology, Vol. 28, pp. 941–942, 2006. [6] Sibson, R. H., Fluid involvement in normal faulting, Journal of Geodynamics, Vol. 29, pp. 469–499, 2000. [7] Morris, A. P., Ferrill, D. A. and Henderson, D. B., Slip tendency and fault reactivation, Geology, Vol. 24, pp. 275–278, 1996.
BookID 214574_ChapID 49_Proof# 1 - 23/04/2011
570 [8] Streit, J. E. and Hillis, R. R., Estimating fault stability and sustainable fluid pressures for underground storage of CO2 in porous rock, Energy, Vol. 29, pp. 1445–1456, 2004. [9] Lisle, R. J. and Srivastava, D. C., Test of the frictional reactivation theory for faults and validity of fault-slip analysis, Geology, Vol. 32, pp. 569–572, 2004. [10] Collettini, C. and Trippetta, F., A slip tendency analysis to test mechanical and structural control on aftershock rupture planes, Earth and Planetary Science Letters, Vol. 255, pp. 402–413, 2007. [11] Barton, C. A., Zoback, M. D. and Moos, D., Fluid flow along potentially active faults in crystalline rock, Geology, pp. 683–686, 1995. [12] Ferrill, D. A., Winterle, J., Wittmeyer, G., Sims, D., Colton, S., Argmstrong, A. and Morris, A. P., Stressed rock strains groundwater at Yucca Mountain, Nevada, GSA Today, Vol. 9, pp. 1–8, 1999. [13] Stock, J. M., Healy, J. H., HIckman, S. H. and Zoback, M. D., Hydraulic fracturing stress measurements at Yucca Mountain, Nevada, and relationship to regional stress field, Journal of Geophysical Research, Vol. 90, pp. 8691–8706, 1985. [14] Wiprut, D. and Zoback, M. D., Fault reactivation, leakage potential, and hydrocarbon column heights in the northern North Sea, Hydrocarbon Seal Quantification, edited by Koestler, A. G. and Hunsdale, R., Vol. 11 of NPF (Norwegian Petroleum Society) Special Publication, pp. 203–219, 2002. [15] Zhang, X., Koutsabeloulis, N. and Heffer, K., Hydromechanical modeling of critically stressed and faulted reservoirs, American Association of Petroleum Geologists Bulletin, Vol. 91, pp. 31–50, 2007. [16] Zoback, M., Barton, C., Finkbeiner, T. and Dholakia, S., Evidence for fluid flow along critically-stressed faults in crystalline and sedimentary rock, Faulting, Faults Sealing and Fluid Flow in Hydrocarbon Reservoirs, edited by Jones, G., Fisher, Q. and Knipe, R., pp. 47–48, University of Leeds, London, 1996. [17] Takatoshi, I. and Kazuo, H., Role of stress-controlled flow pathways in HDR geothermal reservoirs, Pure and Applied Geophysics, Vol. 160, pp. 1103–1124, 2003. [18] Ferrill, D. A. and Morris, A. P., Fault zone deformation controlled by carbonate mechanical stratigraphy, Balcones fault system, Texas, AAPG Bulletin, Vol. 92, pp. 359–380, 2008. [19] Ward, W. C. and Ward, W. B., Stratigraphy of the middle part of Glen Rose Formation (Lower Albian), Canyon Lake Gorge, Central Texas, USA, Cretaceous Rudists and Carbonate Platforms: Environmental Feedback, edited by Scott, R. W., Vol. 87 of SEPM Special Publication, pp. 193–210, 2007. [20] Seber, G. A. F. and Wild, C. J., Nonlinear Regression, John Wiley & Sons, Inc., Hoboken, New Jersey, 2003. [21] McFarland, J., Uncertainty Analysis for Computer Simulations through Validation and Calibration, Ph.D. thesis, Vanderbilt University, 2008. [22] McFarland, J. M., Mahadevan, S., Romero, V. J. and Swiler, L. P., Calibration and uncertainty analysis for computer simulations with multivariate output, AIAA Journal, Vol. 46, No. 5, pp. 1253–1265, 2008. [23] McFarland, J. M., Urbina, A. and Mahadevan, S., A top-down approach to the calibration of computer simulations, Proceedings of the International Modal Analysis Conference XXVII, Orlando, FL, February 2009. [24] Kennedy, M. C. and O’Hagan, A., Bayesian calibration of computer models, Journal of the Royal Statistical Society B, Vol. 63, No. 3, pp. 425–464, 2001. [25] Lee, P., Bayesian Statistics, an Introduction, Oxford University Press, Inc., New York, 2004. [26] Haario, H., Saksman, E. and Tamminen, J., An adaptive Metropolis algorithm, Bernoulli, Vol. 7, No. 2, pp. 223–242, 2001.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Effectiveness of Modeling Thin Composite Structures Using Hex Shell Elements
Ricardo M. Garcia, Jr. D. Gregory Tipton Sandia National Laboratories PO Box 5800 Albuquerque, NM 87185 NOMENCLATURE CUBIT FEA FEM MAC Salinas SNL
SNL Automated Mesh Generation Toolkit Finite Element Analysis Finite Element Model Modal Assurance Criterion SNL Solid Mechanics Code Sandia National Laboratories
ABSTRACT This paper investigates the effectiveness of modeling thin composite structures with hex shell elements for structural dynamics simulation. The current finite element modeling method for an existing three-layer composite aerospace structure uses solid 8-noded hex elements. It is relatively expensive in terms of the number of degrees of freedom and element count. A finer mesh typically results in a more accurate solution, however, the computation time increases. Modal analysis was used to test if a single layer of hex shell elements for each material could replace multiple layers of solid hex elements, enabling computational savings. Element aspect ratio was varied on a solid hex model of a frustum part to optimize the technique. The hex shell modeling technique was then applied to the existing three-layer composite structure. The analysis results, when compared to validation data obtained from tests performed on the actual hardware, exhibit very satisfactory agreement. A single layer of hex shell elements are capable of providing solutions that are equivalent to multiple layers of hex elements. A considerable savings in element count and solution equations result. A broader understanding of modeling options for future, more efficient methods of modeling composite shell structures is also obtained. 1. INTRODUCTION Traditional methods for modeling composite shell structures using finite elements can be expensive in terms of the number of degrees of freedom and element count. This can be worth the computational effort depending on the use of the model. For long simulations or large sets of simulations, however, a reduced model may be preferred to provide tractable run times. In finite element modeling, a finer mesh typically results in a more accurate solution, but the computation time increases. Performing a mesh convergence and element size study can provide guidance to produce a mesh that satisfactorily balances accuracy and computing resources. The objective of this paper is to investigate the effectiveness of modeling thin composite structures with hex shell elements. The approach is to compute the natural frequencies and mode shapes of a composite shell structure for various mesh densities. “Truth” results are generated by creating several meshes with 8-noded hex elements and extrapolating to the converged solutions. Hex shell meshes of the structure with varying aspect ratio are then compared to the “truth” model. It is shown that utilizing three layers of hex shell elements in a three-layered composite structure can produce similar frequencies to those of multiple solid hex elements. A reduction in element count and computation time for a converged solution is the result.
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_51, © The Society for Experimental Mechanics, Inc. 2011
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CUBIT, a Sandia National Laboratories automated mesh generation toolkit, was used to produce the meshes for this study [5]. It is a full-featured software toolkit for robust generation of two- and three-dimensional finite element meshes (grids) and geometry preparation. Salinas, a Sandia National Laboratories developed solid mechanics code, was used to perform the analysis for this investigation [3]. It provides a massively parallel implementation of structural dynamics finite element analysis, required for high fidelity, validated models used in modal, vibration, static, and shock analysis. Salinas has been extensively verified and solves the equations correctly for this class of problem. The model for the shell structure used in this study is composed of three material layers. The external layer is an orthotropic fiber-reinforced composite. A “bond” layer, composed of filled rubber, acts as a glue to connect the inner layer to the outer layer. The internal layer is a metal substructure. Although the fiber-reinforced composite is orthotropic, it was a reasonable assumption to model it as isotropic. The modulus of elasticity and Poisson’s ratio are similar in the two planar directions. A simple conical shell, shown in Figure 1, is modeled as a layered frustum and is used as the validation problem for this investigation. Only a quarter section of the shell structure is modeled, and symmetry conditions are imposed on the “cut” edges. No displacement was allowed normal to the surface of the “cut”, but lateral displacements were permitted. All other faces of the shell are free.
Figure 1: Layered shell structure geometry. 2. HEX SHELL ELEMENTS Hex shell elements (sometimes referred to as solid shell elements or thick shell elements) are available in Salinas and provide a shell element capability with a hex element topology [1,4]. These elements provide an intermediate capability between a typical shell element and a continuum element. These elements are based on the assumed natural strain formulation, but with modifications to account for constant stress through the thickness instead of constant strain. This allows for multiple material layers to be modeled within the element. These elements assume transverse isotropy which is believed to be a valid assumption for the shell structures considered here. Practically speaking, these elements allow for the modeling of thin structures with hex-like elements, but without concern for the bad aspect ratio and the number of elements through the thickness. Because hex shells have an inherent thickness direction, it is important to be able to identify that direction. In addition, assumptions about extracting a mid-plane to mesh are irrelevant with a hex shell, thus the geometry is preserved. Hex shells can also be used in conjunction with regular hex elements for complicated structures. It is important to note that extracting a mid-plane to mesh is a time consuming, arduous and ambiguous process. While this approach works well for simple part geometries with uniformly thin walls, it does not capture the true phenomena occurring in the model. Significant accuracy can be lost on parts with a moderate to high level of detail, variable wall thickness and/or thick and bulky areas. Bulky parts with varying wall thickness cannot be accurately represented with mid-planes and require a more advanced solution. Solid mesh elements fill the entire volume of the part geometry, without the modifications and assumptions associated with a mid-plane. This results in a much better representation of the original part, and therefore much more accurate simulation results.
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Because some shell structures vary in thickness along the cross-section, modeling each material separately is advantageous. If a single hex shell is used with three material layers, the model would have to be split up into many element blocks along the axis of the structure to capture the changing thickness. By modeling each material layer with separate hex shell element blocks, thickness changes are captured naturally with the element geometry. 3. DEVELOPING A TRUTH MODEL The shell structure shown in Figure 1 was first meshed with 8-noded hex elements to establish a baseline or “truth” model. An initial mesh size was chosen to give 1 element through the thin middle layer with a 3:1 aspect ratio (Figure 2a). To establish converged results, this mesh was refined twice, splitting the elements in half in each direction. The first of these refinements is depicted in Figure 2b. The second refinement resulted in 4 elements through the thin middle layer with a 3:1 aspect ratio.
bond layer
metal substructure
fiber-reinforced composite
3:1 aspect ratio (bond layer)
(a)
(b)
Figure 2: Baseline and first level refined models meshed with traditional 8-noded hex elements. The first twenty modes of vibration were computed with the baseline, 2x2x2 refinement, and 4x4x4 refinement meshes. Richardson extrapolation was performed to estimate the converged frequencies to establish the “truth” results. The converged natural frequencies are listed in Table 1.
Modes of Vibration
Converged Frequencies (Hertz)
Modes of Vibration
Converged Frequencies (Hertz)
1
0
11
2029.044
2
116.662
12
2393.301
3
192.982
13
2473.548
4
505.836
14
2516.158
5
689.991
15
2519.69
6
1014.711
16
2688.391
7
1250.349
17
2794.034
8
1641.02
18
2843.779
9
1945.861
19
2850.309
10
2028.199
20
2961.505
Table 1: Converged frequency results for the baseline model.
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For completeness, coarser meshes were also analyzed, and all the hex results are plotted in Figure 3. The frequencies have been normalized by the extrapolated (converged) results presented in Table 1. As can be seen, the frequencies are very nearly converged with the finest mesh refinement.
Salinas Hex Runs 1.030
Normalized Frequency (Hz)
1.024
1.018
1.012
1.006
1.000 0.00
0.10 Hex Mode 1 Hex Mode 6 Hex Mode 11 Hex Mode 16
Hex Mode 2 Hex Mode 7 Hex Mode 12 Hex Mode 17
0.20 0.30 Element Edge Length Hex Mode 3 Hex Mode 8 Hex Mode 13 Hex Mode 18
0.40 Hex Mode 4 Hex Mode 9 Hex Mode 14 Hex Mode 19
0.50 Hex Mode 5 Hex Mode 10 Hex Mode 15 Hex Mode 20
Figure 3: Normalized frequency vs. element edge length for the baseline model. 4. HEX SHELL MODEL RESULTS With converged “truth” results in hand, the shell structure was meshed with 1 hex shell element in each material layer. Aspect ratios for elements in the thin middle layer of 24:1 to 1.5:1 were analyzed. Three of these meshes are depicted in Figure 4.
(a)
(b)
(c)
Figure 4: Hex shell mesh refinement with (a) 24:1 aspect ratio in the middle layer, (b) 12:1 aspect ratio, and (c) 1.5:1 aspect ratio. Table 2 shows the solutions obtained for the hex shell meshes. As can be seen, the results converge as the aspect ratio is lowered. Figure 5 plots the frequencies as a function of aspect ratio. These frequencies are normalized by the extrapolated (converged) values presented in Table 1. Corresponding to the positive association between frequency error and increasing element aspect ratio, an upward trend is observed.
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Salinas Hex Shell Runs Frequency (Hertz) Double - 1.5:1
3:1
5:1
7:1
9:1
12:1
15:1
18:1
21:1
0
0
0
0
0
0
0
0
0
24:1 0
116.649
116.669
116.707
116.765
116.84
116.988
117.193
117.419
117.659
118.045
192.92
192.956
193.041
193.137
193.255
193.429
193.667
193.893
194.092
194.507
505.992
506.208
506.614
507.281
508.141
509.859
512.248
514.944
517.885
522.744
690.182
690.444
690.959
691.706
692.646
694.373
696.758
699.338
702.049
706.715
1015.48
1016.36
1017.98
1020.7
1024.21
1031.25
1041.16
1052.56
1065.3
1086.95
1251.46
1252.44
1254.3
1257.3
1261.16
1268.74
1279.38
1291.47
1304.82
1327.66
1643.28
1645.8
1650.42
1658.23
1668.36
1688.9
1718.14
1752.44
1791.78
1860.97
1948.93
1951.74
1957
1965.81
1977.22
2000.23
2032.99
2071.2
2092.87
2119.76
2029.34
2030.6
2033.11
2036.91
2042.08
2051.3
2065.05
2079.38
2114.69
2150.86
2029.86
2031.67
2035.09
2040.23
2047.02
2059.32
2077.23
2096.16
2114.87
2191.07
2398.53
2404.36
2415.09
2433.39
2457.24
2486.18
2494.36
2502.77
2510.57
2526.56
2473.5
2474.13
2475.57
2477.77
2480.77
2506.16
2522.39
2523.48
2524.47
2526.78
2519.64
2519.73
2519.97
2520.25
2520.63
2521.32
2576.9
2652.18
2694.54
2700.3
2520.89
2524.21
2530.45
2540.28
2553.09
2577.55
2612.76
2662.03
2696.46
2772.19
2688.38
2688.51
2688.89
2689.35
2690.04
2691.19
2693.06
2694.91
2762.23
2853.89
2800.81
2807.36
2819.54
2840.3
2845.2
2846.18
2847.74
2849.31
2850.66
2903.03
2843.84
2843.92
2844.24
2844.63
2858.06
2863.48
2871.74
2880
2887.38
2945.29
2850.6
2851.26
2852.76
2854.97
2867.39
2922.79
3002.84
3076.55
3104.41
3169.57
2963.04
2965.86
2971.67
2980.11
2992.07
3012.4
3044.42
3098.59
3165.22
3195.44
Table 2: Hex shell mesh results for different element aspect ratios.
Salinas Hex Shell Runs 1.140
Normalized Frequency (Hz)
1.112
1.084
1.056
1.028
1.000 0
5 Hex Shell Mode 1 Hex Shell Mode 6 Hex Shell Mode 11 Hex Shell Mode 16 Converged
9 Hex Shell Mode 2 Hex Shell Mode 7 Hex Shell Mode 12 Hex Shell Mode 17
14 Aspect Ratio Hex Shell Mode 3 Hex Shell Mode 8 Hex Shell Mode 13 Hex Shell Mode 18
18 Hex Shell Mode 4 Hex Shell Mode 9 Hex Shell Mode 14 Hex Shell Mode 19
23 Hex Shell Mode 5 Hex Shell Mode 10 Hex Shell Mode 15 Hex Shell Mode 20
Figure 5: Normalized frequency vs. aspect ratio for the hex shell meshes. The modal frequencies calculated for the hex shells are within 14% of the converged results for the first twenty modes (coarsest aspect ratio, 24:1). For a more reasonable aspect ratio like 9:1, the frequency errors are less than 3% of the converged results with a considerable decrease in the number of elements. When compared to the 4x4x4 fine hex mesh, the element count was reduced from approximately four and a half million to a little
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more than two thousand elements. The percent error in frequency for the 9:1 aspect ratio model is shown in Figure 6 for the first twenty modes.
Percent Error Between 9:1 and ʘexact 3.00
2.50
Percent Error
2.00
1.50
1.00
0.50
0.00 1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Mode
Figure 6: Percent error in frequencies for the 9:1 aspect ratio hex shell model. The results above show that the natural frequencies predicted by the hex shell models are equivalent to those of the “truth” model. In Figure 7, the modes from the 9:1 aspect ratio hex shell model are compared to the finest hex model through the MAC calculation. As can be seen, the mode shapes are all equivalent with a minimum value of 0.99 on the diagonal.
Figure 7: MAC between the finest hex model modes and the 9:1 aspect ratio hex shell modes. The hex shell modeling technique was applied to an existing aerospace structure for which modal test data was available [2]. Based on the hex shell results presented above, an aspect ratio of 9:1 was chosen to model the structure. This aspect ratio appeared to provide a good balance between element count and solution accuracy. The model results are compared to the modal test data in Table 3 and Figure 8 below. As can be seen, the maximum frequency error from the test results is 6.69%. Less than 10% error was considered acceptable for the structure. It should be noted that this aerospace structure is more complicated than the simple truth model with which the hex shell modeling technique was developed. However, the frequency error is acceptable for the purpose of this model.
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Shell Structure Modal Frequency Comparisons Flexible Body Mode No.
Mode Description
Hex Shell Frequency (Hertz)
Test Frequency (Hertz)
% Error - Hex Shell & Test
7
2,0 ovaling
601.081
581
3.46
8
2,0 ovaling - orthogonal
602.062
588
2.39 3.36
9
3,0 ovaling
1393.33
1348
10
3,0 ovaling - orthogonal
1393.64
1350
3.23
11
2,1 ovaling
1616.68
1653
-2.20
12
2,1 ovaling - orthogonal
1616.74
1670
-3.19
13
1st bending
1619.92
1647
-1.64
14
1 bending - orthogonal
1622.12
1647
-1.51
15
3,1 ovaling
2076.22
2139
-2.94
16
3,1 ovaling - orthogonal
2076.81
2141
-3.00
17
4,0 ovaling
2398.75
2372
1.13
18
4,0 ovaling - orthogonal
2423.07
-
-
st
st
19
1 torsional
2424.58
2317
4.64
20
4,1 ovaling
2973.8
3187
-6.69
21
4,1 ovaling - orthogonal
2973.86
-
-
22
2nd bending
3108.19
3136
-0.89
23
2nd bending - orthogonal
3108.44
3142
-1.07
24
1st axial
3112.06
3128
-0.51
Table 3: Shell structure modal frequency comparisons.
6.00
4.00
Percent Error
2.00
0.00 1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
-2.00
-4.00
-6.00
-8.00 Mode
Figure 8: Percent error between hex shell and test models. 5. CONCLUSIONS This study shows that the Salinas hex shell elements are capable of providing solutions that are acceptably accurate to the full 8-noded hex elements for a class of structures of interest. A single layer of hex shell elements can replace multiple layers of 8-noded hex elements. A considerable savings in element count and solution equations result from using hex shell elements. A reduction factor of 2250:1 in elements was demonstrated when comparing the 4x4x4 fine hex mesh to the 9:1 aspect ratio model. Not only are there fewer elements required through the thickness of a particular material layer, but the mesh density can be lower without concern for bad
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aspect ratios. The aspect ratio of the hex shell elements can be much larger than the solid hex elements and provide equivalently accurate solutions. 6. ACKNOWLEDGEMENTS Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. 7. REFERENCES [1]
Felippa, C. A., “The SS8 Solid Shell Element: Formulation and a Mathematica Implementation,” Center for Aerospace Studies, University of Colorado, CU-CAS-02-03, 2002.
[2]
Mayes, R. L., Miller, A. K., Holzmann, W. A., Tipton, D. G., Adams, C. R., “A Structural Dynamics Model Validation Example with Actual Hardware,” IMAC-XXVII Proceedings, February 2009.
[3]
Reese, G., Segalman, D., Bhardwaj, M. K., Alvin, K., Driessen, B., Pierson, K., Walsh, T., “Salinas User Notes,” Sandia National Laboratories, SAND99-2801, Albuquerque, NM, 2009.
[4]
Sze, K. Y., Yao, L.-Q., Cheung, Y. K., “A Hybrid Stabilized Solid Shell Element with Particular Reference to Laminated Structures,” Proc ECCOMAS 2000, Barcelona, Spain, 2000.
[5]
“CUBIT 11.1 User Documentation,” Sandia National Laboratories, Albuquerque, NM, 2009.
BookID 214574_ChapID 52_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
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BookID 214574_ChapID 52_Proof# 1 - 23/04/2011
597
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BookID 214574_ChapID 52_Proof# 1 - 23/04/2011
598 Comparison of reconstructed component table response with field data
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BookID 214574_ChapID 52_Proof# 1 - 23/04/2011
599
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BookID 214574_ChapID FM_Proof# 1 - 23/04/2011
BookID 214574_ChapID 53_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
A Dual Approach to Substructure Decoupling Techniques
S.N. Voormeeren and D.J. Rixen Delft University of Technology, Faculty of Mechanical, Maritime and Materials Engineering Department of Precision and Microsystem Engineering, section Engineering Dynamics Mekelweg 2, 2628CD, Delft, The Netherlands [email protected]
ABSTRACT In recent years, the structural dynamic community showed a renewed interest in dynamic substructuring (i.e. component coupling) techniques, especially in an experimental context. In this context the problem of propagation of uncertainty due to measurement errors was also investigated. In this paper the reverse problem is addressed: the decoupling (or identification) of a substructure from an assembled system. This problem arises when substructures cannot be measured separately but only when coupled to neighboring substructures, a situation regularly encountered in practice. Using a dual approach to substructure (dis)assembly, three substructure decoupling techniques will be derived in a unified way. Moreover, their accuracy due to measurement errors will be investigated by performing an uncertainty propagation analysis. The techniques are applied to a simulated experiment.
NOMENCLATURE Y Z u f g λ B L C E + S s σ() Δ
1 1.1
– – – – – – – – – – – – – – –
Receptance FRF matrix Dynamic stiffness FRF matrix Response vector External force vector Connection force vector Lagrange multiplier General Boolean matrix Boolean localization matrix Compatibility Boolean matrix Equilibrium Boolean matrix Generalized (pseudo) inverse Belonging to subsystem S Belonging to set of DoF s Standard deviation Confidence interval
INTRODUCTION Substructure Decoupling
Dynamic substructuring (DS) techniques have been well established over the past decades. These techniques consist in constructing the structural dynamic model of a large and complex system by assembling the dynamic models of its simpler components (also called subsystems or substructures). In recent years, the structural dynamic community showed a renewed interest in these substructure coupling techniques, especially in the context of experimental applications [6, 17].
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_53, © The Society for Experimental Mechanics, Inc. 2011
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One of the issues that has been investigated in this respect was the effect of random measurement errors on the accuracy of the coupled system description [8, 20]. However, sometimes one has to consider the reverse problem, namely how a substructure model can be found from the assembled system. This is a relevant issue for subsystems that cannot be measured separately, but only when coupled to their neighboring substructure(s). This can for example be the case for very delicate subsystems or substructures in operational conditions [12]. To illustrate the problem at hand, consider the subsystems A and B shown in figure 1 (a); when assembled they form system AB. In a dynamic substructuring analysis, the dynamics of assembly AB are obtained by coupling the dynamic models of A and B. In substructure decoupling, the reverse problem is solved. In this case, it is assumed that the dynamic models of the assembled system AB and the substructure A are known (e.g. from measurements). Based on this information, the aim is to find the dynamics of subsystem B as a “stand alone” component, that is, completely decoupled from subsystem A. Practical applications of substructure decoupling can be imagined in structural monitoring and vibration control techniques, where monitoring and controlling of individual (critical) components in an assembly can be very valuable. However, as outlined in [4], quite a number of challenges remain in the practical implementation of decoupling techniques. One important issue is the sensitivity of decoupling techniques to small (measurement) errors, especially around the antiresonances of the known component [18].1 This paper aims at writing the different decoupling techniques in a common framework in order to understand how they differ and how they are related, and if possible deriving more robust decoupling techniques using a so called “dual” (dis)assembly approach. 7ZAR Z{ ZS{BZ}j$tU `y `
`
`K
` \
y
= \4
wZt,tBbtD
` 4
`K
+ \ w,tBbtD
W 7 ; WK
4 w,tBbtD
7 L /
7ZAR Z{ ZSI{BZ}j$tU
= 4 wZt,tBbtD
\
– \4
` L ; `K
w,tBbtD
wsD
4
\ w,tBbtD wAD
Figure 1: Substructure coupling (dynamic substructuring) and substructure decoupling (a); Finding the uncoupled response of subsystem B (b).
1.2
Problem Description
To illustrate the problem of substructure decoupling more thoroughly, let us consider the situation depicted in figure 1 (b). Suppose one is interested in the uncoupled dynamics of component B. To this end, the dynamics of the assembled system AB and the component A are assumed to be known in the form of frequency response functions or in short FRFs (assuming the systems are linear and time invariant). For the sake of illustration, suppose that we want to obtain the response uo of component B at DoF o, while it is excited by a force fi at DoF i, without the influence of neighboring subsystem A. Both DoF are internal to subsystem B and hence are part of the DoF set ub . The situation is depicted in figure 1 (b). In general, the decoupling problem can now be described as follows: 1 Note that in substructure decoupling the sensitivity is highest around the anti-resonance frequencies of the known subsystem(s), while in substructure coupling (DS) the sensitivities are found to be highest around the resonance frequencies of the subsystems [20].
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• The force fi excites the system AB at DoF i internal to component B. • As a result, the assembled system AB shows a response uAB • Now take only the part of the response of AB associated to component B and realize that in addition to the excitation force fi , subsystem B in the assembly AB is also subjected to (connection) forces of component A. • Additional forces opposing these connection forces should thus be applied to the assembly AB in order to let B behave without “feeling” the influence of A. • Using the FRFs of uncoupled system A, one can determine these interface connection forces loads by imposing to the uncoupled model the coupled responses of subsystem A. Summarizing, one can now formulate the decoupling problem as finding the behavior of substructure B as part of the assembled system AB when additional forces are applied at the interface such that substructure B experiences no connection forces from subsystem A. Hence, substructure B behaves as if it were decoupled from A. The decoupling problem can be expressed in terms of equations by starting with the FRF description of the assembled system AB: uAB = Y AB f AB − g AB ⎡ AB ⎤ AB AB Yaa Yac Yab ua fa 0 (1) AB AB AB uc fc − gc = ⎣ Yca Ycc Ycb ⎦ AB ub fb 0 Yba YbcAB YbbAB and the FRF matrix of subsystem A is assumed to be known too: uA = Y A f A + g A A A
A
A 0 ua Yaa Yac fa = + A gcA uA Yca YccA fcA c
(2)
Here u is the substructure vector of degrees of freedom (DoF), Y the substructure receptance matrix and f the external force vector. The vector g represents the additional disconnection forces (with non-zero entries only at the interface DoF) felt from the coupling/decoupling of neighboring components. The subscripts a, b and c denote “internal to subsystem A”, “internal to subsystem B” and “coupling”, respectively; the superscripts A, B and AB denote the two subsystems and the assembled system. The explicit frequency dependency is omitted for clarity. To find the decoupled response ub of subsystem B to a force fb , we can start by writing the third equation in eq. (1) as (since fa = 0): ub = YbbAB fb − YbcAB gc . In this equation only the connection forces gc are unknown. To find an expression for these forces, use can be made of the compatibility condition. This condition states that the displacements at the interface DoF that need to be coupled (or decoupled) must be compatible, i.e.: uc = u A c
(3)
An expression for the interface forces can now be found by using the response of the interface DoF from eqs. (1) and (2): AB YccA gcA = uA = YcbAB fb − YccAB gc c = uc
Next, we will express that applying the forces predicted by the local problem on A to the interface of AB must leave the interface DoF of B as if they would be free, or in other words that the connection forces between the substructures are in equilibrium: gc + gcA = 0 Hence, the connection forces can now be expressed as: −1 AB gc = YccAB − YccA Ycb fb .
(4)
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The above expression can be plugged into the expression for ub . The resulting uncoupled frequency response functions Ybb of subsystem B, in terms of the FRFs of A and AB, then become: −1 AB ub = YbbAB − YbcAB YccAB − YccA Ycb fb = YbbB fb (5) Using this approach, the complete decoupled FRF matrix of subsystem B can be reconstructed. There is however another, more systematic, way of obtaining this decoupled FRF, namely starting from a so called “dual” formulation. This approach will be discussed in the next section. 1.3
A Dual Formulation
In the above discussion, the decoupled receptance matrix of component B was found in a rather “ad hoc” manner, using the receptance matrices of A and AB (eqs. (2) and (1)) employing the interface compatibility and equilibrium conditions. A more systematic approach can be taken when starting from a dynamic stiffness representation of the subsystems by describing system AB as ⎡
AB Zaa AB ⎣ Zca AB Zba
AB Zac AB Zcc AB Zbc
AB Zab AB Zcb AB Zbb
Z AB uAB ⎤ ua ⎦ uc ub
= f AB − g AB fa 0 fc − gc = fb 0
(6)
and subsystem A as
A Zaa A Zca
A Zac A Zcc
Z A uA = f A + g A
A A
0 ua fa = + . gcA uA fcA c
(7)
Here u is the same DoF vector as before, Z the dynamic stiffness matrices, f the external force vector and g again the vector of connection forces (with non-zero entries only at the interface DoF). The compatibility and equilibrium conditions can be expressed as before, i.e. as in eqs. (3) and (4). However, two Boolean matrices can be introduced to write these conditions more compactly and allow a more systematic description of the problem. The first is a signed Boolean matrix B, operating on the substructure interface degrees of freedom. Using the partitioning of DoF as written in eqs. (6) and (7), this matrix can be: 0 −I ] B = B AB B A = [ 0 I 0 The B matrix has a number of rows equal to the total number of connections defined between the substructures, while the number of columns equals the total number of DoF of the substructures to be (dis)assembled. Using this Boolean matrix, the compatibility condition can be conveniently expressed as
AB uAB A Bu = B = uc − uA (8) B c = 0. uA The second Boolean matrix L localizes the interface DoF of the substructures in the global set of DoF and is similar to the localization matrices used in the assembly of individual elements in finite element models. In this case, the L matrix is ⎡ ⎤ I 0 0 0 AB ⎢ 0 0 0 I ⎥ L ⎢ ⎥ L= = ⎢ 0 I 0 0 ⎥, LA ⎣ 0 0 I 0 ⎦ 0 0 0 I so that the equilibrium condition can be stated as: ⎡ T
L g=
L
AB T
AT
L
g AB gA
⎢ ⎢ ⎢ =⎢ ⎢ ⎣
0 0 0 0 0 gc + gcA
⎤ ⎥ ⎥ ⎥ ⎥=0 ⎥ ⎦
(9)
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Interestingly, L actually represents the nullspace of B or vice versa (see [7]): L = null (B) B T = null LT
(10)
In a primal approach, the interface conditions are satisfied by introducing one unique set of interface DoF. This way, the interface compatibility is a priori satisfied and as a result the interface connection forces are eliminated, thereby satisfying the equilibrium condition as well.2 This approach was implicitly taken in the previous section. In a dual approach however, the equilibrium condition is chosen to be satisfied a priori. This is obtained by choosing the interface forces in the form g = BT λ Here, λ are Lagrange multipliers, corresponding physically to the interface force intensities. By choosing the interface forces in this form, they act in opposite directions for any pair of dual interface degrees of freedom, due to the construction of Boolean matrix B. The equilibrium condition in eq. (9) thus becomes LT g = LT B T λ = 0. Because LT is in the nullspace of B T , see eq. (10), this condition is always satisfied. The compatibility condition as shown in eq. (8) should however be taken into account explicitly as an additional equation. The decoupling problem can therefore be formulated in a dual way as: ⎧ T ⎪ Z AB uAB + B AB λ = f AB ⎪ ⎨ T Z A uA − B A λ = f A ⎪ ⎪ ⎩ AB AB B u + B A uA = 0 One can transform this to matrix vector notation as: ⎡ AB ⎤⎡ ⎤ ⎡ ⎤ T Z 0 B AB uAB f AB T ⎣ 0 Z A −B A ⎦ ⎣ uA ⎦ = ⎣ f A ⎦ AB λ 0 B BA 0 However, this system of equations is non-symmetric. By multiplying the second equation by minus 1, the system becomes symmetric: ⎡ AB ⎤ ⎡ ⎤ T ⎤⎡ Z 0 B AB uAB f AB T ⎣ 0 (11) −Z A B A ⎦ ⎣ uA ⎦ = ⎣ −f A ⎦ AB A λ 0 B B 0 This last relation clearly shows that the decoupling of a subsystem from a total system is equivalent to a dual assembly of a negative dynamic stiffness for the substructure that one wants to subtract (here substructure B). The actual uncoupled FRFs of B can now be found by eliminating the Lagrange multipliers. At first, start by writing explicitly the substructure DoF as: −1 T uAB = Z AB f AB − B AB λ (12) uA = −Z A
−1
T
−f A − B A λ
Substitution in the compatibility condition and solving for λ gives: −1 −1 T −1 T −1 B AB Z AB f AB λ = B AB Z AB B AB − B A Z A B A Here it is assumed that f A = 0; since this component is “subtracted” there is no excitation at its DoF. Substitution in the expression for uAB (eq. (12)) gives the decoupled responses: −1 −1 −1 T −1 T −1 T −1 uAB = Z AB − Z AB B AB B AB Z AB B AB − B A Z A B A B AB Z AB f AB 2 More
details on primal and dual substructure assembly can be found in [7].
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It should now be realized that the dynamic stiffness matrices are the inverse of the receptance matrices, i.e Z AB
−1
= Y AB
and Z A
−1
= Y A.
Hence one can write: −1 AB AB AB AB T AB AB AB T A A AT AB AB u = Y −Y B B Y B −B Y B B Y f AB
(13)
A clear physical interpretation can be given to the above expression. To this end, let us express the above equation as: T
uAB = Y AB f AB − Y AB B AB Zint uint where
−1 T Zint = B AB Y AB B AB + B A Y A B A uint = B AB Y AB f AB
This form of the decoupling problem can now be interpreted as follows: • The term Y AB f AB represents the response of the assembled system AB to an external excitation f AB ; • This also leads to interface displacements uint ; • However, these interface displacements are due to the combined stiffness of A and B. Therefore, a corrected interface stiffness Zint must be calculated to eliminate the influence of substructure A; • The adjusted interface stiffness times the interface displacements, Zint uint leads to a correction force at the interface DoF; T
• This force correction is spread to the other subsystem DoF by multiplication by Y AB B AB . Next, we can insert the expressions for the receptance matrices of systems A and AB in (13) and calculate the products with the Boolean matrices. This gives: ⎤ ⎡ AB ⎤ ⎞ ⎛⎡ AB AB AB Yaa Yac Yab Yac ua fa −1 AB A AB AB uc = ⎝⎣ Yca (14) YccAB YcbAB ⎦ − ⎣ YccAB ⎦ Ycc − Ycc Yca YccAB YcbAB ⎠ fc AB ub fb Yba YbcAB YbbAB YbcAB If one now tries to obtain the same transfer function as before, i.e. the response ub to an excitation force fb , one can start by noting that again fa = 0 but also fc = 0 since the interface connection forces are present in the Lagrange multipliers and no external excitation is assumed at the interface DoF. Insertion in the previous equation and extracting the third row then gives the expression for the uncoupled FRFs Ybb of B: −1 AB ub = YbbAB − YbcAB YccA − YccAB Ycb fb (15) Evidently, the expression for YbbB is exactly equal to the one found earlier in eq. (5). So, starting from a dual formulation in terms of dynamic stiffness FRFs, the same decoupled (receptance) FRFs can be obtained as starting from a receptance representation of the systems. Finally, note that the responses ua in eq. (14) correspond to the responses of the internal DoF of A assembled system AB when additional interface forces are applied and are generally not of interest. 1.4
Paper Outline
The idea outlined above will be further expanded in the next section. From the dual formulation, three methods for substructure decoupling will be derived. Section 3 thereafter presents an uncertainty analysis of these techniques in order to investigate the sensitivity of the methods to small (measurement) errors in the FRFs. The results of a case study are presented in section 4, while the paper is ended by some conclusions and recommendations in section 5. Note that all the discussions in this paper consider the case of decoupling two substructures for the sake of illustration. Nevertheless it is straightforward to generalize the approach for any number of substructures that need to be coupled and/or decoupled.
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607 W7 ; WK
`L ; `K
7 L . \4
W 7 ; WK
.\
`L ; `K
7 L .4
Figure 2: Substructure decoupling from a dual perspective.
2
A DUAL FRAMEWORK FOR SUBSTRUCTURE DECOUPLING
Based on the dual formulation of the previous section, one can now go one step further in by realizing that the total system AB itself can be written as a dual assembly of systems A and B, so: ⎡ ⎤⎡ ⎤ ⎡ ⎤ T ZA 0 BA uAB fAAB A ⎢ T ⎥⎢ ⎢ AB ⎥ AB ⎥ Z AB = ⎢ ZB BB ⎥ ⎣ 0 ⎦ ⎣ uB ⎦ = ⎣ fB ⎦ 0 BA BB 0 λAB This can be inserted in eq. (11) and writing (from now on it will be assumed that f A = 0): ⎡ ⎤ ⎡ AB ⎤ ⎡ T T ⎤ uA ZA 0 BA 0 CA fAAB T ⎢ 0 ⎥ ⎢ AB ⎥ ⎢ AB ⎥ Z B BB 0 0 ⎢ ⎥ ⎢ uB ⎥ ⎢ fB ⎥ ⎢ BA BB ⎥ ⎢ AB ⎥ ⎢ 0 0 0 ⎢ ⎥⎢ λ ⎥=⎢ 0 ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ 0 ˜ A −C AT ⎦ ⎣ uA ⎦ ⎣ 0 ⎦ 0 0 −Z 0 λ CA 0 0 −C A 0
(16)
Here the Boolean matrix B AB has been replaced by [C A 0 0] (zeros have been added at the Lagrange multipliers λAB ) and B A = −C A , in order to emphasize that the Boolean matrix used for the decoupling of substructures is not necessarily the same as the one used (implicitly) for coupling. Note that the dynamic stiffness matrix of the separate ˜ A , while the dynamic stiffness of A in the total system AB is denoted by Z A . Taking component A now is denoted by Z the above expression as a starting point, three approaches to the decoupling problem can be formulated. These will be discussed next. 2.1
Standard Decoupling
The easiest way of solving the decoupling problem of eq. (16) is by choosing C A = B A . This means that compatibility and equilibrium are only required at the interface DoF between substructure A and assembled system AB. Basically, in this approach the connection forces between the substructures (that are used to eliminate the influence of A in the response of AB) are determined using the minimum information needed, namely only the responses on the interface DoF of subsystem A. We will refer to this approach as “standard decoupling”. The Boolean matrix can in this case be expressed as: C A = [0ca
Icc ]
The subscripts c and a respectively refer to the number of interface and internal DoF of A. Consequently, the Boolean matrix acting on AB is found as: C AB = [0ca
Icc
0cb ]
The decoupled FRFs of B can be found by inserting this choice for the Boolean matrix in eq. (16). For the sake of simplicity, the assembled system AB is expressed simply as Z AB . This leads to the description of the decoupling problem in the form of eq. (11), i.e.: ⎤ ⎡ ⎡ AB ⎤ T ⎤⎡ Z 0 C AB uAB f AB T ⎣ 0 (17) −Z A C A ⎦ ⎣ uA ⎦ = ⎣ 0 ⎦ AB A 0 λ C C 0
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By eliminating the Lagrange multipliers λ and solving for the degrees of freedom uB , the decoupled FRFs of B are found. Not surprisingly, since the same Boolean matrix is used, the result will be exactly the same as found in the previous section in eq. (14). In practice however, when dealing with measured FRF matrices, measurement errors are inevitable. As a result the separately measured FRFs of component A are slightly different from the ones (implicitly) measured when system A is ˜ A ≈ Z A . Exactly these small discrepancies between the measured part of AB. Hence, eq. (16) is slightly modified since Z components can cause the standard decoupling method to fail on a practical problem, as they amplify to large errors on the decoupled FRFs [2]. 2.2
Decoupling with Additional Internal DoF
Compared to the standard approach to decoupling as described above, a somewhat more clever approach can be taken. The idea is very simple: in addition to using the information at the interface DoF to determine the connection forces, one might also use the knowledge of the internal DoF of subsystem A. As their responses are also (partly) due to the connection forces, this can be valuable additional information for determining those forces. This is believed to reduce the ˜ A ≈ Z A and therefore enhances the estimation of the decoupled FRFs. This approach problems described above when Z has been termed “extended interface” decoupling in [5]. In this approach (some of) the internal DoF are also taken into account in the compatibility and equilibrium conditions. In the limit case where all internal DoF of subsystem A are used, the Boolean matrix for component A becomes identity:
Iaa 0 A C = 0 Icc Consequently, the Boolean matrix acting on AB is found as:
Iaa 0 0 AB C = 0 Icc 0 The decoupling problem is now found by inserting these Boolean matrices in eq. (17). We can proceed by eliminating the Lagrange multipliers and solving for the uncoupled responses, as in section 1.3. This gives the following result:
ua uc ub
⎛ = ⎝Y AB
⎡
AB Yaa AB − ⎣ Yca AB Yba
⎤ AB AB Yac Yaa YccAB ⎦ AB Yca AB Ybc
AB Yac YccAB
−
A Yaa A Yca
A Yac YccA
−1
AB Yaa AB Yca
AB Yac YccAB
AB Yab YcbAB
⎞
fa ⎠ fc fb (18)
Indeed, the complete FRF matrix Y A is now used to find the uncoupled responses of B. Solving the decoupling problem with additional internal DoF as outlined above also raises the question what internal DoF should be taken into account. An answer to this question is not very easy to find. Of course, when performing decoupling using measured data one can use some indicators (e.g. coherence functions) to filter out badly measured FRFs, but currently no general criteria exist. ˜ A ≈ Z A . In case Z ˜ A = Z A , the Furthermore, it should be noted that the above approach only works when indeed Z problem becomes singular at all frequencies, as was already observed in [5]. Physically, this is due to the fact that additional Lagrange multipliers are introduced to satisfy equilibrium at the internal DoF. However, due to the compatibility condition being enforced exactly at all DoF, these Lagrange multipliers become redundant when the FRFs of substructure A are measured as being the same in AB and in A alone. In [5] it is proposed to overcome the singularity by applying truncated SVD techniques to perform the inversion. 2.3
Non-Collocated Compatibility and Equilibrium Conditions
The decoupling methodology can be further generalized by realizing that a certain freedom exists in the choice of DoF on which to enforce compatibility and equilibrium. In other words, it is not required to enforce compatibility and equilibrium on
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the same (number of) DoF. Physically this translates to applying forces at some set of DoF in order to satisfy equilibrium at another set of DoF. As long as the controllability properties of these DoF sets are correct, this is perfectly possible. This idea can be translated into equations by taking different Boolean matrices for the compatibility and equilibrium conditions, as: ⎡ AB ⎤ ⎡ ⎤ T ⎤⎡ Z 0 E AB uAB f AB T ⎣ 0 (19) −Z A E A ⎦ ⎣ uA ⎦ = ⎣ 0 ⎦ AB 0 λ C CA 0 Here E are the Boolean matrices governing the equilibrium while C are the matrices enforcing compatibility. Solving the above equation for the uncoupled responses as before then gives + AB AB AB AB T AB AB AB T A A AT AB AB u = Y −Y E C Y E −C Y E C Y f AB , (20) T T where + denotes the (Moore-Penrose) pseudo-inverse, since the term C AB Y AB E AB − C A Y A E A is now no longer necessarily a square matrix. The trick is now to make a smart choice for both interface conditions such that the influence of possible errors in measured FRFs of A and AB are minimized. Such a minimization may be obtained by choosing the Boolean matrices for the enforcing the compatibility as
Iaa 0 Iaa 0 0 CA = , and C AB = 0 Icc 0 Icc 0 while for the Booleans that govern the equilibrium condition we choose: E A = [ 0ca
and E AB = [ 0ca
Icc ]
Icc
0cb ] .
This choice for the Boolean matrices corresponds to enforcing compatibility at all DoF of A (interface and internal) while the equilibrium condition is satisfied by introducing Lagrange multipliers only at the interface DoF. Hence, we have a + c compatibility conditions and only c Lagrange multipliers to solve. As before we can insert these Boolean matrices in eq. (19), eliminate the Lagrange multipliers and solve for the uncoupled responses. This gives: ⎤ ⎡ AB ⎤ ⎞ ⎛⎡ AB AB AB AB A + AB
fa Yaa Yac Yab Yac ua AB AB Yac Yac Yaa Yac Yab AB ⎠ fc uc = ⎝⎣ Yca − YccAB YcbAB ⎦ − ⎣ YccAB ⎦ AB A AB AB Y Y Y Y YcbAB AB AB AB AB cc cc ca cc ub fb Yba Ybc Ybb Ybc The result is indeed decoupling from an overdetermined set of equations, sometimes called the “mobility approach” in literature [4, 18, 5]. It is easy to see that in this method the decoupling is performed by solving for the interface forces in a least squares sense. Using the non-collocated compatibility and equilibrium conditions, many variations can be imagined. Note however that in order to end up with a solvable set of equations, it should always hold that rank (C ) ≥ rank (E ) ≥ c, where c is the number of interface (coupling ) DoF. Another interesting possibility is to choose for the compatibility Boolean matrices CA = [ 0
Icc ]
C AB = [ 0
,
Icc
0 ]
and for the equilibrium matrices E A = [ Ica
0cc ]
,
E AB = [ Ica
0cc
0cb ]
This corresponds to applying forces at the internal DoF of A such that the interface compatibility condition can still be satisfied. This requires that the interface DoF of A are controllable from inputs at its internal DoF. Using the above choice for the Boolean matrices, one obtains the following expression for the uncoupled responses: ⎤ ⎡ AB ⎤ ⎞ ⎛⎡ AB AB AB Yaa Yac Yab Yaa ua fa −1 AB A AB AB AB AB AB AB AB ⎠ fc uc = ⎝⎣ Yca Ycc Ycb ⎦ − ⎣ Yca ⎦ Yca − Yca Yca Ycc Ycb AB AB ub fb Yba YbcAB YbbAB Yba In this way, the inversion of (possibly sensitive) driving point FRFs on the interface can be avoided. Moreover, it should be noted that knowledge of the driving point FRFs on the interface is not required in case fc = 0 and one is not interested in uc (i.e. only in the internal FRFs of B). This can be beneficial in practical applications, where driving point FRFs of the interface DoF may be hard to obtain.
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3
UNCERTAINTY ANALYSIS
Although from the previous sections it is evident that the theory of substructure decoupling is rather straightforward, the practical application of decoupling techniques remains troublesome. This is a problem experienced as well in the field of substructure coupling techniques (dynamic substructuring) [6]. In particular, decoupled substructure models seem to be very sensitive to errors in the (experimental) description of the other subsystem(s) and assembled system. From a general point of view, two kinds of errors can be made in (sub)system measurements and modeling. The first type, called bias errors, are systematic errors which lead to measured values being systematically offset. Careful design of an experiment and/or model creation allows one to reduce the influence of those errors. The second kind of error, which is addressed here, is more random of nature. Random errors are fluctuations that can be evaluated through statistical or interval analysis. When performing dynamic measurements, all sorts of random errors (“measurement noise”) can be encountered, such as round-off errors in A/D conversion, sensor noise, varying impact locations from a hammer test, etc. From a broader perspective, one can also count uncertain parameter values (e.g. geometric dimensions or material properties) as a random error, which is especially important when coupling and decoupling a mix of numerical models and measurements. In contrast to systematic errors, random errors are generally due to factors that cannot be controlled. Therefore they introduce uncertainty, following the definition of uncertainty of Hazelrigg [10]: “In an experiment, when the sample space contains more than one element with non-zero probability, there is uncertainty.” An important question is how these uncertainties in the assembled and subsystem descriptions propagate, and possibly amplify, in the substructure decoupling process. In dynamic substructuring, common belief is that small errors in a subsystem interface description can lead to significant discrepancies in the coupled system’s representation, due to the numerical conditioning of the subsystems’ interface flexibility matrix [1, 13, 3, 16]. Since this matrix needs to be inverted in the coupling process, ill conditioning could severely magnify the small errors in subsystems. Similar problems for decoupling techniques are described in [18, 4, 5]. In this paper an uncertainty propagation analysis will be performed to quantify this effect for the decoupling techniques, allowing a comparison of the sensitivity of the three methods to small errors. Such an uncertainty analysis can be a valuable tool in the light of model verification and validation. The remainder of this section is organized as follows. First, the general theory of uncertainty propagation is outlined in section 3.1, thereafter these methods will be applied to the decoupling techniques in 3.2.
3.1
Theory of Uncertainty Propagation – Moment Methods
The study of uncertainty propagation comprises the determination of a function’s uncertainty based on the uncertainties of its input variables. Different approaches exist to investigate the uncertainty propagation from a number of inputs to an output [11]. In the uncertainty propagation analysis in this paper the so called ‘moment’ method will be used, an efficient, sensitivity based method for uncertainty analysis. The approach taken here is analog to [20, 8]. For the sake of compactness, the theory will be summarized in this section, details can be found in the aforementioned papers. In statistics, the ‘moments’ are properties that characterize a variable’s probability distribution. The most common statistical moments are the following four ‘central’ moments (taken about the mean): 1. The first moment corresponds to the mean of the distribution. 2. The second moment represents the variance. 3. The third moment is the skewness, expressing the symmetry of a distribution. 4. The fourth moment is the kurtosis, describing the distribution’s ‘peakiness’. In many cases however, not all four (central) moments are required to characterize a probability distribution. A normal or Gaussian probability distribution for example is completely defined by its first two moments. A Gaussian distribution is often a good approximation of the random external influences on a measurement, and will therefore be used in the uncertainty propagation method derived here. Indeed, in [19] it was found that the random influences on a real-life structural dynamic measurement closely obey this distribution.
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Assume now a set of n stochastic input variables xi assembled in vector x [x1 · · · xn ] , which have known mean values x ¯i and standard deviations σ(xi ). Let g be a function of the variables in x. The moments of the function g(x) can then be calculated from a truncated Taylor series expansion about the mean value of the input variables. Hence, the approach taken here is a special type of sensitivity analysis. In this paper it is assumed that the input uncertainties are small and Gaussian distributed, and that all functions considered are continuous and can be linearized around the mean value ¯ In that case a first order Taylor series expansion suffices to obtain approximations for the first of the input variables x. and second moments. These approximations are usually called first order, first moment (FOFM) and first order, second moment (FOSM) approximations, respectively. Approximating the function g(x) by a first order Taylor series around the mean values of the input variables gives the first moment of the function g(x) as: ¯ E[g(x)] = g(x).
(21)
Using some statistics [9] and assuming that the variables are uncorrelated, one finds the variance (the second moment) of the function as: 2 n ! ∂g Var[g(x)] = Var[xi ]. (22) ∂xi i=1 The zero correlation assumption might be somewhat crude, as it is not expected that random errors on measured structural dynamic signals (e.g. forces and accelerations) will be fully uncorrelated. However, this assumption is still made as it considerably simplifies the subsequent analysis. Note that this assumption can be regarded valid as long as the noise on the measured mechanical signals is dominated by uncorrelated random influences (e.g. sensor resolution). When the noise on the signals is ‘mechanical’ of nature (e.g. vibrations from the environment, fluctuations in applied excitation, etc.), the errors on the measured signals will be highly correlated due to the physical structure. Hence, the excitation should have a deterministic character, the resulting practical implications have been addressed in [8]. In practice it is convenient to express the second moment in terms of the standard deviation, which has the same unit as the function itself. Since the standard deviation is the positive square root of the variance and the standard deviation in the input variables was defined as σxi , the standard deviation of the function g(x) is found to be " # n 2 #! ∂g $ σ(g(x)) = σ(xi ) . (23) ∂xi i=1 This last equation forms the starting point for the uncertainty propagation derivation for the decoupling techniques, although it first needs to be generalized in case the function g is a vector/matrix function. The derivative of a matrix with respect to any of its entries may be written as ∂G Pij , ∂Gij
(24)
where matrix Pij is a ‘Boolean’ type of matrix with the same size as matrix G and the elements of Pij are all zero except for entry (i, j), which equals one. With this definition one can now express equation (23) in matrix form as "⎧ ⎫ "⎧ ⎫ # %2 ⎬ # #⎨! n ! m #⎨! n ! m ⎬ ∂G # # 2 σ(G) = $ σ(Gij ) =$ {Pij σ(Gij )} , (25) ⎩ ⎭ ⎩ ⎭ ∂Gij i=1 j=1 i=1 j=1 where G has dimension n-by-m. Here the curly-bracket notation {· · ·}, indicates that the square and square root operations must be performed elementwise. Note that expression (25) thus simply states that the standard deviation of a matrix is the sum of the standard deviations of its entries. Another helpful result needed for the upcoming uncertainty propagation analysis is the derivative of the inverse matrix G−1 to its elements Gij [14]: ∂G−1 ∂G −1 = −G−1 G = −G−1 Pij G−1 . ∂Gij ∂Gij
(26)
With some matrix algebra and the definition of the Moore-Penrose pseudoinverse it can be easily shown that it also holds that: ∂G+ ∂G + = −G+ G = −G+ Pij G+ . ∂Gij ∂Gij
(27)
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3.2
Uncertainty Analysis of Decoupling Techniques
Now the theory of uncertainty propagation using moment methods has been outlined, the decoupling methods discussed in section 2 can be analyzed. The assembled and subsystem FRFs serve as input for the decoupling techniques, it is therefore assumed that the uncertainty on these FRFs is known. How to obtain the uncertainty on individual FRFs from measured excitations and responses is described in [20]. Furthermore, the uncertainty on the FRFs is assumed to be specified in terms of confidence intervals instead of in terms of standard deviations. These intervals are denoted by Δ and are specified here at the 95% level. Describing uncertainty using confidence intervals allows to express the uncertainty on the mean FRFs and includes the influence of repeated measurements [20].3 Note that the theory from the previous section is equally valid for confidence intervals. Basically, the uncertainty analysis is to a large extent similar as the one performed in [20], since the set of equations that form the starting point are largely similar. Here, we will perform the uncertainty analysis on the most general form of the dual decoupling problem, i.e. the “non-collocated” approach in eq. (20). From this equation, the decoupled FRFs of B can be expressed as: + T T T Y dec = Y AB − Y AB E AB C AB Y AB E AB − C A Y A E A C AB Y AB (28) Since the “standard” and “extended interface” methodologies only differ from the “non-collocated” in the choice of the Boolean matrices, the uncertainty analysis is also valid for these approaches. In order to calculate the confidence interval of the decoupling equation, one should first realize that the decoupled FRFs are a function of the FRFs in Y AB and Y A : Y dec = Y dec Y AB , Y A Therefore, the derivative of Y dec to the FRFs in these matrices is needed. To find the derivative, it should be noted that there are no ‘duplicate’ FRFs in the matrices Y AB and Y A . This means that, for example, when deriving Y dec to an FRF in Y A , the first term in eq. (28) will be zero. The derivative is now found as: ∂Y dec ∂Y AB ∂Y AB ABT ∂Y AB ABT ∂Y A AT ∂Y AB = − E D2 + D1 C AB E D2 − D1 C A E D2 − D1 C AB , (29) AB AB AB A ∂Yij ∂Yij ∂Yij ∂Yij ∂Yij ∂YijAB where use was made of the product rule on the second term in equation (28) and D1 and D2 are defined as + T T T Δ D1 = Y AB E AB C AB Y AB E AB − C A Y A E A + T T Δ D2 = C AB Y AB E AB − C A Y A E A C AB Y AB
(30)
By combining the above and equation (25), one finds the confidence interval on the decoupled FRFs of subsystem B
ΔY¯ dec =
=
") # # * * + dec ∂Y $ i
") # # * * . $ i
j
j
AB ∂Yij
Pij −Pij E
ΔY¯ijAB
AB T
,2
+
* * + ∂Y dec k
l
A ∂Ykl
D2 +D1 C AB Pij E
AB T
ΔY¯klA
,2
-
D2 −D1 C AB Pij
/ . / * * 2 2 T ΔY¯ijAB + D1 C A Pkl E A D2 ΔY¯klA k
l
Note again that the uncertainties on the receptance matrices of AB and A (ΔY¯ijAB and ΔY¯klA ) are assumed to be known. Furthermore, note that the square and square root operations must be performed elementwise as indicated by the curly-bracket notation. 3 This can be interpreted as follows: after 100 measurements the average FRF magnitude at a certain frequency is found, for example, as Y = 10.0, while √ the standard deviation is 0.2.√One can therefore say with 95% confidence that the true mean FRF magnitude is between 10 − 1.96 · 0.2 100 = 9.61 and 10 + 1.96 · 0.2 100 = 10.39. The factor 1.96 originates from the fact that the confidence intervals is stated at the 95% level, which corresponds to 1.96 times the standard deviation. Now suppose one would take another 300 measurements, giving a total of 400 measurements. Even though the average and standard deviation might √ not have changed, the 95%√confidence interval has now narrowed so that the true mean magnitude is somewhere between 10 − 1.96 · 0.2 400 = 9.80 and 10 + 1.96 · 0.2 400 = 10.20.
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4
CASE STUDY
In this section the results of a case study are presented. The simple problem used for this study is shown in figure 3 and consists of two lightly damped mass-spring-damper systems (modal damping < 1%). Subsystem A has 7 degrees of freedom, subsystem B possesses 4 DoF and the systems are coupled at a 2 DoF interface.
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The remainder of this section is organized as follows. The next section presents the application of the decoupling techniques to the simple decoupling problem of figure 3. Thereafter, section 4.2 shows some results from the uncertainty analysis.
4.1
Calculation of Decoupled FRFs
˜A ≈ ZA To resemble a practical situation, the FRFs of A in AB and the FRFs of A separately are not exactly equal, i.e. Z as in eq. (16). The disturbance on the FRFs of A was a normally distributed random error on the amplitude with a 95% confidence interval of ±5% and a normally distributed random error on the phase with a 95% confidence interval of ±2o . This was done to resemble real-life measurement situations, where external disturbances result in both amplitude and phase uncertainty. Note that a different random error was taken for each frequency; in total 200 frequency points were evaluated. The results of the decoupling using the three methods are shown in figure 4 for an arbitrary FRF of B; Decoupled FRF YB1B4 - Magnitude
30
True Standard Extended Non-collocated
20
0
Phase [rad]
Magnitude [dB]
10
-10 -20
3 2 1 0 -1
-30
-2
-40
-3
-50
0
10
20 30 Frequency [Hz]
40
50
Decoupled FRF YB1B4 - Phase
4
-4
0
10
20 30 Frequency [Hz]
40
50
Figure 4: Decoupled FRF of subsystem B obtained with three decoupling techniques vs. true FRF of B.
the “true” FRF is also shown. The following compatibility and equilibrium conditions were chosen: • “Standard” method: compatibility and equilibrium only at the two interface DoF.
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• “Extended interface” method: compatibility and equilibrium at all DoF of A, so at seven DoF. • “Non-collocated” method: compatibility at all (seven) DoF of A, equilibrium only at the (two) interface DoF. As one can see, the three decoupling methods seem to perform identical for this system. All three methods show some spurious peak around 6 Hz, but seem to resemble the true FRF quite nicely at the other frequencies. Note however that due to the dB-scale the results may seem better then they are and using the FRFs for subsequent manipulations (e.g. for a modal identification or a substructuring analysis) might give problems. To zoom in on the actual error, an uncertainty analysis has been performed. This will be discussed next.
4.2
Uncertainty Analysis
˜ A ≈ Z A , with the same error on A as mentioned above, in order to avoid The uncertainty analysis is performed with Z conditioning problems with the “extended interface” method. Additionally, confidence intervals have been assumed on the FRFs of AB and A of the same size as before, i.e. a 95% confidence interval of ±5% on the amplitude and ±2o on the phase. With this data the uncertainty analysis as outlined in the previous section has been performed. The result is shown in figure 5. The same FRF of B is considered as before and the relative confidence intervals on the magnitude (the confidence interval on the magnitude divided by the magnitude of the FRF) and the confidence intervals on the phase are plotted for the three methods. As can be seen, the “standard” and “extended interface” methods perform identical, they
Standard Extended Non-collocated
Relative error | ΔY / Y| [%]
1200 1000 800 600 400 200 0
0
10
20 30 Frequency [Hz]
40
50
Error on decoupled FRF YB1B4 - Phase
3.5 Phase error ∠(Y + ΔY) - ∠ Y [rad]
1400
Error on decoupled FRF YB1B4 - Magnitude
3 2.5 2 1.5 1 0.5 0
0
10
20 30 Frequency [Hz]
40
50
Figure 5: Comparison of the uncertainty of the three decoupling techniques on the magnitude (relative) and phase of an decoupled FRF.
produce exactly the same uncertainty on the decoupled FRF. An unambiguous reason for this finding cannot yet be given. Furthermore, this is in contrast to the findings in [5], where a clear difference was reported between the performance of the standard method and the extended interface method, in favor of the latter. This is believed to be due to the fact that in this paper a truncated SVD inversion is used instead of a normal inverse. This implicitly leads to an approach similar as for the non-collocated method, since some (the smallest) singular values are thrown away which physically corresponds to forcing some Lagrange multipliers to zero and thus implicitly gives an overdetermined system. Note that the standard and extended interface methods yield relative confidence intervals on the amplitude of decoupled FRFs of more then 1000% at some frequencies. This is an error amplification of more than a factor 200. More or less the same holds for the phase angle, where the input error of ±2o is amplified to phase errors of 2.5 radians or approximately ±150o . Note that in the region between 20 and 30 Hz the phase error is wrongly shown due to an “unwrapping” error. The “non-collocated” method on the other hand seems to generally give much smaller confidence intervals on the decoupled FRF. At some frequencies, the method performs up to ten times better (i.e. it produces relative errors that are ten times smaller) then the other two methods. Still it gives pretty large confidence intervals at some frequencies. The effect of these large confidence intervals is shown in figure 6. Here the average decoupled FRF, obtained without noise,
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is shown with the confidence interval shown as a grey band. The likelihood that a decoupled FRF is in this band is 95%. One can see that for the standard and extended interface decoupling techniques at certain frequencies the intervals become so large, that “spurious peaks” can appear [15]. This can lead to problems when, for example, a modal identification is performed on the decoupled subsystem. The non-collocated decoupling technique suffers from this problem to a lesser extent. Decoupled FRF YB1B4 with Confidence Interval 30
20
20
20
10
10
10
0 -10 -20 -30
0 -10 -20 -30
Average FRF
-40 -50
Magnitude [dB]
30
Magnitude [dB]
Magnitude [dB]
30
0
20 40 Frequency [Hz]
60
-50
-10 -20 -30
Average FRF
-40
CI - 'Standard'
0
0
20 40 Frequency [Hz]
Average FRF
-40
CI - 'Extended Int.'
60
-50
CI - 'Non-collacted'
0
20 40 Frequency [Hz]
60
Figure 6: Uncoupled subsystem FRF with its confidence interval (CI) shown as a band of possible outcomes.
5
CONCLUSIONS & RECOMMENDATIONS
In this paper a dual framework was presented for substructure decoupling techniques. From this framework, three different types of decoupling methods were derived using different Boolean matrices for the compatibility and equilibrium conditions imposed on the structures that need to be decoupled. A “standard” decoupling method was found when equilibrium and compatibility were only at the interface DoF; an “extended interface” method was found when both conditions were also enforced at internal DoF of the known subsystem. A third method was proposed in which equilibrium and compatibility were chosen at non-collocated DoF, such that an the decoupled FRFs are determined in a least squares sense. Furthermore, this method allows to compute the internal responses of the decoupled subsystem without knowledge of the driving point FRFs on the interface. The three methods were subjected to a uncertainty analysis based on a sensitivity method for finding statistical moments in order to assess their accuracy when small, random errors are present in the dynamic systems’ descriptions. Subsequently, both approaches were applied to a numerical decoupling problem. This case study showed that the standard and extended interface decoupling methods perform identical in terms of uncertainty propagation. Both methods can greatly amplify input errors. The non-collocated method generally performs much better (up to a factor 10), this method thus seems preferable over the other two methods. Given the possible improvements in accuracy of the decoupled FRFs, it might therefore be worthwhile measuring additional internal DoF in a practical decoupling problem. However, in order to make decoupling techniques truly robust, further research has to be performed. For example, research is needed in order to establish criteria for selecting the additional (internal) DoF that need to be taken into account. One could for instance think of choosing optimal compatibility and equilibrium conditions at each frequency. Further research is also required to investigate the influence of errors on the phase and amplitude of the input FRFs, to tell the effect of both types of errors on the decoupled FRF.
REFERENCES [1] Bregnant, L., Otte, D., and Sas, P. FRF Substructure Synthesis. Evaluation and Validation of Data Reduction
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Techniques. In Proceedings of the Twentieth International Conference on Noise & Vibration Engineering (ISMA) (Leuven, BE, September 1995), Katholieke Universiteit Leuven. [2] Carne, T., Griffith, D. T., and Casias, M. Support Conditions for Free Boundary-Condition Modal Testing. In Proceedings of the Twenty Fourth International Modal Analysis Conference, St. Louis, MO (Bethel, CT, February 2006), Society for Experimental Mechanics. [3] Carne, T. G., and Dohrmann, C. R. Improving Experimental Frequency Response Function Matrices for Admittance Modeling. In Proceedings of the Twenty Fourth International Modal Analysis Conference, St. Louis, MO (Bethel, CT, February 2006), Society for Experimental Mechanics. [4] D’Ambrogio, W., and Fregolent, A. Promises and Pitfalls of Decoupling Techniques. In Proceedings of the Twenty Sixth International Modal Analysis Conference (Bethel, CT, February 2008), Society for Experimental Mechanics. [5] D’Ambrogio, W., and Fregolent, A. Decoupling procedures in the general framework of Frequency Based Substructuring. In Proceedings of the Twenty Seventh International Modal Analysis Conference (Bethel, CT, February 2009), Society for Experimental Mechanics. [6] de Klerk, D. Dynamic Response Characterization of Complex Systems through Operational Identification and Dynamic Substructuring. PhD thesis, Delft University of Technology, Delft, the Netherlands, March 2009. [7] de Klerk, D., Rixen, D., and Voormeeren, S. General Framework for Dynamic Substructuring: History, Review and Classification of Techniques. AIAA Journal 46, 5 (May 2008), 1169–1181. [8] de Klerk, D., and Voormeeren, S. Uncertainty Propagation in Experimental Dynamic Substructuring. In Proceedings of the Twenty Sixth International Modal Analysis Conference, Orlando, FL (Bethel, CT, February 2008), Society for Experimental Mechanics. Paper no. 133. [9] Dekking, F., Kraaikamp, C., Lopuha¨a, H., and Meester, L. A Modern Introduction to Probability and Statistics – Understanding Why and How, first ed. Springer Texts in Statistics. Springer-Verlag, London, England, 2005. [10] DeLaurentis, D. A., and Mavris, D. N. Uncertainty Modeling and Management in Multidisciplinary Analysis and Synthesis. In 38th AIAA Aerospace Sciences Meeting & Exhibit (January 2000). Paper AIAA 2000-0422. [11] Faragher, J. Probabilistic Methods for the Quantification of Uncertainty and Error in Computational Fluid Dynamics Simulations. Australian Government, Deparment of Defence, Defence Science and Technology Organisation, Victoria, Australia, October 2004. URL: http://dspace.dsto.defence.gov.au/dspace/bitstream/1947/4214/1/DSTO-TR-1633%20PR.pdf. [12] Ind, P., and Ewins, D. Impedance based decoupling and its application to indirect modal testing and component measurement: A numerical investigation. In Proceedings of the Twenty First International Modal Analysis Conference, Kissimmee, FL (Bethel, CT, February 2003), Society for Experimental Mechanics. [13] Lim, T., and Li, J. A Theoretical and Computational Study of the FRF-Based Substructuring Technique applying enhanced Least Square and TSVD Approaches. Journal of Sound and Vibration 231 (April 2000), 1135–1157. [14] Petersen, K. B., and Pedersen, M. S. The matrix cookbook, February 2008. [15] Rixen, D. How measurement inaccuracies induce spurious peaks in Frequency Based Substructuring. In Proceedings of the Twenty Sixth International Modal Analysis Conference, Orlando, FL (Bethel, CT, February 2008), Society for Experimental Mechanics. Paper no. 87. [16] Sanliturk, K., and Cakar, O. Noise elimination from measured frequency response functions. Mechanical Systems and Signal Processing 19 (2005), 615–631. [17] Sj¨ ovall, P. Identification and Synthesis of Components for Vibration Transfer Path Analysis. PhD thesis, Chalmers University of Technology, G¨ oteborg, Sweden, October 2007. [18] Sj¨ ovall, P., and Abrahamsson, T. Substructure System Identification from Coupled System Test Data. Mechanical Systems and Signal Processing 22 (June 2007), 15–33. [19] Voormeeren, S. Coupling Procedure Improvement & Uncertainty Quantification in Experimental Dynamic Substructuring – Validation and Application in Automotive Research. Master’s thesis, Delft University of Technology, Delft, the Netherlands, October 2007. [20] Voormeeren, S., de Klerk, D., and Rixen, D. Uncertainty Quantification in Experimental Frequency Based Substructuring. Mechanical Systems and Signal Processing 24, 1 (2010), 106–118. doi:10.1016/j.ymssp.2009.01.016.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Using component modes in a system design process Guillaume Vermot des Roches 1 , Jean-Philippe Bianchi1 , Etienne Balmes1,2 , Remi Lemaire3 , Thierry Pasquet3 1 : SDTools, 2 : Arts et Metiers ParisTech, 3 : Robert Bosch France (SAS), Division Chassis System Brakes
ABSTRACT Classically Component Mode Synthesis has been used to couple models from different sources (mixed test and analysis, different companies, different software, ...) or reduce models. The focus of this paper is on using component modes in a design cycle. Component modes are the natural representation in the lowest part design cycle. In NVH applications, it is well known that coupling at the system level makes understanding of the impact of component design changes difficult. The paper details and uses the disjoint component synthesis method. Like classical CMS this synthesis method considers reduced component models and couples these models to obtain a system level synthesis. The first novel aspect is that a physical interface is assumed to exist between components. Synthesis thus becomes a trivial application of model assembly. The second novel aspect is that components are reduced using component modes and the trace of nominal system modes, thus allowing exact predictions for the nominal design. Illustrations of the methodology are given for an automotive brake application. INTRODUCTION Numerical prototyping is widely used in industrial design processes to allow optimization and limit the cost of validation through experimental testing. A typical organization of the design is so called V-model shown in figure 1. In NVH applications, performance can often only be evaluated in the system verification and validation step. It is thus quite difficult to specify requirements or component level verifications. &RQFHSW
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Figure 1: V-model for product development The purpose of the present work is to introduce a methodology that, given a base system model (current design), leads to a rapid iteration between component level properties and system level predictions. The existence of a base model ensures that the component coupling with its environment is appropriate and will allow system level predictions of performance. The first issue that will be addressed is to build a method that does not require computationally intensive evaluations of performance to validate the impact of component changes. The second issue is to allow the evaluation of design changes in terms of component frequencies and damping rather than detailed model geometry and properties.
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_54, © The Society for Experimental Mechanics, Inc. 2011
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To simplify the integration of components into a system model, section 1.1 introduces the idea of coupling components using physical interfaces rather than the assumption of continuity classically introduced in Component Mode Synthesis (CMS) methods [1, 2] . With this disjoint component synthesis method, components have fully distinct Degrees Of Freedom (DOF) and one distinguishes component and coupling matrices. Earlier work on this approach can be found in Refs. [3] to cope with incompatible meshes, [4] for structural dynamics modification applications, [5] for damping allocation. As components are disjoint, their reduction is a direct application of Ritz analysis. The only requirement is to build a basis approximating the component motion. Section 1.2 shows that reduced component models can be built using the trace of system responses thus leading to coupled responses that are exact for the nominal design and very accurate otherwise. In such models, it is not necessary to represent all possible interface deformations as classically done using static interface or constraint modes [1] . In a number of cases, this results in much smaller reduced models. Finally, using disjoint components gives a wide range of options for defining component DOF. Section 1.3 thus discusses how component free/free modes can be used as Rayleigh Ritz vectors. Component mode frequency and damping are then used as design parameters in a specification phase. The models being very small, system performance evaluations are typically very fast. The use of the proposed methods is then illustrated in section 2 on the case of a state of the art automotive brake model. 1 1.1
CMS PROCEDURES FOR DESIGN Coupling components with physical interfaces
Component Mode Synthesis methods [1, 2] divide structures into components and generate coupled predictions by introducing assumptions on the interfaces between components. Most of the literature on CMS considers that the fundamental assumption for coupling is the fact that displacement is continuous at interfaces. The approach, considered here, assumes that the components are physically disjoint and that coupling occurs through interfaces that have a physical extent. Since the components are disjoint, the associated DOF {qci } are always distinct. Matrices associated with full or reduced components are non zero for a single component, while interfaces can have nonzero values coupling multiple components. Their diagonal blocks are non null on interface component DOF. In the case of two components one thus has Z1 0 ZI11 ZI12 [ZT ] = [Zel ] + [ZI ] = + (1) 0 Z2 ZI21 ZI22 Figure 2 shows the matrix connectivities for a sample 3 component/2 interface model. The block structure of the component and interface models is indicated through colors. In the application of section 2, the interface matrices correspond to tangent contact/friction coupling models.
(a) Conceptual model
(b) FE elastic matrix
Figure 2: Finite element stiffness matrices
(c) FE interface matrix
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Since DOF associated with components are distinct, model reduction is simply performed by generating an assumed basis (classical Rayleigh Ritz reduction) for each component response {qci } ≈ [Tci ] {qciR } Thus yielding a reduction basis defined on the whole model ) - qc1 Tc1 = qc2 0
)
0 Tc2
(2)
qc1R qc2R
(3)
Displacements are by nature continuous since they are defined on a single component, and system predictions are readily obtained by reducing the coupling matrices T
[ZI ]ijR = [Tci ] [ZI ]ij [Tcj ]
(4)
It is important to note that block size for component interface is equal to the number of vectors (2). The number of interface DOF between components thus has no effect on the reduced model size. This feature is a strong deviation from what would be obtained with classical reduction methods, such as the Craig-Bampton method [1] , which tend to have very high numerical costs in cases with large interfaces. In the coupling matrix reduction, accounting for block operations on components that are actually coupled often leads to major performance gains. 1.2
Reduced models with exact modes
In the proposed setting, computing the nominal model modes is considered to be straightforward. This is nowadays the case using either classical iterative method (Lanczos, Implicitly Restarted Arnoldi, ...) or automated multi-level substructuring methods [6, 7] . The key point in design procedures is rather the ability to accurately perform parametric studies. It is then critical to compute the system modes several hundred times for reanalysis, which requires the use of reduced models to be performed in reasonable computation times. In classical CMS methods [1] , one assumes component independence : the data from one component should be independent of the data from other components. While this made sense for the purpose of getting faster computations or coupling test and analysis derived models, this assumption is a poor idea in a design cycle. Indeed model precision is altered despite the fact that the nominal design gives appropriate coupling information. It is thus proposed to replace component modes by the trace of the global system modes [Φ] on each component ci [Tci ] = Φ|ci 1:N Orth. (5) s
The clear advantage of this reduction basis is that both reduced and full models share the same modes. Such dynamic behavior equivalence allows to retain a good accuracy for modal based computations like forced response or complex mode computation. If multiple system configurations are of interest, one can use the trace of multiple mode sets ( Φ|ci (p1 ) Φ|ci (p2 ) . . . ). Variable rotation speeds are thus considered in [8] , where the issue of generating bases from vector sets is also addressed. Fixed basis reduction for variable configurations can be applied to all or specific components. For compatibility with existing software, it is also possible to include static interface or constraint modes [1] . For interfaces with a large number of DOF this is however often very detrimental for the reduced model size. With the present methodology, the component model really only needs as many base vectors as there are independent shapes in the trace of system level modes (for example a very stiff component only needs 6 shapes to describe its rigid body modes).
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1.3
Using component modes as DOF and design parameters
In a detailed design phase, component parameters to be optimized are known and the CMS methodology detailed above gives a rapid way to evaluate system level performance. But in a specification phase, one typically seeks to understand how the component behavior impacts system response.
The idea proposed here is to analyze the impact of changes in frequency and damping of the free/free component modes. These represent a series of simple scalar parameters and introducing detailed design changes to shift the frequency or damp specific modes is often a clear process. A similar idea with fixed interface modes is used to study mistuning of bladed disk assemblies in Ref.[8] . To obtain accurate predictions, one now seeks to retain exact modes for both the system and components. For each component i, one thus uses a reduction basis that combines component modes in free/free conditions [φci ] and the trace on the component of exact system modes Φ|ci [Tci ] = [φci ] Φ|ci Orth.
(6)
In practice, Φ|ci may not be orthogonal to the component free/free modes, one thus generates an orthonormal basis by some appropriate method [8] . The component modes are kept explicitly and are not modified by the orthonormalization process since by definition they are orthonormal to start with. The difference between the subspace of system modes and the component modes is also kept with the only differences linked to numerical round off errors in the orthonormalization process. Since the component basis is orthonormal, the reduced component mass is identity and the stiffness is diagonal with the first terms corresponding to the square of component mode pulsations. Damping is a full matrix if a damping model exists (non-proportional damping) but in a design phase it is typically assumed to be diagonal (modal damping). Parametric studies on component mode frequencies or modal damping ratio thus simply correspond to changes on the associated diagonal terms of the reduced component models. This process will be illustrated in section 2.2. As shown in figure 3, the interface matrices are now fully populated of blocks corresponding to connected components. When the component basis contains more vectors than interface DOF with another component, specific basis reordering can significantly reduce interface coupling and thus lead to sparse reduced models.
(a) Conceptual model
(b) Reduced stiffness matrix
Figure 3: Reduced stiffness matrices
(c) Reduced interface matrix
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2 2.1
APPLICATION TO AN AUTOMOTIVE BRAKE SYSTEM Test case
For the illustrations shown in this paper, one considers a state of the art brake model shown in figure 4. The model is an assembly of 8 components in contact with each other finely meshed to comply with the need to compute its modes up to 16kHz for squeal simulations, thus leading to the use of 600,000 DOF. Static computations of the steady state shape of the brake are performed using ABAQUS. Component and interface models are then imported into SDT[9] whose the superelement utilities are used to manage further computations.
Figure 4: Full brake model The components are in contact with each other through 11 interfaces representing several thousand DOF. The reduction basis of equation (3) is computed for 8 different components. The reduced model obtained is sparse and contains only 1,300 DOF (for 316 component modes and 250 global modes). With such a small size, only a few seconds are then needed to compute reduced global system modes. Figure 5 shows the relative frequency error between reduced and full system real modes. The error is within numerical precision for the whole range.
Figure 5: Reduced real mode relative frequency error to full modes The non-linearities induced by contact and friction yield system instabilities responsible in particular for brake noises (groan, moan, squeal, ...). These instabilities are non destructive and an end goal of the present work is time simulation of the full brake assembly (see [10] for details). This paper focuses on an earlier part of the design cycle, where performance is analyzed using unstable complex modes and structural modifications are tested for their impact. Section 2.2 illustrates the effectiveness of the proposed reduced model for reanalysis studies. Section 2.3 shows how component modes can be used to guide design.
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2.2
Reanalysis in design phase
In a design phase, the ability to rapidly evaluate the impact of modifications is a key advantage of the proposed method. These modifications can be physical (one will consider a modulus or mass change here) or correspond to component modes (which will be illustrated in the next section). The uses of reanalysis are classically perturbation/sensitivity, uncertainty and model updating analyses. The low cost allows the evaluation for a large number of intermediate values. This makes tracking of poles practical as will be illustrated. The two examples given in this section are chosen to illustrate the robustness of the method for significant changes. The first example shows the effect of the disc Young modulus on the global system modes. Such shift is trivial to apply to a component with an isotropic material. Since the stiffness matrix is proportional to the modulus, the impact of a modulus change is a simple multiplicative coefficient applied to the block of the reduced elastic matrix of figure 2.
Df and DDf
Figure 6 shows the prediction accuracy for the system real modes with a +10% shift of the Young modulus. Predicted Δf −Δfcompt and exact frequency shifts Δf = fnom − fmod are plotted along with the shift error ΔΔf = pred . The error Δfcompt on predicted shifts is clearly small on the whole frequency range.
10
2
10
0
10
−2
10
−4
0
2000
4000
6000
8000 10000 Frequency [Hz]
12000
14000
16000
Figure 6: Frequency shift prediction error for a +10% disc Young Modulus shift. — Δfpred , — Δfcompt , — ΔΔf Figure 7 shows two complex mode prediction showing high disc participation. It appears that changing the modulus leads to stable/unstable transitions, which is of clear interest in a squeal study. A variation from -20% to +30% of Young modulus is considered and compared to full order computations at -20%, +10% and +20% modulus shifts. The correspondence is very good and clearly demonstrates the ability to predict complex pole patterns found in stability analysis.
Figure 7: Complex mode evolution prediction for a disc Young modulus variation. red : full recomputation, yellow prediction corresponding to recomputed points, green : nominal point, +: prediction The second example presents the effect of mass modification on a component. The brake anchor has a so-called handle which drives the anchor rigidity in torsion. The effect of the handle inertia can be studied to set anchor handle modes at frequencies where no interaction involving anchor torsion occurs. The effect of adding a 9 gram point mass at the anchor handle center is considered.
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Figure 8 shows the additional mass effect on all the anchor modes, a variation from nothing to -6% is observed. To apply the mass modification, a ΔManchor matrix is computed and projected on the anchor reduction basis which can then be added to the system mass.
Figure 8: Anchor handle mass modification. Left, original mode #11 and #13 (highly impacted by handle design modification). Right, anchor real modes frequency variation The error on real mode frequencies is presented in figure 9 following the same evaluation than for the disc Young modulus variation. Frequency shifts due to the modification are well predicted. It can be noticed that significant frequency shifts (with high anchor modal participation) show the smallest errors on frequency shift. 2
Df and DDf
10
0
10
−2
10
−4
10
0
2000
4000
6000
8000 10000 Frequency [Hz]
12000
14000
16000
Figure 9: Frequency shift prediction error for a mass variation of the anchor handle. — Δfpred , — Δfcompt , — ΔΔf 2.3
Sensitivity to component modes
The previous section focused on the structural modification effect predictions. This section focuses on the methodology used to identify components needing modifications. Since component modes appear as explicit DOF in the reduced system matrices, the sensitivity of the stiffness matrix to a frequency change is a matrix with only one non zero element on the diagonal at position p. The sensitivity of a complex pole to a change in the p frequency of a component mode is thus simply given by the product of the complex mode components associated with that DOF . T / . T/ ∂λj = ψjL ψj p p ∂p
(7)
The sensitivity of all system to all component modes of the reduced model is easily computed in less than 60 seconds. Since the focus is on instabilities, one only considers the damping sensitivity of unstable modes. Figure 10 shows a global sensitivity map. The plot gives a direct illustration of which modes of which component impacts the instabilities.
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Component mode
624
hub knuckle piston caliper anchor inner pad outer pad disc 0
10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 Assembled mode
Figure 10: Sensitivity map of the first 150 system modes to all component modes. The larger the marker, the hotter the color the higher the sensitivity The most unstable system mode (#55, shown in figure 11) is identified as inducing squeal and must therefore be altered. A zoom in on its sensitivity study is shown in figure 12. One sees that 3 component modes are of main interest, being the ninth knuckle mode, first outer pad mode and second inner pad mode . A modification of the first outer pad mode could then be considered to stabilize the squealing mode.
Figure 11: Brake squealing mode, partial restitution plot
Disc Outer Pad Inner Pad Anchor Caliper Piston Knuckle Hub
Sensitivity
1500 1000 500 0
5
10 15 20 Non Rigid Component Mode Number
25
Figure 12: Mode #55 damping sensitivity to all component modes 3
CONCLUSION
This paper introduced the disjoint component mode synthesis method and showed how this approach gives a clear mechanism to evaluate the impact of component modes on the system response. This evaluation is done using very compact models and can thus be used in a for systematic sensitivity studies that are needed in a design phase.
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REFERENCES [1] Craig, R. J., A Review of Time-Domain and Frequency Domain Component Mode Synthesis Methods, Int. J. Anal. and Exp. Modal Analysis, Vol. 2, No. 2, pp. 59–72, 1987. [2] de Klerk, D., Rixen, D. and Voormeeren, S. N., General Framework for Dynamic Substructuring : History, Review and Classification of Techniques, AIAA Journal, Vol. 46, No. 5, pp. 1169–1181, 2008. [3] Balmes, E., Use of generalized interface degrees of freedom in component mode synthesis, International Modal Analysis Conference, pp. 204–210, 1996. [4] Corus, M., Am´elioration des m´ethodes de modification structurale par utilisation de techniques d’expansion et de ´ r´eduction de mod`ele, Ph.D. thesis, Ecole Centrale Paris, 2003. [5] Roy, N., Abbadi, Z. and Balmes, E., Damping Specification of Automotive Structural Components via Modal Projection, PDF, September 2008. [6] Bennighof, J., Kaplan, M., Muller, M. and Kim, M., Meeting the NVH Computational Challenge: Automated Multi-Level Substructuring, International Modal Analysis Conference, pp. 909–915, 2000. [7] Gao, W., Li, X., Yang, C. and Bai, Z., An Implementation and Evaluation of the AMLS Method for Sparse Eigenvalue Problems, ACM Transactions on Mathematical Software, Vol. V, pp. 1–27, September 2007. [8] Sternch¨ uss, A., Multi-level parametric reduced models of rotating bladed disk assemblies, Ph.D. thesis, Ecole Centrale de Paris, 2009. [9] Balmes, E., Bianchi, J. and Lecl` ere, J., Structural Dynamics Toolbox 6.2 (for use with MATLAB), SDTools, Paris, France, www.sdtools.com, Sep 2009. [10] Vermot des Roches, G., Balmes, E., Pasquet, T. and Lemaire, R., Time simulation of squeal phenomena in realistic brake models, Proceedings of the International Conference on Advanced Acoustics and Vibration Engineering (ISMA), pp. 3007–3019, 2008.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Computation of Distributed Forces in Modally Reduced Mechanical Systems Wolfgang Witteveen, Linz Center of Mechatronics GmbH., Altenbergerstr. 69, 4040 Linz, Austria, Phone: +43 (0)70 2468 6112, [email protected] Nomenclature Ȧ
()
n r s j nx nz
M K K Ext M
red
K red K
Ext,red
F Fred A A
red
ĭ Ĭ p Ĭ
quantity expressed in frequency domain number of DOF of FE model number of regarded modes number of force shapes iteration number number of points in circumferential direction number of points in axial direction mass matrix of FE model stiffness matrix of FE model external stiffness matrix of FE model mass matrix of reduced model stiffness matrix of reduced model external stiffness matrix of reduced model force shape matrix of FE model force shape matrix of reduced model EHD coefficient matrix reduced EHD coefficient matrix mode matrix force modes pressure modes
T1 G x G xĭ G x G q G k(t) G g G Gf Ĭ Gf fred G b G bred G p t ti Ȧ h x z Ș p
transformation matrix vector of nodal DOF of FE model
G
modal approximation of x
G
second time - derivative of x generalized coordinates of reduced model vector of scaling functions vector of force-mode coordinates external force vector of FE model modal approximation of
G f
force vector of reduced model EHD perturbation vector reduced EHD perturbation vector pressure at FD mesh grid points time discrete time instant frequency lubrication gap height circumferential coordinate axial coordinate effective dynamic fluid viscosity pressure
1. Abstract The consideration of state depended and distributed loads, like contact and friction forces, can be of remarkable computational effort in the framework of a modally reduced mechanical system. This is caused by the common strategy, in which such forces are expressed in the coordinates of the high dimensional unreduced space. In case of a Finite Element (FE) model the latter approach involves three steps. First, the physical degrees of freedom (DOF) are determined out of the DOF of the reduced (modal) system. In a subsequent step, the FE node forces have to be computed based on certain physical laws in the high dimensional space of the FE model. Finally, the reduced force vector is determined by the projection of the physical force vector into the modal space. This paper is devoted to a more efficient computation of such loads. It will be shown that a consequent computation in the reduced (modal) space requires the well known displacement trial vectors (commonly called ‘modes’) as well as ‘force trial vectors’, which we will denote as ‘force – modes’.
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_55, © The Society for Experimental Mechanics, Inc. 2011
627
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628 It will be shown that any base of displacement trial vectors leads to an associated base of force trial vectors and all system relevant loads can be computed by a superposition of these force-modes. Consequently, the force computation can be done in the low dimensional space of the reduced system. This paper is subdivided into three main sections. In the first part, the motivation for this research project is outlined. In the second section the theory is presented and finally a numerical example from elastohydrodynamics demonstrates the methods potential. 2. Introduction and Motivation Industrial FE models do have many nodal DOF, typically up to several millions. It is practical impossible to do time integration based on such models. Model reduction via the restriction that the final deformation has to be a superposition of displacement trial vectors (‘modes’) is a common strategy in order to decrease the number of DOF to be manageable. Since the midst of the last century a lot of research has been done on the question how these modes have to be determined. Among the huge number of publications three mechanics focused reviews are to be found in Refs. [1] - [3]. For mechanical systems, the principle of these methods can be summarized by the procedure outlined in equations (1) to (3):
G K xG Mx
G f
(1)
and the (n x n) stiffness matrix K . The nodal The linear FE model is characterized by the (n x n) mass matrix M G DOF (mostly translations and rotations) are collected into the (n x 1) vector x and the external forces are G
G
contains the second represented by the (n x 1) vector f . The symbol n denotes the total number of DOF and x G derivative of x with respect to the time t. In a first step a base of trial vectors, commonly called ‘modes’, is determined. The resulting displacement is restricted to a superposition of these modes. This introduction of a trial space can be expressed as
G G x | xĭ
G, ĭq
(2)
G contains in its columns the r selected modes and the (r x 1) vector q where the (n x r) matrix ĭ contains the G Gĭ generalized coordinates. For the sake of a clearness we will distinct further one between x and x , where the Gĭ G G Gĭ (n x 1) vector x is an approximation of x based on (2). Substituting x in equation (1) by its approximation x from equation (2) and applying the principle of virtual work leads to the reduced model equation G G G K q q (3) M fred . red red This system is of order r due to the fact that the reduced mass and stiffness matrix M red and K red are of the G dimension (r x r). Consequently, the reduced load vector fred is of the dimension (r x 1). These reduced quantities are determined by
TMĭ , M ĭ red TKĭ and K red ĭ G G Tf . fred ĭ
(4) (5) (6)
The actual model reduction is based on the fact that many technical applications can be approximated quite satisfactory by a number of modes r which is significantly smaller as the number of DOF of the FE model (r << n).
and leads to various Intense research in the last decades was mainly focused on the transformation matrix ĭ approaches for its computation, see the literature above. As far as the author overlook the literature, there was no particular focus on the computation of
G
the common projection fred
G G f and fred beside
G T f . This may be caused by the fact that an efficient force computation for at least ĭ
two different types of external forces is easily possible and a considerable range of applications may be covered by them, namely
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629 1. An efficient computation is possible in case of a state independent and imposed external force form of
G f
G (t) Fk
G f in the
(7)
G is a time invariant force shape matrix and the (s x 1) vector k contains time where the (n x s) matrix F depended scaling functions. The reduced force vector can be effectively computed by G G fred Fredk(t) (8) T F is time invariant and needs to be computed just once. with the (r x s) matrix F ĭ red G 2. For external forces which depended linearly on the system state, an efficient computation of fred is possible as well. Such forces can be written as
G f
G K Ext x
(9)
where (n x n) Matrix K Ext
G G contains the linear stiffness acting on x B . Using (2) and projecting f onto the
modal space leads to
G fred
G K Ext,redq
(10)
where the (r x r) matrix K Ext,red
TK ĭ can be added to K on the left hand side of the reduced ĭ Ext red
equation of motion. In case of general state depended loads, the common approach can be characterized by the set of equations
G G G K q q M fred , (11) red red Gĭ G x ĭq , (12) G Gĭ (13) f f( x ,....) and G G T f (14) fred ĭ G Gĭ where f( x ,....) denotes a general non-linear relationship between the nodal force vector f and state variables like deformations, velocities or other parameter.
G
Note, that all kind of distributed forces can be treated as external force f on the right hand side of the equation of motion. In case of forces, which depend nonlinearly on the system state, it is a common strategy to impose the
G
nonlinear portion as external force f on an underlying linear system, see [4] and [5] and the references there. There are different approaches to deal with the system (11) to (14). On the one hand side it can be considered as a single system of nonlinear algebro – differential equations. Commercially available software packages, on the
G
other hand, often provide interfaces, so that the force fred can be computed in user written subroutines. Figure 1 contains a qualitative flow chart for such an iterative time integration process for the discrete time instant ti. However, we will focus on that kind of problem.
Figure 1: Qualitative flow of iterative process at time ti
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630 The grey box in Figure 1 contains the computational procedures, which needs to be done at each iteration in order to consider the nonlinear loads in a mode based time integration of that type. A virtual experiment may outline the principle disadvantage of that approach. Let us consider a structure with n = 107 DOF. The structure contains a bolted joint and the corresponding contact area has b = 105 DOF. The contact is modelled using a penalty approach and just one mode (r=1) is considered. The selection of a single mode is unrealistic but possible and this particular choice will underline the point of the present experiment. The computational effort according to the grey box in Figure 1 is two times r times b floating point multiplications (fpm) in order to swap between the two spaces. Furthermore, r times b floating point summations (fps) and floating point comparisons (fpc) are necessary in order to detect contact areas. Finally, contact forces have to be computed for the penetrating surface nodes. The result of the latter effort for the virtual experiment under consideration is 2*105 5
G
5
fpm, 10 fps, and 10 fpc. The procedure ends up with a single value in fred for the reduced equation of motion (3) of order 1. The latter procedure becomes very inefficient in case of a moderate or high number of b which is common for problems with distributed state depended loads like contact problems [5] or in elasto – hydrodynamics, see [6] and [7]. The next section introduces a new approach for a more effective computation of such distributed forces. 3. Force – Trial Vectors (‘Force – Modes’)
G G G K ĭq Mĭq
Substituting x in the equation of motion (1) by the approximation (2) leads to
G f
(15)
and the transformation of the latter equation into the frequency domain gives
K Ȧ M ĭ 2
Ȧ
G q
Ȧ
G f,
(16) Ȧ
where Ȧ represents the frequency and the left superscript () denotes that the corresponding quantity is expressed in the frequency domain. Equation (16) can be interpreted as the equation of motion in the frequency domain which needs to be fulfilled at each time instant. Beside this common point of view the latter equation can be interpreted as a condition for the force
G f , namely Ȧ ȦG Ĭ q
Ȧ
G f,
G contains r ‘force – modes’ which are scaled by the (r x 1) vector Ȧ q where the (n x r) matrix Ĭ . Ȧ ĭ Ĭ K Ȧ 2M
(17)
Ȧ
(18)
contains a r-dimensional space of force-modes which spans all the force distributions which are Ĭ G , and, consequently, the computation of f an be done in an r dimensional relevant for a certain mode base ĭ
Note, that
Ȧ
subspace instead of the n dimensional space of the FE model.
is obtained by a generalized eigenvalue problem based on K and a diagonal matrix M a If the mode matrix ĭ is obtained which is equal to the displacement modes times a scalar frequency independed ‘force-mode’ base Ĭ factor. In case of other displacement trial vectors, like ‘Joint Interface Modes’ (see [5]) or Constraint Modes (see
of equation (18) are a function of the frequency, which is not very convenient in terms Ĭ Ȧ of time integration. In a first step the exact force-modes Ĭ are approximated by a (n x r) force-mode base Ĭ [2]) the ‘force-modes’
Ȧ
where the inertia effects are neglected: Ȧ
|Ĭ Ĭ
. Kĭ
(19)
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631 In analogy to (2) an approximation
G GĬ f|f
GĬ G f of the force f in the form of
G Ĭg (20) G is introduced, where g results from a reformulation of the general law (13) so that the solution space is restricted to a superposition of the available force modes in the form of
G g
G g( x ĭ ,....) .
(21)
Note, that equation (13) has n DOF, whereas equation (21) just has r DOF. Using the latter definitions, system (11) to (14) can be rewritten to
G G G K q q M fred , red red Gĭ G x ĭq , G G g g( x ĭ ,....) and G G fred K redg
(22) (23) (24) (25)
G GĬ where equation (25) is obtained by substituting f in relationship (14) with f from the approximation (20).
The advantages of the system (22) to (25) in contrast to the one of (11) to (14) are the reduced computational effort for x the solution of the nonlinear equation (24) which has r DOF instead of equation (13) which has n DOF and x
G
T in (14), which commonly is a diagonal matrix instead of ĭ the computation of fred in (25) because K red
is a full matrix If the system (22) to (25) can be treated as a single system, another obvious simplifications is possible, namely
G G G q R q g, G G x ĭq and G G g g( x,....)
(26) (27) (28)
is defined as where the (r x r) matrix R
R
1 ªK red º M red ¬ ¼
(29)
For the sake of a better understanding of the advantages outlined before, a quasi-static numerical example is presented in the next section. 4. Example Equation (20) can be used for the reduction of the solution space in case of forces which are characterized by field equations. This will be demonstrated using an example from elasto-hydrodynamics. The computation of the pressure distribution inside a lubrication gap is based on the Reynolds differential equation in the form of
w § h3 wp · w § h3 wp · ¨ ¸ ¨ ¸ wx © K wx ¹ wz © K wz ¹
6
w(hU) wh 12 , wx wt
(30)
where h is the lubrication gap height, x the circumferential coordinate, z the axial coordinate, Ș the effective dynamic fluid viscosity, t denotes the time and p holds the pressure at x und z. The domain in which equation (30) has to be solved can be discretized by a Finite Difference (FD) mesh with nx x nz grid points. Refer to [8] for the details on the discretization procedure.
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632 Following [8] the discretization leads to a system of linear equations in the form of
G G b, Ap
(31)
G where the ((nx x nz) x 1) vector p holds the pressure at the FD mesh grid points. The ((nx x nz) x (nx x nz)) matrix G and the ((nx x nz) x 1) vector b contain time varying quantities which can be computed out of the dynamic fluid A viscosity together with geometric and kinematic quantities like the gap height, the rotational speed of the shaft etc..
Figure 2: Elastic bearing block Figure 2 contains a screen shot of the FE model of the elastic bearing block and Table 1 contains all relevant geometric data. For more details on the chosen example, see [8]. nx 76 nz 17 bearings nominal diameter 58 mm bearing width 40 mm speed of rotation 3000 rpm -9 dynamic oil viscosity 5.8 10 Pa s absolute radial play 0.03 mm relative eccentricity 0.95 Table 1: Data for elasto-hydrodynamic example For the sake of simplicity, a FD mesh is defined which coincide with the mesh of the FE model. In a first step the FE model of the full bearing block is reduced to the radial DOF of the bearing shell. This has been done by a static reduction method. Consequently, the number n of the equation of motion (1) for this example is n = nx x nz= 76 x 17 = 1292. In a next step the full mode matrix is determined by solving an eigenvalue problem utilizing the stiffness matrix.
. Note, that a quasistatic example is The first 50 eigenvectors have been considered for the modebase ĭ presented because the emphasis of this section is to demonstrate the save of computational effort, when the space of force distributions is reduced by the force – modes. Consequently, the mass term of the reduced equation of motion (3) is neglected.
G
has been computed according to equation (19). The pressure p on the FE nodes The force-mode matrix Ĭ needs to be integrated in order to get nodal forces for the FE model. This is done by the (n x n) transformation
. matrix T 1 G f
G T1p
(32)
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633
and to For the lubrication surface linear solid elements have been used. Therefore it is possible to invert T 1 compute the pressure out of node forces as
-1 G ªT1 º f . ¬ ¼ G GĬ Substituting f by f of equation (19) leads to G pG p Ĭ g
G p
(33)
(34)
Where the (n x r) matrix
p Ĭ
-1 ªT1 º Ĭ ¬ ¼
(35)
represents a pressure – mode base computed by the force – modes. By using the solution space of (34) the system of equations (31) can be reduced to
G G g A bred , red
(36)
with the (r x r) reduced coefficient matrix A red G bred
G pb . Ĭ
p ºT p ªĬ ¬ ¼ AĬ and the (r x 1) reduced perturbation vector
Computation based on grid dimension (equation (31)))
Computation based on reduced equation (36) utilizing force-modes
Figure 3: Radial elastic deformation in mm of bearing block Note, that in common approaches, where force-modes have not been used, equation (31) has to be solved which is of much higher dimension as equation (36). A comparison of the radial elastic deformations of the two approaches can be seen in Figure 3. The approach which utilizes force-modes needed in MATLAP a factor 8 less cpu time per iteration. Please note, that this factor will increase with the third power in case of a finer FD mesh which would be definitely necessary in
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634 case of lubrication grooves, bores an the like. This is based in the fact, that the dimension of equation (31) is determined by the number of FD grid points while the dimension of equation (36) is equal to the number of regarded modes. Furthermore it has been observed, that the convergence characteristics of the force-mode based approach is
Gj
much better. The termination criteria in a simple fix point iteration has been set to 5% difference in the norm of p
G j-1
and p , where j is the iteration number. The force-mode based approach needed 31 iterations while the other one needed 154 iterations. This is probably caused by the increased consistency of the new approach where the reduced quasi-static equation of motion (3) and equation (36), which characterises the oil film properties, are using a corresponding space of solutions while the common approach uses a reduced space for equation (3) and a unreduced space for equation (36).
Computation based on grid dimension (equation (31)))
Computation based on reduced equation (36) utilizing force-modes
Figure 4: Pressure distribution inside the lubrication gap It can be seen in Figure 4 that even if the elastic deformations of Figure 3 are almost the same, the pressure (=force) distributions are significantly different. The proposed method just computes the deformation relevant
GĬ
forces f while the common approach contains the non relevant forces as well. Note, that the non-zero pressure constraint has been realised by applying the so called Gümbel boundary condition, where all negative pressures are set to zero. 5. Conclusion It has been shown that a particular mode base corresponds with a certain base of ‘force – modes’, which spans the full space of the forces which are relevant for the reduced system. A static approximation of these force modes has been used in order to minimize the solution space of the Reynolds differential equation in a final example, where state depended elasto – hydrodynamic forces have been
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635 computed. It has been demonstrated, that the procedure leads to accurate results by a significant save of computational effort. 6. Acknowledgement Support of the authors by the Linz Center of Mechatronics GmbH (LCM), the Engineering Center Steyr (MAGNA Powertrain) and the K2 Austria Center of Competence in Mechatronics (ACCM) is gratefully acknowledged. 7. Bibliography [1] [2] [3] [4] [5] [6] [7] [8]
Noor A. K., Recent advances and applications of reduction methods, Appl. Mech. Rev., Vol. 47, No. 5, pp. 125 – 146, 1994 Craig R. J., “Coupling of substructures for dynamic analyses - An overview,” AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference and Exhibit 41st, Atlanta, GA, 2000, AIAA2000-1573 Zu Q. Q., Model order reduction techniques: with Applications in Finite Element Analysis, Springer London, ISBN-13: 978-1852338077, 2004 Witteveen W., Modal based computation of jointed structures, Dissertation, Johannes Kepler University Linz, Austria, 2007 Witteveen W., Irschik H., Efficient Mode-Based Computational Approach for Jointed Structures: Joint Interface Modes, AIAA Journal, Vol. 47, No. 1, pp. 252-263, 2009 Boedo S., Booker J.F., A Mode Based Elastohydrodynamic Lubrication Model With Elastic Journal and Sleeve, Journal of Tribology, Vol. 122, pp. 94-102, 2000 Kumar A., Goenka P. K., Booker J.F., Modal Analysis of Elastohydrodynamic Lubrication: A Connecting Rod Application, Journal of Tribology, Vol. 112, pp. 524-534, 1990 Plank A., Modal- Elastohydrodynamic Simulation of Journal Bearings, Diploma Thesis, Vienna University of Technology, 2009
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Substructuring using Impulse Response Functions for Impact Analysis
Daniel J. Rixen Delft University of Technology, Faculty of Mechanics, Maritime and Material Engineering Department of Precision and Microsystems Engineering, Engineering Dynamics Mekelweg 2, 2628 CD Delft, The Netherlands [email protected]
ABSTRACT In the present paper we outline the basic theory of assembling substructures for which the dynamics is described as impulse response functions. The assembly procedure computes the time response of a system by evaluating per substructure the convolution product between the impulse response functions and the applied forces, including the interface forces that are computed to satisfy the interface compatibility. We call this approach the Impulse Based Substructuring method since it transposes to the time domain the Frequency Based Substructuring approach. In the Impulse Based Substructuring technique the impulse response functions of the substructures can be gathered either from experimental test using a hammer impact or from time-integration of numerical submodels. In this paper the implementation of the method is outlined for the case when the impulse responses of the substructures are computed numerically. A simple bar example is shown in order to illustrate the concept. Future work will concentrate on including in the assembly measured substructure impulse responses. The Impulse Based Substructuring allows fast evaluation of impact response of a structure when the impulse response of its components are known. It can thus be used to efficiently optimize designs of consumer products by including impact behavior at the early stage of the design.
Keywords: Experimental Substructuring, assembly, impulse response functions, time integration, impact, impulse based substructuring
NOMENCLATURE dof FRF IRF FBS IBS u f H(t) (s) Ns B λ M , K, C dt n []i β, γ
degrees of freedom Frequency Response Functions Impulse Response Functions Frequency Based Substructuring Impulse Based Substructuring array of degrees of freedom array of external forces matrix of Impulse Response function pertaining to substructure s number of substructures in the system signed Boolean matrix defining compatibility constraints Lagrange multipliers on interface mass, stiffness and damping matrix of a linear(ized) system time-step size pertaining to time-step n component i of an array parameters of the Newmark time-integration scheme
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_56, © The Society for Experimental Mechanics, Inc. 2011
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1
INTRODUCTION
Substructuring techniques allow combining the dynamics of different components. The dynamics of the components is described either using experimental data (Frequency Response Functions in the frequency domain) or with a numerical model (in terms of system matrices, computed Frequency Response Functions or component modes). See for instance 5 for a review of substructuring concepts. In the Frequency Based Substructuring approach one can assemble the Frequency Response Functions (FRFs) of substructures that were measured or that were obtained by numerical simulation. This technique can provide accurate results on simple systems 11 and gives interesting qualitative information when applied to complex engineering systems such as cars 2,8 . When the substructures are described by FRFs obtained experimentally tremendous care must be taken to ensure a high degree of accuracy, for instance it must satisfy reciprocity, passivity and artifacts like additional mass effects or location/orientation errors in the sensors must be very small. In practice such errors often introduce in the assembled FRFs spurious peaks 10 and non-physical properties 1 that renders the obtained assembled model useless. To clean-up the measured FRFs of the substructures before assembly one can apply modal identification techniques in order to fit a pole-residue model to the data. When combining identified modes of the substructures with a numerical model of the measured components high quality assembled models can be obtained 7 . Obtaining high quality measured FRFs is delicate since, unless slow and costly sine sweeps are used, the dynamic properties in the frequency domain are obtained through several processing steps (anti-aliasing filters, windowing, Fourier transforms) which will unavoidably alter the information contained in the measurements. Furthermore using modal identification and FRF synthesis to obtain clean FRFs is a very labour-intensive and error-prone process, and it assumes that a pole-residue model can be fitted. When high damping is present, proper identification of the poles and residues becomes difficult and if non-viscous damping (visco-elastic damping) is present the pole-residue model cannot correctly represent the frequency domain response of the components. Finally we note that if substructuring is used to simulate impact responses, working in the frequency domain requires considering a large frequency band which makes all the Frequency Based Substructuring strategies expensive and badly suited. Applying substructuring techniques to simulate impact responses of structures is a very attractive idea since it would allow to efficiently predict impact behavior at the early design stage in many fields. For instance when designing the structure of a mobile phone or a notebook , many COTS (components of the shelf) are used. If the dynamics of the COTS are characterized properly either numerically or experimentally, one could use susbtructuring techniques to rapidly optimize the housing, frame and connections (screws, rubber pads ...) and thereby guarantee an improved life-time and reliability of valuable mass-produced appliances. Since Frequency Based Substructuring is not well suited for setting up a model for impact simulation (see discussion above), we propose in this paper an alternative substructuring technique. We use the same concepts as in other substructuring approaches (i.e. admittance representation of the components and dual assembly) but consider for the substructures directly the impulse response functions in the time domain measured for the input and interface degrees of freedom, instead of the modal properties or the FRFs. The method will be called Impulse Based Substructuring or IBS. This paper outlines the basic principle of the method and shows that the theory can be easily applied when the impulse response functions are computed through direct time integration of a numerical model. Future work will investigate the combination of numerical and experimental sub-models in the Impulse Based Substructuring strategy.
2
THEORY OF IMPULSE RESPONSE SUPERPOSITION FOR PARTITIONED PROBLEMS
Let us call H(t) the matrix of the responses to a unit impulse at t = 0 for a linear system that is initially at rest. In other words a coefficient [H(t)]ij of the impulse response matrix represents the response of degree of freedom (dof) i to a unit impulse on dof j. The response of the linear system to an applied force f (t) can then be evaluated by the convolution product (Duhamel’s integral) between the impulse response function matrix and the applied forces: 0 t u(t) = H(t − τ )f (τ )dτ (1) 0
This is a classical result of time analysis of linear systems, usually obtained using Laplace transforms. This convolution product can be interpreted as follows: the response at time t is an infinite sum of the responses to the infinitesimal impulses f (τ )dτ before time t (see figure 1). Each impulse at time τ gives a contribution through the impulse response
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from τ to t, that is H(t − τ ). φ(τ )
Δτ
τ
Figure 1: Forcing function as a series of impulses The impulse responses can be obtained either experimentally or numerically. If obtain by measurements, an impact hammer can be used: 1 t+ • One evaluates the impulse I = 0 δ(t)dt, where δ(t) is the measured force and t+ the duration of the force assumed to be very short for a hammer impact. In fact the hammer impact is only an approximation of a Dirac function, but if the duration t+ is much smaller then the characteristic time of the expected response u(t) to the applied force f (t) in (1) the obtained impulse response can be used in the convolution product. In other words the duration of the hammer impulse must be much shorter then the period of the frequency of the modes having a significant contribution to the response. • Compute the impulse response by scaling the measured time response to the hammer impact by the impulse I. The impulse response can also be obtained from a closed-form solution of a mathematical model or by time-integration of a numerical model. In the later case, the time-step must be small enough so that the initial impulse can be considered as an impulse for the dynamic response one wants to compute with the Duhamel integral. See the next section for further discussion on how to compute the impulse response of a numerical model.
Let us now assume that the problem has been decomposed into N s sub-structures. The response of each substructure can be obtained using the convolution product (1), but for the solutions to be the responses of the substructures as part of a full system the coupling forces on the interface between the substructures must be included in the forcing function. The interface forces are unknown beforehand, but we know that those interface forces coupling the interface dofs must be such that the interface is compatible in the assembled problem. Calling B (s) the signed Boolean matrices localizing the interface dofs (see for instance 3,9 ), the compatibility condition on the interface, namely the condition stating that the dofs on each side of the interface are equal, can be written as s
N !
B (s) u(s) = 0
(2)
s=1
Hence the extension of Duhamel’s integral (1) to a partitioned problem writes ⎧ 0 t ⎪ (s) (s) (s) (s)T ⎪ u (t) = H (t − τ ) f (τ ) + B λ(τ ) dτ ⎪ ⎨ 0
Ns ! ⎪ ⎪ ⎪ B (s) u(s) (t) = 0 ⎩
(3)
s=1 T
where B (s) λ represent the interface forces, namely the reactions associated to interface compatibility constraint, λ being the Lagrange multipliers.
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The Impulse Base Substructuring (IBS) method proposed in this paper relies on (3): if the impulse response functions are known per substructures it allows computing the impulse response (or the response to any external force) for the assembled problem. It is thus clearly a transposition to the time domain of the Frequency Based Substructuring (FBS). In practice the formulation (3) must be applied by discretizing the time integral in order to evaluate the convolution product. Indeed the impulse response H(t) is generally known only for discrete time instances, either from measurements or from numerical modelling. A way to evaluate the integral is explained in the next section.
3
IMPULSE BASED SUBSTRUCTURING WITH NUMERICAL MODELS
In this section we will show how the IBS method can be applied, in particular how Duhamel’s integral can be discretized in time. Here we will assume that the dynamics of the substructures are described by a numerical model. First we will explain how an impulse response is computed, then we outline how impulse response superposition can be applied for a single (non-decomposed system). We will extend the method to assemble several numerical models described by their IRFs and finally illustrate the strategy with a simple application example.
3.1
Impulse Response computation
For a numerical model the impulse response is obtained by numerical time-integration. The linear dynamic equilibrium at time tn can generally be represented by the matrix equation ¨ n + C u˙ n + Kun = fn Mu
(4)
where M , C, K are the linear(ized) mass, damping and stiffness matrices, u is the set of degrees of freedom, and f are the applied forces. The subscript n indicates the time-step at which the accelerations, velocities and displacement are considered. Let us call dt the time-step size (assumed for simplicity to be constant during the time-integration) such that t = n dt = tn . Given the initial conditions u0 , u˙ 0 , the initial acceleration can be computed by ¨ 0 = M −1 (f0 − Ku0 − C u˙ 0 ) u
(5)
To solve (4) one needs to approximate time derivatives by well chosen finite differences. In structural dynamics one classically uses the Newmark time-integration scheme 4,6 stating that un u˙ n
¨ n−1 + βdt2 u ¨n = un−1 + dtu˙ n−1 + (0.5 − β)dt2 u ¨ n−1 + γdtu ¨n = u˙ n−1 + (1 − γ)dtu
(6) (7)
where β and γ are parameters used to build integration schemes with different properties. For instance when γ = 1/2, β = 1/4 one obtains an implicit and unconditionally stable scheme (equivalent to the trapezoidal integration rule). If γ = 1/2, β = 0 the scheme is explicit but conditionally stable (equivalent to the central difference). Replacing the discretized time derivatives (6,7) in the dynamic equation (4) ¨ n = fn − K u ˜n − Cu ˜˙ n M + γdtC + βdt2 K u
(8)
˜ n and u ˜˙ are the predictors where u ˜n u ˜˙ n u
=
¨ n−1 un−1 + dtu˙ n−1 + (0.5 − β)dt2 u
=
¨ n−1 u˙ n−1 + (1 − γ)dtu
For an unit impulse at time t = 0 the dynamic response can be computed in three different ways.
Initial velocity step To compute the impulse response we can first compute the velocity jump at time t = 0 due to a unit impulse. Integrating the dynamic equation in an infinitesimal interval [0− , 0+ ] results in the momentum
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equation
0 M (u˙ 0+ − u˙ 0− ) =
0+
0−
f (t)dt
and since just before t = 0 the system is at rest, the initial velocity resulting from a unit impulse on dof j is M u˙ 0 = 1j
(9)
where 1j is a vector with a unit coefficient for dof j. Hence the impulse response can be computed by setting the applied force f (t) to zero and starting the time integration with the initial conditions u0 u˙ 0 ¨0 u
= 0 = M −1 1j = M
−1
(10)
(−C u˙ 0 )
then continuing the integration with (8).
Initial applied force Another manner to compute the impulse response is to use the initial conditions and applied force u0 u˙ 0
= =
0 0
f0
= =
1j 0
fn>0
(11) ¨ 0 = M −1 1j hence u
Note that, in the time integration scheme, this is equivalent to an impulse generated by a force being suddenly unity at time t = 0 and decreasing linearly to 0 at time t = dt: the IRF so obtained is in fact for an impulse equal to dt/2. The numerical impulse response is thus the time response obtained for the settings (11) divided by dt/2. In the limit where the time-step goes to zero this method is obviously converging to the impulse response as computed with the initial velocity step (see above). If the Newmark time integration scheme γ = 1/2, β = 1/4 is used the impulse response obtained by this method is actually identical to the response obtained with an initial velocity jump even for a finite time-step size.
Applied force at the second time-step The two methods outlined above to compute the impulse response require factorizing the mass matrix. This is not a problem when the mass matrix is diagonal (as found for an explicit time integration), but for a consistent (non-diagonal) mass matrix the factorization cost could be significant. It that case one can avoid factorizing the mass matrix by applying a unit force on the second time-step, namely u0 u˙ 0 ¨0 u f1
= = = =
0 0 0 1j
(12) (13) fn=1 = 0
This computation represents in fact a force increasing linearly to 1j between t0 and t1 , then decreasing to zero between t1 and t2 . The related impulse value is thus dt and the unit impulse response is obtained by dividing the obtained time response by dt. Again In the limit where the time-step goes to zero this method is equivalent to the two strategies explained above. For a finite time-step the obtained impulse response is slightly different.
3.2
Impulse superposition for a single structure
The impulse responses computed at time tn by one of the methods described in the previous section are stored in the Impulse Response matrix Hn . The coefficient [Hn ]ij is the response for dof i at time t = n dt to an unit impulse at time t = 0 on a dof j. So H, containing the full time history of the impulse responses, can be seen as a three-dimensional matrix of dimension N × p × nmax , calling N the number of degrees of freedom of the system (or the number of output considered), p the number of excitation (input) locations, and nmax the number of time-steps
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for which the impulse response has been computed. Note that this is similar to the Frequency Response Functions (FRFs) of a system, except that for the FRFs the third dimension is the frequency. Obviously, theoretically speaking, the FRFs of a system are the Fourier transform of the IRFs. The time response for a general applied force f (t) can then be computed by approximating the convolution integral (1) by the finite sum n−1 ! un = Hn−i fi dt (14) i=0
Note that H0 is not present in this series since the displacement response to an impulse is null at the instant when the impulse is applied, meaning that H0 = 0. The graphical interpretation of the discretized convolution (14) is given in figure 2.
f
u τ
H(t-τ) t-τ
t Figure 2: Discretization of Duhamel’s integral
In figure 2 it is seen that the applied force, in the time-integration scheme, is approximated by piece-wise linear forces between the time steps. Hence during a time step from tn to tn+1 the applied forces are coming for one half from fn and for the other half from fn+1 . Let us then consider in figure 2 the response at time t1 . According to (14) the response at t1 is solely due to the impulse created by the force at t0 , while in fact the force at time t1 also produced an impulse between t0 and t1 . Hence one can say the the response at t1 is due to an impulse equal to It0 ,t1 = f0 dt/2 + f1 dt/2 =
f0 + f1 dt 2
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indicating that to evaluate the impulse the force should in fact be taken at the middle of the time step. Therefore it is more accurate to consider the following discretization of Duhamel’s integral: un =
n−1 !
Hn−i (fi + fi+1 ) dt/2
(15)
i=0
In practice however, since the time step dt is small, the difference between (15) and (14) is negligible. 3.3
Impulse superposition and assembly of substructures
The matrices of the impulse responses for each substructure can be computed as indicated in the section 3.1. In order to assemble the substructures the IRFs between all interface dofs are required, in addition to the IRFs for the dofs where the external forces f (s) are applied. The convolution product and the compatibility condition of (3) can then be discretized as ⎧ n−1 ! (s) (s) ⎪ ⎪ (s) (s)T ⎪ u = H f dt + B λ ⎪ i n−i i ⎨ n i=0
s
(16)
N ⎪ ! ⎪ ⎪ ⎪ B (s) u(s) = 0 ⎩ s=1
where λi are the Lagrange multipliers related to the compatibility condition. They represent the impulse between the interface dofs needed to ensure the interface compatibility.1 In the dual assembly formula (16) the solution at time tn is determined by the external forces and the interface impulses for t = 0 up to t = n − 1. Hence the compatibility condition at time tn determines the interface impulse at time tn−1 . Let us rewrite (16) as ⎧ (s) (s) (s)T ˜ (s) ⎪ λn−1 n + H1 B ⎨ uns = u N ! (17) ⎪ B (s) u(s) = 0 ⎩ s=1
where
(s) ˜n u
is the predicted displacement when λn−1 = 0, namely T (s) (s) (s) (s) Hn−i fi dt + B (s) λi + H1 fn−1 dt
(18)
From (17) the Lagrange multiplier is computed by solving the dual interface problem
Ns Ns ! ! (s) (s)T (s) ˜ (s) B H1 B λn−1 = − B (s) u n
(19)
˜ (s) u n =
n−2 ! i=0
s=1
s=1
which is very similar to the dual interface problem of the Frequency Based Substructure (see e.g. 5 ). Equations (19,17) constitute the stepping algorithm for the Impulse Based Substructuring strategy (IBS) proposed in this paper.
3.4
Numerical example
To illustrate the Impulse Based Substructuring technique let us consider the bar structure described in figure 3 excited by a load at its end. The structure is divided in 2 substructures of equal length, each substructure being modeled by 25 bar finite elements (the consistent mass matrices are used here). The bar is made of steel (E = 2.1 1011 Pa, ρ = 7500 kg/m3 ), has a uniform cross-section of A = 3.14 10−4 m2 and each substructure has a length of L = 0.5 m. In the model damping has been introduced by constructing C = 2 10−6 K. 1 Equation (16) can also be written for the forces at half time-step as in (15). This does not modify the basics of the algorithm and provides slightly more accurate results.
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λ
0
f
0
L
L
Figure 3: Example of a beam with two substructures
Assuming the dofs of the substructures are numbered from left to right, the Boolean constraining matrices are B (1)
=
[ 0
···
0 −1 ]
B
=
[ 1
···
0 0 ]
(2)
First we compute the Impulse Response Functions as indicated in section 3.1. Here a unit force at time t = 0 is used. The implicit, unconditionally stable Newmark γ = 1/2, β = 1/4 scheme is used. The time-step is chosen equal to 3hcrit where hcri t is the critical time step, namely the stability limit if the integration scheme would be explicit. This critical time-step is given by the CFL condition and is equal to 4 hcrit = 2/ω
for ω the highset eigenfrequency in the model.
The obtained IRFs are plotted in figure 4 for inputs on the interface and on the end of the bar. On the right of that figure the IRFs are zoomed.
−4
(1)
HLL
1
−4
x 10
1
0.5
0.5
0
0
−0.5
−0.5
−1
0
0.005
0.01
0.015
0.02
−1
x 10
0
0.5
1
1.5
2
−3
(2) 0.015
2
H00
2.5 x 10
−3
x 10
−3
x 10
−3
x 10
1.5
0.01
1 0.005
0
0.5
0
0.005
0.01
0.015
0.02
0
0
0.5
1
1.5
2
−3
(2) 0.015
2
HL0
2.5
x 10
1.5
0.01
1 0.005
0
0.5
0
0.005
0.01
t (s)
0.015
0.02
0
0
0.5
1
1.5
t (s)
Figure 4: IRFs for the bar substructures (zoomed on the right)
2
2.5
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Let us assume that one is interested in computing the dynamic response at the end of the bar when a force is applied on it. The IBS expression (17) and the dual interface problem (19) are (2)
u ˜n[ L]
=
n−2 !
(2) (2) (2) (2) (2) Hn−i[LL] fi[L] dt + Hn−i[L0] λi + H1[LL] fn−1[L] dt
i=0 (1) ˜ n[L] u
(2)
˜ n[L] −u
(20)
λn−1
=
u(2) n[L]
= u˜(2) n[L] + H1[LL] λn−1
(1)
(2)
H1[LL] + H1[00] (2)
First we will apply the IBS technique to compute the response of the full bar to an impulse at its end: applying a unit force at the end of the second substructure, and dividing the obtained response by dt/2 we obtain the IRF shown in figure 5. If this impulse response is computed with a non-decomposed model, exactly the same IRF is found. −4
f ull H2L,2L
1
−4
x 10
1
0.5
0.5
0
0
−0.5
−0.5
−1
0
0.005
0.01
0.015
−1
0.02
x 10
0
0.5
1
t (s)
1.5
2
2.5 −3
x 10
t (s)
Figure 5: IRF for the full bar computed by IBS (zoomed on the right)
(2)
Finally let us apply a step load at the end of the bar (fL (t) = 1 for t ≥ 0). Using again the IBS approach to compute the response based on the impulse responses of the substructures one obtains the response plotted in figure 6. Again the same response would have been found if a model of the complete bar would have been used in a direct time-integration. It is observed that the solution converges in time to the correct static solution. −8
(2) 3 uL (1) (2) u L = u0 2 (1) udx 1
−8
x 10
3 2 1
0 −1
x 10
0
0
0.01
0.02
0.03
t (s)
0.04
−1
0
0.5
1
1.5
t (s)
2
2.5 −3
x 10
Figure 6: Dynamic response to a step load at the end of the bar, computed by IBS (zoomed on the right). The solutions are shown for the end of the bar, the middle point (on the interface between the substructures) and on the first node next to the fixed end.
4
CONCLUSIONS AND FUTURE WORK
In this paper we propose a method that transposes the Frequency Based Substructuring technique to the time domain. It allows computing the response of a system by computing the responses of its substructures with a discretization of the Duhamel integral and enforcing the interface compatibility at every time step. Hence if the substructure impulse responses are known for all interface and input degrees of freedom, the method proposed in this paper allows predicting
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the dynamics of the full system. The method is named Impulse Based Substructuring or IBS. The IBS (in the time domain) is the equivalent of the FBS (in the frequency domain), but we believe that when applied to the computation of impact response and shocks, the IBS can be significantly more effective since it allows using directly the impact responses of the susbstructures. The IBS approach could be used for instance when predicting the impact response of new designs. It can enable engineers to optimize their designs during the very initial design phase, if the impulse response of the components are known either from models or tests. In many cases the designer uses standard components which would then need to be characterized once, the work of the designer being then to optimize the housing and the link between the components. In this paper we outline the basic principle of the method and show that the theory can be easily applied when the impulse response functions are computed through direct time integration of numerical models. Future work will investigate the applicability of the approach when the impulse responses of the substructures are obtained from experimental tests. We will also investigate the combination of numerical and experimental sub-models in the IBS method. Finally we will perform research to combine substructures described by impulse response functions with non-linear components that need to be time-integrated simultaneously with the time-stepping in the IBS for the linear parts.
REFERENCES [1] Thomas G. Carne and Clark R. Dohrmann. Improving experimental frequency response function matrices for admittance modeling. In IMAC-XXIV: International Modal Analysis Conference, St Louis, MO, Bethel, CT, February 2006. Society for Experimental Mechanics. [2] Dennis de Klerk. Dynamic Response Characterization of Complex Systems through Operational Identification and Dynamic Substructuring: An application to gear noise propagation in the automotive industry. PhD thesis, Delft University of Technology, Delft, The Netherlands, March 2009. [3] C. Farhat and F.-X. Roux. A method of finite tearing and interconnecting and its parallel solution algorithm. International J. Numer. Methods Engineering, 32:1205–1227, 1991. [4] M. G´eradin and D. Rixen. Mechanical Vibrations. Theory and Application to Structural Dynamics. Wiley & Sons, Chichester, 2d edition, 1997. [5] D. De Klerk, D. J. Rixen, and S. N. Voormeeren. General framework for dynamic substructuring: History, review and classification of techniques. AIAA Journal, 46(5):1169–1181, 2008. [6] N.M. Newmark. Method of computation for structural dynamics. J. Eng. Mech., 85:67–94, 1959. [7] Dana Nicgorski and Peter Avitabile. Conditioning of frf measurements for use with frequency based substructuring. Mechanical Systems and Signal Processing, (in press), 2009. [8] D. Otte, J. Leuridan, H. Grangier, and R. Aquilina. Prediction of the dynamics of structural assemblies using measured frf-data: some improved data enhancement techniques. In IMAC-IX: International Modal Analysis Conference, Florence,Italy, pages 909–918, Bethel, CT, February 1991. Society for Experimental Mechanics. [9] Daniel Rixen. Encyclopedia of Vibration, chapter Parallel Computation, pages 990–1001. Academic Press, 2002. isbn 0-12-227085-1. [10] Daniel J. Rixen. How measurement inaccuracies induce spurious peaks in frequency based substructuring. In IMAC-XXVII: International Modal Analysis Conference, Orlando, FL, Bethel, CT, February 2008. Society for Experimental Mechanics. [11] D.J. Rixen, T. Godeby, and E. Pagnacco. Dual assembly of substructures and the fbs method: Application to the dynamic testing of a guitar. In International Conference on Noise and Vibration Engineering, ISMA, Leuven, Belgium, September 18-20 2006. KUL.
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Static correction in model order reduction techniques for multiphysical problems Alexander M. Steenhoek Department of Precision and Microsystems Engineering Delft University of Technology, Mekelweg 2, 2628CD Delft, The Netherlands [email protected]
Daniel J. Rixen Department of Precision and Microsystems Engineering Delft University of Technology, Mekelweg 2, 2628CD Delft, The Netherlands [email protected]
Philippe Nachtergaele Open Engineering ` Reu des Chasseurs-Ardennais,B-4031 Liege, Belgium [email protected] ABSTRACT In this paper multiphysical model order reduction methods for thermomechanical problems are investigated. Different variants of modal truncation methods are outlined. A basis built of state space modes of the system represents the behavior of the system but the method requires complicated complex modes. This is why we also introduce bases built from a composition of bases of the separate uncoupled physical fields because these are simpler to build. One can improve on these bases again by inclusion of a correction for the coupling effect, for example based on a derivation from a first order perturbation analysis. These bases shows a good representation on the dynamics behavior, but generally do not give correct static results. However we want at least the static solution to be correct in order to guarantee that if the problem is quasi-static or if it converges to a steady-state static solution, one gets the exact solution. Hence one wants to add a static correction to the solution (a posteriori). That static correction can also be used to enrich the basis. In this contribution an investigation on the static residuals for different bases is performed.
Nomenclature Matrices: M C K S E A L B F Q G Ψ Φ ΔΨ Λ I
2nd order time derivative system matrix 1st order time derivative system matrix 0th order time derivative system matrix Load applied to a system 1st order time derivative state space matrix 0th order time derivative state space matrix Output observation matrix Input control matrix Applied mechanical force Applied heat flux Residual flexibility Basis built on modes of the second order problem Basis built on modes of first order system Correction of modal basis Diagonal eigenvector matrix Identity matrix
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_57, © The Society for Experimental Mechanics, Inc. 2011
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Vectors: u Mechanical DOF: deformation θ Thermal DOF: temperature difference i Input signal r Residual force y Output signal q Degree of freedom x State space variables z Generalized degree of freedom ψ Eigenmode of a second order problem φ Eigenmode of a first order problem η Modal amplitude
1
Scalar values: n Dimensions of full order system k Dimensions of reduced order system ω Frequency T0 Temperature at working point λ Eigenvalue α Modal participation factor ζ Modal damping value Mathematical notation: i Imaginary number O Order of the error nc . . . The uncoupled part of scalar / vector / matrix c . . . The coupled part of scalar / vector / matrix t . . . Transposed of vector / matrix
INTRODUCTION
Multiphysical models are intensively used during the design of high-tech systems and in particular microsystems. However both the complexity and large dimension of the models makes intensive use costly and clarifies the demand for reduced order models that preserve the dominant behavior of the system. This demand especially holds for models describing multiphysical behavior, because these can be very complicated whilst modeling the coupling effects is of key importance for the observed behavior. From several fields of engineering model order reduction techniques are available, but their performance is not straightforward in a multiphysical context. Standard techniques applied to coupled problems are numerically inefficient and therefore in [1] a procedure is proposed to obtain a reduced order model starting from reduction bases that for the uncoupled physics. A reduced order model for the coupled problem is obtained by improving the ability of the uncoupled bases to represent coupling phenomena, which is done by performing a correction to the uncoupled bases that accounts for the coupling with other physics. A specific property of a reduction technique is the ability to predict correctly the static behavior since the steadystate of systems is of prime importance in many engineering problems. In this contribution we will discuss how the correct static solution can be guaranteed in reduced order coupled problems. It gives rise to either a correction to the solution obtained from the reduced models or an augmentation of the reduction basis. This approach is illustrated on a two-way coupled thermomechanical problem, where the static coupling contains only a one-sided coupling however. This brings the possibility to perform the correction sequentially when starting from fully uncoupled bases for a separate mechanical and thermal field.
1.1
A thermomechanical application
In microsystems multiphysical behavior often becomes of interest. On the one hand several systems use multiphysical effects as the key working principle. Think for example of a commonly used actuation principle in a thermomechanical actuator. On the other hand multiphysics can have parasitic side-effects, for example when we think of thermomechanical damping in resonators. The thermomechanical behavior can be described by the following governing equation:
%
%
% % u ¨ u˙ Muu 0 F Cuu 0 Kuu Kuθ u + + = (1) 0 0 Cuθ Cθθ 0 Kθθ θ Q θ¨ θ˙ It expresses the thermomechanical equations of motion written in terms of DOF deformation u and temperature change θ. In this equation we can recognize Muu , Cuu , Kuu , F as mechanical mass, damping and stiffness and external force. Cθθ , Kθθ and Q are heat capacity, thermal conductance and heat flux as can be recognized from the heat equation. The terms that are described until now can be seen to be a description for the uncoupled mechanics or thermal problem. Two terms remain that are recognized to couple the equations. Firstly we recognize the thermal expansion Kuθ by which a temperature difference can influence the deformation. Secondly we recognize
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Figure 1: A thermomechanical micro-actuator
2
x 10
−4
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 −3
x 10
Figure 2: The net bending effect in a micro-actuator
Cθu which describes how the strain rate, the time derivative of deformation, induces heat. From the description of the equations it is seen that a second order problem (the mechanics) is coupled to a first order problem (the thermal field). Furthermore the problem becomes especially interesting for microsystems, because when scaling the dimensions of a problem to the range of microns, the critical time constants of both physical fields can fall in the same range and thus dynamics that originates from each field can be observed. As an illustration we can look at the thermal actuator depicted in figure 1. On the lefthand side we see how two thermal actuators can be configured in a setup. On the righthand side a screen capture is given from finite element software that predicts dynamic behavior such as described in the next section.
1.2
Behavior of the full system
In this section the behavior of the thermomechanical actuator is explained. The working principle of these actuators is that an applied voltage results in resistive heating and generates a distributed heat source. Due to the thermal expansion the applied heat now results in a mechanical actuation. The system can be seen to consist of two beams with different cross-section. The thinner beam has more electric resistance and thus extends more as a result of thermal expansion. The difference in expansion leads to a net bending effect, which is seen in figure 2. The thermomechanical equations of motion can be written in general second order form as: M q¨ + C q˙ + Kq = S
(2)
This describes the entire system and we can investigate a certain response y by measuring the DOF q with Lt . This is the response of the system to a load S that is excited with i and is applied according to B. The possibility to investigate the response of a certain output y of the system with respect to an excitation i applied to the system is now described by the following set of equations: ) S = Bi M q¨ + C q˙ + Kq = S (3) y = Lt q Now we will look at the transient response of this system of the deformation at the tip when a heat step loading is applied. We will find the results of figure 3. Different snap shots of the deformation are depicted. Throughout this
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Figure 3: Response of a thermomechanical micro-actuator
paper we will have in mind that a dynamic model is of importance for instance to model the transient behavior, but for the design of the actuator at the end we are interested in the possible actuation, which is the static deformation. Therefore a correct representation of the static behavior remains of interest.
1.3
First order formulation
A descriptor formulation is often used, for example to easily calculate the solution to equation 3. This formulation can be obtained by writing equation 3 in first order form. In this formulation variables are introduced that write the state of the system at an any time, which are therefore called state variables x. This also induces that in general the control matrix B and observation matrix L need to be extended, leading to Bss and Lss respectively, where underscript ss stands for state space or first order form. The dynamic behavior of the system can now be written in terms of these state variables and their time derivatives. The general form now looks like: )
E x˙ Sss y
= Ax + Sss = Bss i = Ltss x
(4)
For the thermomechanical problem, which was formulated by equation 1 we can also write such a first order form for E x˙ = Ax + S. Moreover, we can write this expression into a symmetric form with the following operations. First t we scale the identity relation for u˙ with −Kuu . Second we use the relation Cθu = T0 Kuθ as is found from [2]. The 1 third operation is to scale the heat equation by T0 . Furthermore from here on we assume that mechanical damping is absent, meaning that Cuu = 0, in order to simplify the analysis. But the developments found from the analysis can be generalized to the situation that mechanical damping is present. These steps result in: ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ 0 ⎤ −Kuu 0 0 u˙ 0 −Kuu 0 u ⎣ ⎦⎣ u 0 Muu 0 0 −Kuθ ⎦ u˙ + ⎣ F ⎦ ¨ ⎦ = ⎣ −Kuu 1 1 1 t ˙ 0 0 C 0 −K − θ θ θθ uθ T0 T0 Kθθ T0 Q (5) Note that in design one can be interested in a small number of outputs (for instance tip displacement of the thermomechanical beam). But engineers are often interested in all physical quantities in order to ensure the reliability of the design and therefore to analyze not only the output of a device but also maximal stresses and temperatures. Therefore for the thermomechanical system we will use y = x, or Lt = I.
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2 2.1
DIFFERENT VARIANTS OF MODAL TRUNCATION METHODS AS A REDUCTION TECHNIQUE The idea of modal truncation
Modal truncation is a reduction technique generally applied in many fields of engineering, for example the field of mechanical engineering. The concept of modal truncation is as follows. Modes ψi give a solution to the homogenous equations of motion of the system and can be used to form a projection or transformation basis Ψ for the problem. For a general second order problem such as given by equation 3, we can use the following transformation: q = Ψη
(6)
This writes the problem in modal coordinates. The orthogonality properties of the modes yield that the frequency response function of the system can be written as a modal summation. For a general unsymmetric problem we can distinguish the so-called left and right modes. In this case the right modes are used for a trial basis and the left modes for a test basis. For convenience we will use identical left and right modes (such as found for symmetric systems) throughout this paper in order to explain general ideas. The transfer matrix relating input to output in the frequency domain can then be written in a modal expansion as: H=
n !
λ2 i=1 i
Lt ψi ψit B − 2iζi λi ω − ω 2
(7)
In modal superposition methods the number of variables in the model can be reduced by representing the dynamic behavior of the system with only a limited amount of modes. This means that the representation of the transfer function is truncated after k terms. H
= ≈
k !
λ2 i=1 i k ! i=1
n ! Lt ψi ψit B Lt ψi ψit B + 2 − 2iζi λi ω − ω 2 λi − 2iζi λi ω − ω 2 i=k+1
t
ψi ψit B
L λ2i − 2iζi λi ω − ω 2
(8)
For the projection basis it implies that instead of the full modal basis Ψ a reduced basis is used that contains only the k modes. Throughout this paper we will indicate Vˆ to be the truncated basis of V that contains only k terms. The truncated set of k corresponding DOF is indicated with η. ˆ This gives: ⎡ ⎤ η1 ⎢ η2 ⎥ ˆ η = [ ψ1 ψ2 . . . ψ k ] ⎢ . ⎥ ≈ q (9) qˆ = Ψˆ ⎣ . ⎦ . ηk When the system is written in first order form we can perform a comparable operation. Now we use modes φi that can be used to form a transformation basis Φ for the problem. For a general first order problem assumed to be symmetric such as given by equation 4, we can use the transformation x = Φz
(10)
to write the problem in modal coordinates. The orthogonality properties again allow to write the frequency response function as a modal summation and truncating this summation after k terms we find the approximate transfer function as: H=
k n ! ! Ltss φi φti Bss Ltss φi φti Bss ≈ −λi + iω −λi + iω i=1 i=1
(11)
where φi and λi are respectively the (generally complex) eigenmodes and eigenvalues of the first order problem. When this transfer function is approximated the corresponding projection basis Φ contains only the k modes, as we can see in: ⎡ ⎤ η1 ⎢ η2 ⎥ ˆ z = [ φ1 φ2 . . . φk ] ⎢ . ⎥ ≈ x x ˆ = Φˆ (12) ⎣ . ⎦ . ηk
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2.2
Different variants of modal truncation for thermomechanical systems
Now that the idea of modal truncation is explained we will discuss 4 variants of modal truncation as a reduction technique for thermomechanically coupled problems, described by equations 1 or equation 5, depending if one considers the second or the first order problem.
2.2.1
Method 1: truncation of a basis of fully coupled modes
The modes are the full modes of the coupled problem, containing both thermal DOF θ and mechanical DOF u. Modal truncation can be applied directly to the thermomechanical system written in state space, resulting a basis of state space modes indicated as Φss , but it is numerically inefficient to obtain these coupled modes. The basis is given by: x ˆ 2.2.2
ˆ ss zˆss = Φ
(13)
Method 2a: truncation of a basis of fully uncoupled modes of the second order problem
Especially when coupling remains small, either the thermal DOF or the mechanical DOF in the coupled mode show a large amount of similarity with uncoupled modes (modes of the corresponding uncoupled physical fields). Generally the uncoupled modes can be calculated directly from the equations of motion written in second order form such as given by equation 3. For the thermomechanical problem given by equation 1 a truncated basis of ku uncoupled mechanical modes Ψ(u) and kθ uncoupled thermal modes Ψ(θ) can possibly provide an efficient basis for the coupled problem in the form of: (u)
(u) ˆ Ψ 0 ηˆ qˆnc = (14) ˆ (θ) η ˆ(θ) 0 Ψ
2.2.3
Method 2b: truncation of a basis of fully uncoupled modes of the first order problem
To be consistent with the fully coupled modes that are calculated when the problem is written in first order form, we can also derive the uncoupled modes for this problem. Again when coupling remains small, either the thermal DOF or the mechanical DOF in the coupled mode show a large amount of similarity with the modes of the corresponding uncoupled physical problems written in first order form (uncoupled state space modes). In other words a truncated basis of ku uncoupled mechanical modes Φ(u) and kθ uncoupled thermal modes Φ(θ) could possibly provide an efficient basis for the coupled problem in the form of: (u)
(u) ˆ Φ 0 η ˆ nc x ˆ = (15) (θ) ˆ η ˆ(θ) 0 Φ Although it seems that using this truncated basis is equivalent to using the truncating basis of the uncoupled modes in the second order, we like to especially point out that when applying the truncated basis in the first order form one also approximates the closure relation in the state-space. In other words here also the relation between velocities and time derivatives of the mechanical DOF is approximated.
2.2.4
Method 3a: truncation of a basis of corrected uncoupled modes
Although for some frequency response functions a mono-physical basis as described above indeed results in a sufficiently accurate reduced order model, especially in the representation of cross-coupling behavior it is observed that reduced models based on the truncation explained so far do not always accurately represent the dynamic behavior of the system.. Therefore a correction of the uncoupled basis was proposed, see [1]. This correction was derived performing a perturbation analysis of the uncoupled basis with system matrix 0 0 0 0 Kuθ Ac = 0 (16) t 0 Kuθ 0
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It results that uncoupled mechanical modes require a correction on their thermal DOF and uncoupled thermal modes require a correction on the mechanical DOF. For a single uncoupled mechanical mode φnc ui and a single uncoupled nc nc thermal mode φnc the corresponding corrections Δφ and Δφ are respectively: u θi θi i Δφnc ui = −
nθ nc t c ! φnc θ φθ A s
s=1
s
nc λnc ui + λθs
φnc ui
Δφnc θi = −
and:
nu nc t c ! φnc u φu A s
s=1
s
nc λnc θi + λus
φnc θi
This correction leads to an improvement of the uncoupled basis, indicated with superscript n 2c, and gives:
2 ˆ (u) ΔΦ ˆ (θ) Φ ηˆ(u) x ˆn2c = 2 ˆ (u) Φ ˆ (θ) ΔΦ η ˆ(θ)
(17)
(18)
These corrected modes generally show good resemblance with all individual DOF of the state space modes. However the full corrections would imply operations with all individual uncoupled modes. Therefore it is often not possible to fully develop this correction and in practice one would build the correction with the modes used that are also used in the truncation basis and therefore would give an approximated correction.
2.2.5
Method 3b: truncation of a simplified basis of corrected uncoupled modes
Under the special circumstances there is a possibility to build corrections with the full set of modes involved. When the mechanical and thermal spectra are well separated we can recognize that either λui or λθj will be dominant, whereas they both appear in the denominator of the correction term of the uncoupled bases. When the spectra of the uncoupled physical fields are separated we can therefore apply a simplification, see ??. The simplified corrected basis, indicated with superscript n 3c, is given by: 3 3 ˆ (u) ΔΦ ˆ (θ) Φ η ˆ(u) n 3c x ˆ = (19) 3 3 ˆ (u) ˆ (θ) ΔΦ Φ ηˆ(θ) The simplification gives rise to the ability to express the summation in the correction as inverse stiffness or inertia matrices of the uncoupled physical fields. This will be expressed for the two possible extremes of separations of the spectra. Fast mechanical response compared to thermal response For this situation we expect λu to be much larger than λθ and the correction to the uncoupled bases can be approximated with: −1 Δφnc ui ≈ −Eθθ Aθu
1 nc φui λnc u
and
And the simplified corrected basis is given by: ˆ (u) Φ n 3c x ˆ = −1 ˆ u E −1 Aθu Φ ˆ (u) −Λ θθ
−1 nc Δφnc θi ≈ −Auu Auθ φθi
ˆ (θ) −A−1 uu Auθ Φ (θ) ˆ Φ
3
ηˆ(u) 3 η ˆ(θ)
(20)
(21)
Slow mechanical response compared to thermal response For this situation we expect λθ to be much larger than λu and the correction to the uncoupled bases can be approximated with: −1 nc Δφnc ui ≈ −Aθθ Aθu φui
and
Δφnc θi ≈ −
1 −1 Euu Auθ φnc θi λnc θ
And the simplified corrected basis, again indicated with superscript n 3c, leads to:
(u) 3 −1 ˆ (u) ˆ −1 Euu ˆ (θ) Φ −Λ Auθ Φ ηˆ n 3c θ x ˆ = 3 ˆ (u) ˆ (θ) −A−1 Φ η ˆ(θ) θθ Aθu Φ
(22)
(23)
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3
THE IDEA OF STATIC CORRECTION
In this section we explain the principle of static correction of modal truncation basis for a general second order problem. In section 4 we will perform this correction specifically to the thermomechanical problem.
3.1
Static relation between response and excitation for a second order problem
Suppose we have a system whose dynamics is prescribed by a generalized set of equations such as was written in equation 3. The static solution for this system can be easily obtained from: ⎧ ⎨ S = Bi q = K −1 S and Hstatic = Lt K −1 S (24) ⎩ y = Lt q Note that the same static solution can be found from the static relation between response and excitation, that is the static gain of the frequency response function of this system, which can be expressed as: H
n ! Lt ψi ψ t B i
=
λ2i
i=1 t
= LK
−1
n !
M ψi ψ t B
(25)
i=1
3.2
Static residualized reduction of a second order problem
In previous analyzes we mentioned that the transfer function is a frequency response function. In order to calculate it, for each frequency the value for transfer function needs to be evaluated, which explains the urge to truncate the system after a limited amount of modes. A relatively cheap extension to the modal truncation given by equation 8 is to statically account for the neglected modes instead of just deleting them: H
≈
k ! i=1
n ! Lt ψi ψit B Lt ψi ψit B + 2 λi − j2ζλi ω − ω 2 λ2i
(26)
i=k+1
The static residual r for the truncated modes can easily be recognized to be: r
=
n ! Lt ψi ψit B λ2i
i=k+1
= Lt K −1
n !
M ψi ψit B
(27)
i=k+1
From the latter expression we recognize that the correction accounts for the static solution, as seen from K −1 , of the forces that arise from the inertia or acceleration force, seen from M , of the truncated modes. The correction is inconvenient to calculate because it requires all modes to be available. Combining the equations enables to express this static correction of inertia forces contributed by the higher order modes as: r
= Lt K −1 B − Lt K −1
k !
M ψi ψit B
i=1
= Lt GB
(28)
In this result we used the residual flexibility G, calculated as
G = K
−1
I−
k ! i=1
M ψi ψit
(29)
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in order to express a term that represents the inverse of the stiffness matrix after deflation with the modes in the truncation basis. The obtained static residual can be applied to calculate the correct static result with the reduced order model. A further extension was proposed, see for example [3] or [4], to use this static correction also as a vector in the reduction basis. This approach is known as the Modal Truncation Augmentation. Using 28 as an a posteriori correction for the reduced solution or as a enrichment of the basis guarantees correct static results calculated with the reduced order model.
3.3
Static residualized reduction of a first order system
Analogous to the derivations in the previous section applied to a general second order problem, we can also do static correction for problems written in first order form, such as the descriptor formulation. This generalized description assumes that the system dynamics is prescribed by a generalized set of DOF’s given by equation 4. The static solution for this system is: ⎧ ⎨ Sss = Bss i x = −A−1 Sss and Hstatic = −Ltss A−1 Bss (30) ⎩ y = Lt x ss Again the same static solution can be found from the static gain of the frequency response function of this system and can be expressed as: Hstatic
= −
n ! Ltss φφt Bss λi i=1
= −Ltss A−1
n !
Eφi φti Bss
(31)
i=1
A relatively cheap extension to the modal truncation given by equation 11 is again to statically account for the neglected modes instead of just truncating them: H
=
k n ! ! Ltss φi φti Bss Ltss φi φti Bss + −λi + iω −λi i=1
(32)
i=k+1
The static correction for the truncated modes is calculated from the static residual r of the modal truncation and can be recognized to be: r
=
n n ! ! Ltss φi φti Bss = −Ltss A−1 Eφi φti Bss −λi
i=k+1
(33)
i=k+1
This static residual can be calculated more conveniently by combining equation 33 and equation 31, which gives:
k ! t −1 t r = −Lss A I− Eφi φi Bss (34) i=1
4
STATIC CORRECTION APPLIED TO DIFFERENT THERMOMECHANICAL REDUCTION BASES
In section 3 the idea of static correction was explained for general systems. In this section we will look how the static correction can be applied to thermomechanical systems for which the reduction bases were introduced in section 2. Before the reduced models are manipulated such that they give the correct static solution, we will first discuss this correct static solution.
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4.1 4.1.1
The correct analytical static result The correct analytical static result for the second order thermomechanical problem
The thermomechanical problem given by equation 1 is described in second order form. The static solution can be easily calculated from:
u θ
=
Kuu .
Kuθ Kθθ
−1
F Q
(35)
Because the stiffness matrix is weakly coupled (one-way coupling) we can solve the static solution sequentially. First calculate the static solution for the temperature θ as: −1 θstatic = Kθθ Q
(36)
The static solution for the displacement can be calculated next, where we substitute the result from equation 36. This leads to: ustatic
−1 = Kuu (F − Kuθ θ)
−1 −1 −1 = Kuu F − Kuu Kuθ Kθθ Q
(37)
We can recognize both a static displacement due to the applied force F as a static displacement due to the applied heat Q. The total static result for the set of DOF can be found to be:
−1 −1 −1 u Kuu F − Kuu Kuθ Kθθ Q = (38) −1 θ static Kθθ Q 4.1.2
The correct analytical static result for the first order thermomechanical problem
The derivation of most of the reduction bases is performed when the coupled problem is expressed in first order form. The thermomechanical problem is described in first order form by equation 5. Now we can look whether this gives the same static result as we found for the problem expressed in second order form. The static result is calculated as: ⎡ ⎤−1 ⎡ ⎤ ⎡ ⎤ −1 −1 −1 . Kuu . . Kuu F − Kuu Kuθ Kθθ Q u ⎦ . Kuθ ⎦ ⎣ F ⎦ = ⎣ u˙ 0 = ⎣ Kuu (39) 1 1 t −1 . Kuθ K Q θ static Kθθ Q θθ T0 T0 The static system matrix A expresses the coupling of the entire vector of state variables x. Because this vector also contains the mechanical velocity u, ˙ the coupling from mechanical velocity to the thermal field is also expressed in A. This again implies that the matrix now contains two-sided coupling and we cannot perform the sequential approach such as used before. However, the result from 39 can be compared to the static result obtained from the second order system, because x and θ are identical to the results from equation 37 and equation 36 respectively. In equation 39 we find that statically x˙ = 0 is calculated, which is the expected result. 4.2 4.2.1
The static residual for different thermomechanical reduction methods The static residual for method 1: fully coupled modes
For reduction method 1 we used a basis that consists of the fully coupled modes calculated from the first order system. Therefore we can directly apply equation 31 to calculate the static residual, indicated with r, which gives:
r
= −A
−1
I−
k ! i=1
Eφi φti
. F Q
(40)
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Note that only the fully coupled modes are correct modes for the coupled problem. Therefore they obtain the original orthogonality properties with respect to the system matrices A and E. Other reduction basis may give a sufficient representation of the coupled problem, but are not correct modes for the coupled problem. Therefore they do not preserve the orthogonality properties. However, often these methods do contain orthogonality properties with submatrices of the coupled problem and this needs to be fully exploited when calculating the static correction.
4.2.2
The static residual for method 2a: fully uncoupled modes of the second order problem
With this method we use a basis of uncoupled modes of the second order problem. The static correction cannot be calculated from equation 26 (re-written for an unsymmetric problem), because as stated above the basis does not consist of correct modes of the coupled problem. The modes in the bases do have orthogonality properties with respect to the system matrices that can be identified to belong to the uncoupled problems. Alternatively however, because equation 26 cannot be applied, we can perform a sequential approach such as used to obtain the analytical static solution before. This means that we express the static residual in terms of the contribution of the individual DOF corresponding to the separate physical fields. Recall that in method 2a we used the basis given by equation ˆ zˆ = F ˆ that is also expressed in a 15, from which we find that the reduced order model writes a static relation K one-way coupled form as: (u)t
(u) (u)t ˆ ˆ (u) Ψ ˆ (u)t Kuθ Ψ ˆ (θ)t ˆ Ψ Kuu Ψ z Ψ F = (41) (θ) (θ)t (θ)t (θ)t ˆ ˆ ˆ z . Ψ Kθθ Ψ Ψ Q Following the sequential approach we first look for the (incorrect) static solution for temperature θ that the reduced order model gives. Note that we can use the orthogonality of Ψ(θ) with respect to Kθθ . This gives: θˆstatic
kθ (θ) (θ)t −1 ! ψi ψi Q −1 ˆ (θ) (θ) ˆ (θ) (θ)t ˆ ˆ = Ψ Ψ Kθθ Ψ Ψ Q= λθi
(42)
i=1
A static error is recognized because, compared to the correct result in equation 36, only part of the spectral expan−1 sion of Kθθ is performed. From this result we see that the static residual is identical to that of a truncated transfer function of a first order system as is given by equation 34. The static residual can therefore be expressed with use of a thermal residual flexibility matrix Gθθ as: rθ:Q→θ
= Gθθ Q
(43)
We will look at the mechanical static residual for u next. From equation 38 we recognized that the correct static solution u consist of a contribution by F and a contribution by Q and therefore we can also expect a static residual for both contributions and for convenience we will treat them separately. The (incorrect) static mechanical solution due to an applied force F calculated with the reduced model is: u ˆstatic:F
ˆ (u)
= Ψ
ku (u) (u)t −1 ! t ψ ψi F (u) −1 (u) (u) i ˆ Kuu Ψ ˆ ˆ Ψ Ψ F = λ2ui i=1
(44)
−1 Compared to the correct static solution as seen in equation 37, we see that the spectral expansion of Kuu is not complete and therefore this part of the static residual is identical to that of a truncated transfer function as is given by equation 28. The static residual is expressed using the mechanical residual flexibility matrix Guu as:
rF →u
= Guu F
(45)
The (incorrect) static mechanical solution due to an applied force Q can be calculated as follows: u ˆstatic:Q
(θ) −1 ˆ ˆ uu = −Ψ(u) K Kuθ zstatic −1 −1 ˆ (u) ˆ (u) Ψ ˆ (u)t Kuu ˆ (u)t Kuθ θˆstatic = −Ψ Ψ Ψ
(46)
Compared to the correct result found in equation 37 we see that 2 effects will introduce a static error. The first source −1 is due to the incomplete spectral expansion of Kuu . The second type of error arises from the fact that θˆstatic is used
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instead of θstatic , where from equation 42 we already found this solution contains the static residual expressed in equation 43. This first source introduces a static residual containing the mechanical residual flexibility: rQ→u:source1
−1 = Guu Kuθ θstatic = −Guu Kuθ Kθθ Q
(47)
The second source introduces a following static residual containing the thermal residual flexibility: rQ→u:source2
−1 −1 = Kuu Kuθ rθ = −Kuu Kuθ Gθθ Q
(48)
Altogether we can recognize the 4 types of contributions to a static residual. All of them are related to the fact that the truncated basis leads to incomplete spectral expansion of the stiffness matrices. The results are summarized in 1. Again all these partial corrections can be used to correct a posteriori the solution obtained from the truncated Indication rQ→θ rF →u rQ→u,source1 rQ→u,source2
Expression kθ * (θ) (θ)t −1 Kθθ I − Cθθ ψi ψi Q
Explanation Error in θ as a result of Q, due to incomplete −1 expansion of Kθθ
i=1
−1 Kuu
ku * (u) (u)t I− Muu ψi ψi F i=1
ku * (u) (u)t −1 −1 −Kuu I− Muu ψi ψi Kuθ Kθθ Q i=1
−1 −1 −Kuu Kuθ Kθθ
I−
kθ * i=1
(θ) (θ)t Cθθ ψi ψi
Q
Error in u as a result of F , due to incomplete −1 expansion of Kuu Error in u as a result of Q, due to incomplete −1 expansion of Kuu Error in u as a result of Q, due to incomplete −1 expansion of Kθθ
TABLE 1: Table of expected residuals for a basis of uncoupled modes of the second order problem. basis, or to enrich the truncated basis.
4.2.3
The static residual for method 2b: fully uncoupled modes of the first order system problem
Comparable to the method described just before this method uses a basis of uncoupled modes. However for this method these are calculated from a first order problem. Again the static correction cannot be calculated from equation 31, because the basis does not consist of correct modes of the coupled first order problem, but the uncoupled modes of this first order problem contain similar information as those of the second order problem. Because also the correct static solution of both the second and the first order system are very alike we expect to find a static residual that is also much alike. Before we calculate the (incorrect) static result of the reduced order model by hand, we especially look on what the reduced order static relation looks like. Therefore we first discuss the mechanical uncoupled basis. Because of the thermal problem is a first order problem we see that Φ(θ) = Ψ(θ) . The mechanical problem in first order form however consists of both u and u. ˙ The corresponding modes of the mechanical problem in state space come in conjugate pairs and a single mode looks like:
u ψiu = ηui (49) u˙ ±λui ψiu We suggest to order all modes in the basis such that we first find all modes belonging to different eigenfrequencies and then their complex conjugates: ⎡ ⎤
Ψ(u) Ψ(u) . Ψ(u) Ψ(u) (u) Φ = ⇒ Φ = ⎣ Ψ(u) Λu −Ψ(u) Λu (50) . ⎦ Ψ(u) Λu −Ψ(u) Λu . . Ψ(θ)
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The static relation Az = S expressed with this basis is then given as: ⎡ ⎤ 2Λu Ψtu Kuu Ψu . Λu Ψtu Kuθ Ψθ zu1 t t ⎣ ⎦ . −2Λu Ψu Kuu Ψu −Λu Ψu Kuθ Ψθ zu2 = 1 t t Λu Ψtθ Kuθ Ψu −Λu Ψtθ Kuθ Ψu Ψtθ Kθθ Ψθ zθ T0
⎡
⎤ Λu Ψtu F ⎣ −Λu Ψtu F ⎦ 1 Ψtθ Q T0
(51)
By inspection of this static relation we recognize that the first and second line express the same operation and therefore the correct solution gives zu1 = zu2 . If one implements this solution in the third line, we immediately find t t that the contribution of Λu Ψtθ Kuθ Ψu and −Λu Ψtθ Kuθ Ψu will cancel each other, resulting to a one-way coupled problem again. Note that the solution for the individual DOF can be written as: ⎤ ⎡ Ψ(u) Ψ(u) . u zu1 (u) (u) ⎣ ⎦ u˙ zu2 = Ψ Λu −Ψ Λu (52) . θ zθ . . Ψ(θ) If we now use zu1 = zu2 we see that indeed the solution for u˙ = 0 is found. Because the static relation can be interpreted to be one-way coupled now, we can find the static solution for zu1 and zθ with a sequential solution procedure to be:
zθ
=
zu1 = zu2
=
−1 t t Ψ(θ) Kθθ Ψ(θ) Ψ(θ) Q −1 t t 1 (u)t Ψ Kuu Ψ(u) Ψ(u) F − Ψ(u) Kuθ Ψ(θ) zθ 2
(53)
The static solution is given in a form that we have seen several times before. We can now reason that if we use a truncated basis, but guarantee that we keep the modes in complex conjugate pairs, we can again recognize two identical operations that result to zˆu1 = zˆu2 . This leads to a one-way coupling in the static relation of the reduced order model. As one can verify now we will obtain the same static result as with a basis of uncoupled second order modes. This also induces that we will find the same static residuals for u and θ as were summarized in table 1. The residual for u˙ = 0 means that we indeed guaranteed that similar to equations ?? and ?? some terms dropped out. The interpretation of this from a reduction point of view is that we neatly represented the identity relation that was used to write a second order problem into first order form. Now let’s return to the general case that we do not intentionally put modes in conjugate pairs in the basis. The resulting (incorrect) static solution can then be found to be: ⎡ ⎤ −1 ˆ (u) K ˆ uu − Λ ˆ uK ˆ uθ K ˆ −1 K ˆt ˆ−K ˆ uθ K ˆ −1 Q ˆ ⎡ ⎤ Ψ F uθ θθ θθ ⎢ u ˆ −1 ⎥ ⎢ ⎥ −1 −1 ˆ ˆ (u) t ⎥ ⎣ u˙ ⎦ = ⎢ Λ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ (54) Ψ K − Λ K K K F − K K Q u uu u uθ θθ uθ θθ ⎢ ⎥ uθ ⎣ −1 ⎦ θˆ −1 ˆ −1 ˆ 4 − 1Λ ˆ (θ) K ˆ θθ − 1 Λ ˆ uK ˆ θu K ˆ uu ˆ uK ˆ θu K ˆ uu Ψ Kuθ Q F 2
2
From this result we can see that due to incomplete conjugate sets of modes, several errors are introduced that we know that should not be there, such as a static solution for u˙ arising from both F and Q, a contribution of F in the static solution for θ and also we see several phase errors that are introduced. With a lot of effort we were able to write these residuals analytically, but in practice we will not use such analytical forms. Therefore we do not write these, but suggest to avoid many of these unnecessary errors by always putting the modes in complex conjugate pairs.
4.3
The static residual for method 3a and method 3b: corrected uncoupled modes of the first order problem
In this section we will look at the reduced static solution for bases in which a correction term was introduced. We can immediately recognize one disadvantage, i.e. the fact that the bases do not consist of block diagonal matrices, but also contain off-diagonal terms. According to equation 18, 21 or 23 the corrected uncoupled basis looks like:
Φ(u) ΔΦ(θ) n 2c Φ = = Φnu2c Φnθ2c (55) ΔΦ(u) Φ(θ)
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The absence of a block-diagonal form leads to a fully coupled reduced static relation, which means in other words that the correction modifies the submatrices that we recognized to belong to the uncoupled physical problems, as was probably the intention of the correction. This means however that if one now wants to guarantee the correct static solution, this becomes very intricate because the correction is not derived from an eigenvalue problem. The fully coupled reduced static relation is given by: n2ct
n2ct t Φu AΦnu2c Φnu2c AΦnθ2c zu Φu F = (56) t t t zθ Φnθ2c AΦnu2c Φnθ2c AΦnθ2c Φnθ2c Q and the results for the individual DOF are represented as: ˆ u zu + ΔΦ ˆ θ zθ u ˆ = Φ
and:
ˆ u zu + Φ ˆ θ zθ θˆ = ΔΦ
(57)
We recognize that the results for the DOF depend on the modal amplitudes corresponding to both fields at the same time, as was indeed the intention of the correction and the approximate static result is now obtained from: −1 4uu − A 4uθ A 4−1 A 4θu 4(u) − A 4uθ A 4−1 S 4(θ) + . . . ˆu A Φ S θθ θθ −1 4θθ − A 4θu A 4−1 A 4uθ 4(θ) − A 4θu A 4−1 S 4(u) ˆθ A ΔΦ S uu uu −1 4uu − A 4uθ A 4−1 A 4θu 4(u) − A 4uθ A 4−1 S 4(θ) + . . . ˆu A θˆ = ΔΦ S θθ θθ −1 4θθ − A 4θu A 4−1 A 4uθ 4(θ) − A 4θu A 4−1 S 4(u) ˆθ A Φ S uu uu
u ˆ=
(58)
where the individual terms can be read from appendix A. It is difficult to foresee the influence of the correction terms. There are however a few possibilities that can be used for static residual. A first possibility is to enforce that the corrections are orthogonal to the mono-physical modes. This simplifies the calculation a lot and for example such as written in the work by Tournour, see [5], the pseudostatic contribution of truncated modes can be taken into account. In order to do this the solution for u and θ such as given by equation 1 are expressed for a harmonic system as: Kuu + iωCuu − ω 2 Muu u = (F − Kuθ θ) t (Kθθ + iωCθθ ) θ = Q − iωKuθ u (59) A so-called pseudostatic solution u0 and θ0 can now be written as: u0 θ0
−1 = Kuu (F − Kuθ θ) t = Kθθ Q − iωKuθ u
(60)
The same procedure can be written for the uncoupled modal basis such as expressed in equation 14 and gives: u ˆ0 θˆ0
ˆ (u)t Λ ˆ (u)−1 Ψ ˆ (u) (F − Kuθ θ) = Ψ t ˆ (θ)t Λ ˆ (θ)−1 Ψ ˆ (θ) Q − iωKuθ = Ψ u
This means that we can express the pseudostatic residual now as: −1 ˆ (u)t Λ ˆ (u)−1 Ψ ˆ (u) (F − Kuθ θ) r u0 = Kuu −Ψ −1 t ˆ (θ)t Λ ˆ (θ)−1 Ψ ˆ (θ) Q − iωKuθ r θ0 = Kθθ −Ψ u
(61)
(62)
If we now implement the modal approximations for u and θ in the righthand side of these residuals, we obtain: −1 ˆ (u)t Λ ˆ (u)−1 Ψ ˆ (u) F − Kuθ Ψ ˆ (θ) zθ ru0 = Kuu −Ψ −1 t ˆ (u) ˆ (θ)t Λ ˆ (θ)−1 Ψ ˆ (θ) Q − iωKuθ rθ0 = Kθθ −Ψ Ψ zu (63)
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Another possibility is to ensure also static correctness of the corrections that were suggested by equation 17. From this equation we noticed that theoretically all modes are needed to calculate the desired correction, whereas in practice we just have a limited amount of modes; namely those that were also used as basis for the reduced models. When the correction is calculated with a truncated set of modes we do not obtain the full correction desired. From section 2.2.5 we recognized however that the full spectral expansion of either the inertia matrix or the stiffness matrix can serve as an approximation of the correction. Therefore we now can suggest at least to use this approximation for the modes that are truncated. Comparable to the residual flexibility given by equation 28 we suggest to introduce a residual correction term for a single mode:
Δφnc ui Δφnc θi
5
≈
−
kθ nct t ! φnc θ φθ Kuθ s
s=1
≈
−
s
λnc ui +
φnc ui λnc θs
ku nct c ! φnc u φu A s
λnc us s=1
s
+
λnc θi
−
−1 Kθθ
−1 φnc θi − Auu
I−
I−
kθ ! s=1
ku ! s=1
nct Cθθ φnc θs φ θs
nc Euu φnc us φus
t Kuθ φnc ui
=−
t
Auθ φnc θi = −
kθ nct c ! φnc θ φθ A s
s=1
s
nc λnc ui + λθs
ku nct c ! φnc u φu A s
s=1
s
nc λnc us + λθi
t nc φnc ui − Gθθ Kuθ φui
nc φnc θi − Guu Auθ φθi (64)
CONCLUSIONS AND OUTLOOK
In this paper we investigated different reduction bases for thermomechanically coupled problems. The intention of all bases is to have a modal representation of the complete or a specific part of the thermomechanical equation of motion. Although modes are capable to give a dynamic representation of the system they do not give the correct static result. In order to identify this error and have an opportunity to improve the reduced models, we investigated the static residuals introduced when applying these different variants. The correct static results predicts that an applied heat generates both a static temperature distribution and a mechanical deformation. An applied force only yields a static deformation. Both loads lead to a zero static velocity. When applying the different bases, we observed the following effects. Fully coupled modes give the possibility to express the thermomechanical problem as modal summation. The static residual can easily be implemented by calculation of the static contribution of the truncated modes by calculation of their corresponding modal acceleration. The static residual when using a basis of uncoupled modes of the second problem could be calculated by using the property of one-sided static coupling, such that the residuals could be calculated sequentially. This approach resulted to static errors due to incomplete spectral expansions of necessary inverse matrices. Coupled modes of the problem are the solution to the thermomechanical equation written in first order form. In first order form the static relation writes a two-sided coupling between the thermal and the mechanical DOF. Using a basis of uncoupled modes calculated in first order form gave rise to more contributions of static error because of this two-sided coupling. It resulted to unexpected contribution from force to the thermal DOF and undesired static solution for velocity. It was observed that these errors disappear when modes are used complex conjugate pairs. Bases that include corrections on the uncoupled modes introduced off-diagonal terms in the bases by which the reduced matrices contain to0 much contributions to the static residual to express them analytically. The choice for the specific type of correction possibly diminishes or deteriorates the static results. In future work we plan to use the terms calculated as static residuals as an enrichment to the bases described in this paper. We suggest to use modes in complex pairs, because this is a simple extension and obviously prevents the occurrence of errors on the static results.
ACKNOWLEDGEMENTS We acknowledge the MicroNed program of the Ministry of Economic Affairs of the Netherlands for the financial support.
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A
APPENDIX:TERMS IN THE FULLY COUPLED REDUCED MATRIX
For a basis Φn2c the reduced static relation looks like: n2ct
n2ct t Φu AΦnu2c Φnu2c AΦnθ2c zu Φu F = t t t zθ Φnθ2c AΦnu2c Φnθ2c AΦnθ2c Φnθ2c Q
(65)
where individual terms are: 4uu A 4uθ A 4θu A 4θθ A 4(u)
S 4(θ) S
t ˆ (θ) ˆ (u)t Kuu Φ ˆ (u) + Φ ˆ (u)t Kuθ ΔΦ ˆ (u) + ΔΦ ˆ (u)t Kuθ ˆ (u)t Kθθ ΔΦ ˆ (u) = Φ Φ + ΔΦ t t t t t ˆ (u) Kuu ΔΦ ˆ (θ) + Φ ˆ (u) Kuθ Φ ˆ (θ) + ΔΦ ˆ (u) Kuθ ˆ (θ) + ΔΦ ˆ (u) Kθθ ΔΦ ˆ (θ) = Φ ΔΦ
= =
t ˆ (u) ˆ (θ)t Kuu Φ ˆ (u) + ΔΦ ˆ (u)t Kuθ ΔΦ ˆ (u) + Φ ˆ (θ)t Kuθ ˆ (u)t Kθθ ΔΦ ˆ (u) ΔΦ Φ + ΔΦ t t t t t ˆ (θ) Kuu ΔΦ ˆ (θ) + ΔΦ ˆ (θ) Kuθ Φ ˆ (θ) + Φ ˆ (θ) Kuθ ˆ (θ) + Φ ˆ (θ) Kθθ Φ ˆ (θ) ΔΦ ΔΦ
ˆ (u)t F + ΔΦ ˆ (u)t Q = Φ ˆ (θ)t F + Φ ˆ (θ)t Q = ΔΦ
(66)
REFERENCES [1] A.M. Steenhoek, D.J. Rixen, and P. Nachtergaele. Model order reduction for thermomechanically coupled problems. IMAC XXVII Orlando, 2009. [2] S. Lepage. Stochastic finite element method for the modeling of thermoelastic damping in micro-resonators. Phd Thesis, Liege, 2006. [3] J.M. Dickens and A. Stroeve. Modal truncation vectors for reduced dynamic substructure models. [4] D.J. Rixen. Dual craig-bampton with enrichment to avoid spurious modes. IMAC-XXVII: A Conference & Exposition on Structural Dynamics, 2009. [5] M. Tournour and N. Atalla. Pseudostatic corrections for the forced vibroacoustic response of a structure-cavity system. J. Acoust. Soc. Am., 2000.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Truncation error propagation in model order reduction techniques based on substructuring. Alexander M. Steenhoek Department of Precision and Microsystems Engineering Delft University of Technology, Mekelweg 2, 2628CD Delft, The Netherlands [email protected]
Daniel J. Rixen Department of Precision and Microsystems Engineering Delft University of Technology, Mekelweg 2, 2628CD Delft, The Netherlands [email protected]
ABSTRACT Several model order reduction techniques split a system in components after which these are reduced individually, where the dynamic response of individual components is typically approximated with a modal truncation of component modes. By an appropriate selection (which usually means selecting enough modes) the truncation error is expected to decrease, but generally no guarantee for the associated error found after reassembling the reduced component models into a single reduced model can be given. In this contribution we investigate how the truncation error arising from the applied reduction techniques for a separate component, propagates to the assembled models. This gives insight on how accurate the model description of separate component needs to be to obey a global overall accuracy of the assembled reduced model and can lead to a different selection criterium for the reduced model. This work is based on an error estimator for modal truncation and the work by Voormeeren [1] on error propagation techniques.
Nomenclature Matrices: m 2nd order time derivative system matrix d 1st order time derivative system matrix k 0th order time derivative system matrix S Load applied to a state space system C Observation from a state space system E 1st order time derivative state space matrix A 0th order time derivative state space matrix Z Dynamic stiffness matrix Y Frequency response function V Reduction basis: a projection matrix D FRF coupling function G Arbitrary time-dependent function b Boolean matrix for compatibility l Boolean matrix for force equilibrium Ω Eigenfrequency matrix Φ Matrix of eigenmodes I Identity matrix
Vectors: q Mechanical DOF: deformation x State vector r Residual σ Hankel singular values ε Error function i Input signal u Output signal f Applied excitation g Interface forces h Applied excitation s Load vector c Response sensing vector λ Lagrange multipliers θ Thermal DOF: temperature difference z Modal amplitudes c Output vector φ Eigenmode
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_58, © The Society for Experimental Mechanics, Inc. 2011
663
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Scalar values: ζ Modal damping p Arbitrary parameter k Number of retained modes t Time ω Frequency λ Eigenvalue
1
Mathematical notation: i Imaginary number (a) . . . Corresponding to substructure a (b) . . . Corresponding to substructure b (ab) . . . Corresponding coupled system t . . . Transposed of vector / matrix −1 . . . Inverse of a matrix . . . A norm of a vector/matrix tr . . . Trace of a matrix max . . . The maximum of a vector sup . . . The maximum of a matrix
INTRODUCTION
Model Order Reduction techniques are convenient or even necessary to perform fast reanalysis on large scale models during for example optimization purposes. Amongst all possible reduction methods, different methods have been proposed that can be combined under the name of component mode synthesis methods. These represent a system’s behavior by a reduced model that is composed of an assembly of reduced component descriptions. The model can be made sufficiently accurate by choosing the mathematical basis describing the components large enough. An appropriate selection criterion for the basis or an stopping criterium that guarantees sufficient accuracy can be difficult to find, because it is not generally known how effects in the separate component, such as for example the truncation error, will be visible in the assembled model. Effects can be magnified or can disappear. In the work by Voormeeren [1], it is presented how errors in the frequency response function (FRF) for a component will affect the assembled model. In others words; how does the error of separate substructures work through the other components. The error propagation gives a fruitful inspiration to exploit this principle for model reduction, because one of the characteristic for a reduction method is its ability to have an error estimation. In this contribution we will use the error propagation as follows. We suggest to start with a reduction method applied to a substructure for which an error predictor is available. Then we investigate how the truncation error, introduced when applying the reduction techniques to a separate component, will propagate to the assembled model. This analysis can give insight on how accurate the model description of a separate component needs to be to obey a global overall accuracy for the assembled reduced model. In order to apply this approach we basically need 2 principles: 1. An estimation of the accuracy of the representation of the frequency response function by a reduced order model. 2. An expression for the sensitivity of the assembled model to a change in an individual component. The first topic is addressed in section 2 and the second in section 3. In section 4 we explain how to combine these two topics to improve model order reduction techniques. In section 5 we illustrate the approach with an academic example.
2 2.1
MODEL ORDER REDUCTION APPLIED TO FREQUENCY RESPONSE FUNCTIONS OF SUBSYSTEMS System dynamics and a systems frequency response function
Suppose that we have a simple system or a component for which the system dynamics is described by the following set of equations: ) m¨ q + dq˙ + kq = s s= f ·i (1) u = cq The dynamic response u of the system and the excitation s of the system do not interact directly, but via an intermediate state vector q. This state vector describes the true dynamics of the systems by an equation of motion,
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containing effects due to mass m, damping c, stiffness k and the state vector q and its time derivatives. However the direct harmonic relation between i and u is expressed by the FRF for this system: −1 Y {ω} = c −ω 2 m + iωd + k f (2) The system can alternatively be written in first order form, where the state of the system is written in terms of its state variables. The first order form looks like: ⎧
I 0 q˙ 0 I q ⎪ ⎪ = +S ⎪ ⎪ 0 m q ¨ q˙ ⎪ ) ⎪ −k −d ⎨ Ex˙ = Ax + S 0 S = Bi S= i or : (3) f ⎪ ⎪
u = Cx ⎪ ⎪ q ⎪ ⎪ u= [ c 0 ] ⎩ q˙ and the corresponding transfer function then looks like: Y = C (iωE − A)
−1
(4)
B
A measure for the interaction between states and the excitation is given by the controllability of the system and in a similar way the observability identifies the interaction between state variables and the response of the system. A qualitative measure for both interactions is obtained from the corresponding grammians that can be calculated as: Wc
0t t = e−Aτ BB t e−A τ dτ 0
∪
0t t Wo = eA τ C t CeAτ dτ
(5)
0
The steady state (time-invariant) solution is obtained more conveniently however from the algebraic Lyapunov equations: AWc + Wc At + BBt = 0 At Wo + Wo A + Ct C = 0 2.2
(6)
Norms of a function
For different reasons people are interested in certain measures for functions. For the frequency response function Y we can think of for example a measure of intensity when standard excitations profiles are applied, a measure for the contribution of individual components to this intensity or a measure for the errors after application of reduction techniques. Standardized norms such as found from [2], [3] or [4] are the H2 , H∞ or Hankel norm. They can be calculated for both frequency-dependent or time-dependent functions and will be described as follows. Obviously, for the FRF we are interested in the frequency-dependent form.
2.2.1
H2 norm
The H2 norm for a FRF can immediately be expressed as: 5 5 6 62 ∞ 6 1 1∞ 6* 1 1 ∗ 6 6 Y {ω} 2 = tr(Y (ω) Y (ω)) = 2π 2π 6 ykk (ω)6 −∞
−∞
(7)
k
and gives the sum of squared magnitudes of all elements in the FRF over all frequencies. Therefore it can be interpreted as the average gain of the system over all elements and over all frequencies. For computational effort is worth mentioning that we can find this norm also from: 7 7 (8) Y {ω} = tr (Ct CWc ) = tr (BBt Wo ) ,where Wc and Wo are respectively observability matrix and controllability matrix and can be solved calculated from equation 6.
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2.2.2
H∞ norm
Another possibility is to use the H∞ norm, which for a time-dependent function G(t) can be defined as: u(t) 2 G {t} ∞ = max 89:; i(t) 2
(9)
u(t)=0
And we can derive a similar form for a FRF as: Y {ω} ∞ = max 89:; σmax (Y (ω))
(10)
ω
From this definition we see that the norm can be interpreted as an indication of the highest peak in the frequency response function. This norm is particularly useful as an induced norm, because when we use it as u(ω) 2 < Y ∞ i(ω) 2
(11)
we see that it shows the worst case amplification of an excitation i to response u. 2.2.3
Hankel norm
In a very comparable fashion as performed for the H∞ norm we can introduce the Hankel norm, which is a subset of the the H∞ norm (thus never exceeds the H∞ norm) and brings a measure for the effect of past excitations to future responses. It is defined as: u(t) 2 i(t) = 0 f or : t > 0, G {t} h = sup where : (12) u(t) = 0 f or : t < 0 i(t) 2 This Hankel norm can be easily calculated using the maximum eigenvalue of the product of controllability grammian and observability grammian (calculated from equation 6) as: 7 Y {ω} h = λmax (Wc Wo ) (13) 2.3
Modal representation of an individual system/component
In dynamics we are generally interested in harmonic vibration responses and in many situations eigenmodes of a system have shown to be of great value or importance during dynamic analysis of this system. Due to the orthogonality properties of modes, the system description decouples (when low damping is considered) and decoupled contributions to the frequency response function as given by equation 2 can be expressed as a model summation in the following form: Y=
n !
ω2 i=1 i
cφi φti f − 2iζi ωi ω − ω 2
(14)
According to [2] we can judge the importance of individual modes on their ability to contribute to the representation of the frequency response function. For this we can use some of the norms introduced before, giving an expression for the input and output gain of individual modes. For convenience the system is written therefore written in the following state space form with modal coordinates:
0 Ωq˙ 0 Ω Ωq z˙ 1 0 Ω z1 0 = + = + q ¨ Ω −2ζΩ q˙ z˙ 2 Ω −2ζΩ z2 Bm Φt f (15)
Ωq z1 −1 y= ΦΩ 0 y = [ Cm 0 ] q˙ z2 For a system written in this form the observability matrix Wo and controllability matrix Wc are shown to be diagonally dominant, see [2]. This is interpreted as follows. Due to decoupling properties modes do not exchange much
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energy and therefore energy put in a mode due to excitation (the input gain) is nearly fully conserved in the response of this modes (output gain). This leads to the following convenient approximation of the grammians: Wc =
wc 0
0 wc
and :
Wo =
wo 0
0 wo
(16)
where: wo
2.4
=
1 −1 −1 ζ Ω diag Bm Btm 4
and :
wc =
1 −1 −1 ζ Ω diag Ctm Cm 4
(17)
Model order reduction of an individual system/component
For various reasons people try to reduce large models. One of the methods applied to large complicated structures or models is to divide the problem by identifying separate components in the structure. Now we can focus on these individual components separately and build reduced models for them separately. The advantage of this approach is that reduction methods can be adapted such that specific details of the component’s behavior are also retained in the reduced model. Assembling all different reduced models of the components into a single model then leads to a reduced model for the entire model. Most of the reduction methods are built within the projection framework. This means that the original DOF q of the system are approximated by a projection matrix V consisting of representative shapes for the system and corresponding amplitudes z as: q = Vz
(18)
The main topic when applying model order reduction is the particular choice of projection matrix V such that the reduced model represent the original system best under different circumstances. Often the projection matrices are derived in such that the reduced model approximate the original FRF best according to the desired specifications. Within this projection framework the reduced order model dynamic description of the system is now expressed as: ⎧ ⎧ ˆz + kz ˆ = ˆ ⎨ Vt mV¨ ⎨ m¨ z + Vt dVz˙ + Vt kVz = Vt s ˆ z + d˙ s V t s = Vt fi ⇔ (19) ˆ s= ˆ f ·i ⎩ ⎩ u = cVz u= ˆ cz The reduced frequency response function is therewith defined as: ˆ {ω} Y
−1 ˆ ˆ = ˆ c −ω 2 m ˆ + iωˆ c+k f
(20)
In this work modal truncation is used as a reduction technique, but the approach can be extended to different reduction techniques. In modal reduction techniques a basis is used that consists of modes and we especially use their orthogonality properties to write the frequency response transfer function of a system as a modal summation such as given by equation 14. A projection matrix instead of the full transformation matrix gives an approximation for the frequency response function, because it can be recognized to resemble the transfer function truncated after after k terms in this summation. y=
k ! i=1
n k ! ! cφi φti f cφi φti f cφi φti f + ≈ 2 2 2 2 2 ωi − 2iζi ωi ω − ω ωi − 2iζi ωi ω − ω ωi − 2iζi ωi ω − ω 2 i=k+1
(21)
i=1
In general a reduction method within the projection framework implies that the representation for the transfer function is performed with only k terms out of the n available term. An important choice is which k terms are important to keep and depends the specific requirements that the reduced model needs to meet. There is not one general approach to perform model reduction and the study of model order reduction techniques has become a discipline on its own. In general we can note that because the models are reduced, they cannot contain the full amount of available information. Most techniques focus on efficient use of the information such that
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they for example represent the FRF sufficiently accurate in at least a certain frequency band of interest or for specific excitations or points of observation. In this contribution we choose modal truncation as a reduction technique. We suggest to choose either the k modes with lowest eigenfrequency leading to a basis Vω . We can also select the modes with largest Hankel norms, where the Hankel norms of individual modes can be read from the diagonal values of the grammians calculated in equation 17. Note that the modes in the full basis are the same, but because of a different ordering the truncated basis becomes different. This leads to the possible projection bases: Vω
= [ φω1
φω2
. . . φωk ]
and:
Vσ = [ φσ1
φσ2
. . . φσk ]
(22)
The first basis is expected to give a good approximation for the FRF in the frequency range ω < ωk and the second basis is expected to represent the most dominant effects in the FRF. Apart from the answer we obtain from the reduced model, we would also like to have an estimation on the accuracy of this answer, or equivalent an idea of the range of variations that is to be expected. This would give an indication in which region the reduced model is applicable and where it needs to be improved. In the next section we will have a discussion on the expected error for the model therefore.
2.5
Expected error due to model order reduction
When a reduced model is used this implies that not all information available in the full model is used. In general a solution calculated with a reduced model will differ from the correct solution expected from the full model. In other words, an error on the solution is introduced which is known as the residual. When reduction is performed within the general projection framework, it means that an error is induced on the representation of both the excitation i, the state of the system x and the response u. In this section we are interested in the error on the frequency response ˆ function Y as given by equation 2 which relates excitation to response. In the FRF for the reduced order model Y such as given by equation 20 all of the aforementioned errors are introduced. This residual will be indicated with r and is calculated as the difference between the full and the reduced FRF. Therefore it is a frequency dependent function and can be written as: r {ω}
ˆ = Y−Y
(23)
The residual can be seen as an absolute error function. In order to express accuracy an expression for the error relative to the correct frequency response function can be convenient. This will be indicated with and can be calculated as: {ω}
=
ˆ Y−Y ˆ Y
(24)
For a reduced order model obtained with modal truncation we can see from equation 21 that we neglected the contribution of modes in the summation after the first k terms. For the error we will focus on the absolute error, which leads to the following residual: r=
n ! i=k+1
cφi φti f ωi2 − 2iζi ωi ω − ω 2
(25)
The residual can be seen as a frequency-dependent error function. In order to express how large the error is, we can use the norms introduced before. For a FRF an induced norm such as the H∞ -norm or Hankel-norm seems a logical choice, because this indicates what the maximum error on the response will be for a certain excitation. Furthermore note that if the modes are ordered according to decreasing Hankel norm, which led to basis V2 , we recognize that we immediately have an error norm in the form of: r ∞ = σk+1 ∞ =
Cm,k+1 Bm,k+1 ∞ 2ζωk+1
(26)
This expresses a worst case error prediction and explains the choice for using basis Vσ , because we now have an error bound for the reduced model.
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3
SENSITIVITY OF THE COUPLED TRANSFER FUNCTION WITH RESPECT TO ERRORS IN THE UNCOUPLED SYSTEM
Suppose that we investigate the frequency response function of a coupled system consisting of 2 subsystems, which are identified as substructure a and substructure b.
3.1
Coupled frequency response functions
Both substructures satisfy dynamic equations similar to equation 1, where now due to coupling apart from external force F also internal/connecting forces g are expected. Additional to the expressions for the equations of motion we therefore need compatibility of the DOF at the interface between the substructures and force equilibrium between the substructures. This leads to the following set of equations describing the coupled system: ) mq¨ + dq˙ + kq = f + g bq = 0 (27) lt g = 0 We can fulfill the force equilibrium condition a priori by choosing the interface forces in a specific form such that we can use l = null(b). The forces are then related to Lagrange multipliersλ, physically representing the interface force intensities, by: g = −bt λ
(28)
Using this specific form for the connection forces between the substructures, we can write the system according to [5], [6] and [7] similar to equation 1 in a dual assembled form: ⎤ ⎡ q¨ ⎤ ⎡ q˙ ⎤ ⎡ ma 0 0 da 0 0 ka 0 bta qa fa a a t 0 mb 0 ⎣ q¨b ⎦ + 0 db 0 ⎣ q˙b ⎦ + ⎣ 0 kb bb ⎦ qb = fb (29) ¨ ˙ 0 0 0 0 0 0 λ 0 ba bb 0 λ λ For convenience this can be written in more compact form with use the dynamic stiffness Z, giving: ⎡ ⎤
Za . bta qa Fa x F Z Bt t ⎣ . Za bb ⎦ qb = Fb ⇔ = λ 0 B . λ 0 ba bb .
(30)
Now we would like to find the frequency response function of the coupled problem, that is when the components are connected and form a single structure. It demands an explicit relation between the excitations and the responses and therefore requires condensation of λ. This is obtained with the following steps. The solution for the DOF is first expressed in terms of external forces and the Lagrange multipliers: u = Z −1 f − B t λ (31) Substituting this in the compatibility condition allows to express the Lagrange multipliers explicitly. Note that it is convenient to express this using Z −1 = Y , where we use a matrix of uncoupled transfer function Y . This leads to: −1 λ = bY bt bY f (32) Backsubstitution of this result in equation 31, enables us to write the relation between u and f explicitly and therefore we can express how the frequency response function of the coupled problem Y (ab) is constituted from the matrix of uncoupled frequency response functions Y : −1 Y (ab) = Y − Y bt bY B t bY (33) Next we suggest to introduce function D, which mathematically represents a projector, and physically expresses how coupling influences the coupled frequency response functions. Note that function D depends itself on the frequency response functions of the uncoupled substructures and thereby of frequency. D {ω} = bt bYbt b (34) Note that with this function we can write equation 35 in the very compact form of: Y (ab) = Y − Y DY
(35)
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3.2
Sensitivity of the coupled transfer function with respect to effects in the uncoupled system
In order to analyze the influence of effects in the FRF of the individual components Ya or Yb on the FRF of the coupled system Yab we need to calculate the corresponding sensitivity. This requires the derivative of the coupled FRF Y (ab) to a single entry of the uncoupled FRF’s Yij . In order to find this we first calculate the derivative of the −1 inverse matrix (bY bt ) , which can be expressed as: ∂ (bYbt ) ∂Yij
−1
−1 ∂Y t −1 = − bYbt b b bYbt ∂Yij
This result can be used to calculate the sensitivity of D next, which can now be written compactly again as:
−1 ∂D ∂ (bYbt ) ∂Y t =b b=D D ∂Yij ∂Yij ∂Yij
(36)
(37)
This result again can be used in combination with the chain rule to find the sensitivity of Y (ab) and leads to: ∂Y (ab) ∂Y ∂Y ∂Y ∂Y = − DY − Y D +YD DY ∂Yij ∂Yij ∂Yij ∂Yij ∂Yij
(38)
We would like to stress that the derivative of a matrix with respect to its entities result to a Boolean matrix with identity for the entry on which you operate: ∂Y = Pij ∂Yij
(39)
This enables us to write the first order derivative more compactly as: ∂Y (ab) = Pij − Pij DY − Y DPij + Y DPij DY ∂Yij
(40)
From the results above we see that we approximated the sensitivity with a first order derivative. In certain situations we would like to know the sensitivity more accurate and then we would need higher order derivatives. Note that Pij is constant (does not depend on Y again) and thus derivatives of it will disappear. Now we can give the second order partial derivative of the coupled FRF as: ∂ 2 Y (ab) = ∂Yij ∂Ykl
Pij DPkl DY − Pij DPkl − Pkl DPij + Y DPkl DPij + . . . Pkl DPij DY − Y DPkl DPij DY − Y DPij DPkl DY + Y DPij DPkl
4
(41)
USING ERROR PROPAGATION FOR MODEL ORDER REDUCTION TECHNIQUES
In the previous sections we introduced two concepts. First we discussed the idea of error(-estimation) of the FRF of a substructure as a result of applying a reduction basis to represent this FRF. Secondly we discussed the sensitivity of a coupled FRF to a change in a FRF of the substructure. Combining these ideas would give an estimation of the error on a coupled FRF due to application of a reduction basis in substructures. We see different possibilities to exploit the concepts and suggest 2 approaches. The first approach is that as mentioned before one of the characteristics of a reduction technique is its ability to give an error estimation once an answer is found. Therefore we will try to express the associated error of the coupled FRF and because this is performed afterwards this can be seen as ”a posteriori ” use of the sensitivity and local error predictions. A second approach is to use the knowledge of the sensitivity of a coupled FRF to adapt the selection criterium for the basis of the substructure. This is performed before the model is reduced and is therefore seen as ”a priori” use. Note that in all situations we will use the sensitivity, which is actually a function of the full FRF’s itself. During the discussions in this paper we will assume that this sensitivity can be calculated in full form, where in general this would also be influenced by the truncation.
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4.1
A posteriori use: error prediction
Such as was mentioned before, one of the characteristics of a reduction technique is its ability to give an error estimation once an answer is found. Therefore we will express the error in the approximated coupled FRF associated to approximation error in substructures. Because this is performed after the reduction was performed, this is seen as ”a posteriori ” use of the sensitivity and local error predictions. In section 2.5 it was seen that for an individual substructure a truncation error is expected due to reduction of the FRF that can be seen as a local error and the corresponding error function is frequency dependent. For substructure a we can write this as: ra {ω} = Y a − Yˆ a = ΔY a
(42)
After coupling we expect a similar error to occur for the coupled FRF, giving: rab {ω} = Y ab − Yˆ ab = ΔY ab
(43)
Remember that an arbitrary function h(p) can be approximated with an Taylor expansion as: 6 6 6 6 n n n 2 ! ! ! 6 ∂h 6 1 ∂ h 66 h(p) ≈ h(p) ¯ + (p − p ¯ ) + (pij − x ¯ij )(pkl − p¯kl ) ij ij ∂pij 66 2 ∂pij ∂pkl 66 i,j=1 i,j=1 k,l=1
p ¯
(44)
p ¯
A similar form can be used to relate the error in the coupled FRF to an error in a FRF of a substructure. Using the sensitivities that were introduced with equation 40, we can express the first order Taylor series for the coupled error to be: ΔY (ab) {ω}
! ∂Y (ab) ∂Y (ab) (a) (a) ΔY = ΔYij (a) ∂Y (a) i,j ∂Yij ! (a) = (Pij − Pij DY − Y DPij + Y DPij DY ) ΔYij
≈
(45)
i,j
A more accurate expansion can be found using also the second order derivative in the sensitivity. We will not write all terms explicitly, but the approximation looks like: ΔY (ab) {ω}
≈
! ∂Y (ab) i,j
(a)
∂Yij
(a)
ΔYij +
n n 1 ! ! ∂ 2 Y (ab) (a) (a) ΔYij ΔYkl (a) (a) 2 k,l=1 i,j=1 ∂Yij ∂Ykl
(46)
The obtained expressions can now be used to predict the error of the coupled system. We like to point out that instead of the correct error functions ΔY (a) , we can also use for example an error bound function of an individual substructure such as expressed in equation 26 in order to estimate the associated error bound for the coupled structure. The advantage of this is that the correct evaluation of the error function can induce lots of computational costs.
4.2
A priori use: selection criterium
Different from what was described above, the sensitivity can also be used before the substructures are truncated. We suggested to use modal summation to approximate the FRF of a substructure. The summation means that the FRF is a summation of contributions of individual modes. Instead of selecting modes based on their importance within the substructure (which led to basis Vσ ), we will now judge or select modes based on their expected contribution to the coupled FRF. This is performed as follows. From the modal summation we recognize that the contribution for a single mode φi is it modal participation: ηi
=
φt f ω 2 − 2iζi ωi ω − ω 2
(47)
From this modal participation we can recognize that within the FRF of the substructure itself the largest contribution is expected near ωi . The modal participation is a frequency-dependent function with its maximum at the eigenfrequency. The expected participation of the mode in the FRF of the coupled system can be estimated by combining
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the modal participation with the sensitivity. For a mode that originates from substructure a we can now derive: (ab)
ηi
=
∂Y (ab) ∂φi
∂Y (ab) ∂Y (a) ∂Y (a) ∂φi k ! Y (ab) (a) = η Yij i i,j=1 =
(48)
Now we cannot indicate beforehand at which frequency the maximum contribution is to be found and therefore we would need to use for example one of the norms to indicate which contributions is largest. Then we are able do a selection based on the norms for each individual mode.
5
ILLUSTRATING EXAMPLE
We can illustrate the approach in this paper with an academic mechanical system consisting of 2 substructures as was found from [1]. The system is depicted in figure 1. For this analysis we constrain Substructure A on the lefthand side uA1 = 0 and Substructure B on the righthand side uB4 = 0. We will first investigate the behavior of substructure A as an independent system. By the constraint uA1 = 0 we are left with only 6 free DOF and therefore also 6 modes can be calculated. By making a selection out of these 6 available modes we can find a truncated basis. We suggest to use the following bases. The first basis will be a pretty accurate basis consisting of 5 out the 6 modes according to increasing eigenfrequencies: V1
= [ φ1
φ2
φ3
φ4
φ5 ]
(49)
As a second basis we suggest to pick 3 the modes according to the 3 lowest eigenfrequencies, giving: V2
= [ φ1
φ2
φ3 ]
(50)
As as third basis we also suggest to choose 3 modes, but those with the highest Hankel norms as can be calculated from equation 17 and read from figure 2. For this we choose C = I and B = I because we do not know beforehand which DOF are of interest. This suggests to choose the following basis: V3
= [ φ2
φ3
φ4 ]
(51)
We can calculate the FRF from an arbitrary DOF to any other DOF in the substructure. We suggest to look at the FRF for uA5 to uA3 . In figure 3 the results for the different basis are depicted for this FRF. On the left hand side we see the FRF of the full and the reduced models, in the mid the absolute errors of the different methods and on the righthand we see the relative errors. We will first investigate the concept of error propagation for the first basis, because this is expected to be relatively accurate because just one mode is truncated. After coupling of the substructures we find the results for the FRF for uA5 to aA3 depicted in figure 4. The FRF of both the full system and the reduced system is depicted on the left. In the mid we see the absolute error and on the right hand side the relative error. In the graphs for the absolute and relative errors we show the results obtained by application of equation 45 and 46 and we can see that both the first order and the second order estimation of the error propagation give excellent results. In the results thus far we looked at DOF that are both in substructure a, meaning that we could see how a the FRF changes after coupling to substructure b. Now we can investigate how an FRF from substructure a to substructure b and we suggest to focus on the FRF uA5 to uB3 . Again we show for the full and the reduced model the magnitude of this FRF, the absolute error and the relative error. We also show the estimation of the error propagation obtained equation 45 and 46. The results are depicted in figure 5. We can observe that the estimation differs slightly more that the results for the FRF uA5 to uA3 , but still give good results. Next we will investigate similar results for the basis given by equation 50 and 51. In both bases 3 modes are retained what induces a larger truncation error than was obtained with the basis given by 49, such as we can observe from figure 3. The results for the coupled system are shown in figure 6 and figure 7 for the bases given by equation 50 and 51 respectively. From the figures we see the estimation of the error is now slightly less accurate, which can be explained because the truncation error in the individual substructure has drastically increased. We can also conclude that the second order sensitivity is not essentially better. At last we observe that the results obtained with the basis of equation 51 does not give better results than that of equation 50. The reason for this is that the selection of the most important modes was based on the largest Hankel-norm that was valid for the substructure, which does not tell about the importance of the modes for the coupled FRF. We can possibly improve this by choosing the interface DOF as point of excitation B = bt , but for now suggest to use the a priori selection instead.
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Figure 2: Importance of individual modes of substructure A.
Figure 3: Full and reduced FRF u5 → u3 for substructure A.
Figure 4: Full and reduced FRF for uA5 → uA3 the coupled system and the associated errors.
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Figure 5: Full and reduced FRF for uA5 → uB3 the coupled system and the associated errors.
Figure 6: Reduced system with 3 modes selected on increasing eigenfrequency. (Mode 1,2,3)
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Figure 7: Reduced system with 3 modes selected on decreasing Hankel norm. (Mode 2,3,4)
6
CONCLUSIONS AND OUTLOOK
In this contribution we investigated the accuracy of reduced models consisting of different components, where the truncation error in a reduced coupled system originates from the modal truncation of individual substructures. An error function for modal truncation was easily found to be the modal summation of all modes that are truncated. An estimation for the error in the coupled system can be found with use of a propagation method that calculates the sensitivity of the FRF of a coupled system to a change in an entry for the uncoupled FRF. From an academic example we observed that the sensitivity is able to predict the error in the coupled system, at least when the FRF of the substructure is already sufficiently accurate. A second order expansion for the sensitivity did not bring a huge improvement of the results, while it computational costs are much higher. We also observed that a basis to reduce substructure that is selected on the Hankel norms is supposed to give best results for this substructure. This does not hold for the ability to represent the coupled system however and we suggested to perform a selection using the expected modal participation in the coupled system. For future work we suggest to investigate approximate error predictors instead of the correct, but dynamic error function for the modal truncation error. Furthermore we used the full substructure FRF’s to calculate the sensitivity, whereas in practice we often have only truncated FRF’s available. For this reason and also to reduce computational costs when generating the sensitivities, we suggest to look at the possibilities to use truncated description for the sensitivities.
ACKNOWLEDGEMENTS We acknowledge the MicroNed program of the Ministry of Economic Affairs of the Netherlands for the financial support.
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REFERENCES [1] SN Voormeeren, D. de Klerk, and DJ Rixen. Uncertainty quantification in experimental frequency based substructuring. Mechanical Systems and Signal Processing, 24(1):106–118. [2] W. Gawronski. Advanced structural dynamics and active control of structures. Springer Verlag, 2004. [3] A.C. Antoulas. Approximation of large-scale dynamical systems. Society for Industrial Mathematics, 2005. [4] S. Skogestad and I. Postlethwaite. Multivariable Feedback Control. Wiley, 1996. [5] D. de Klerk, DJ Rixen, and SN Voormeeren. General framework for dynamic substructuring: history, review, and classification of techniques. AIAA JOURNAL, 46(5):1169, 2008. [6] D de Klerk. Dynamic Response Characterization of Complex Systems through Operational Identification and Dynamic Substructuring. Phd Thesis, Delft, The Netherlands, 2009. ´ [7] M. Geradin and D. Rixen. Mechanical vibrations. Wiley, 1997.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Investigation on the Use of Various Decoupling Approaches
David Cloutier, Dr. Peter Avitabile Structural Dynamics and Acoustic Systems Laboratory University of Massachusetts Lowell One University Avenue Lowell, Massachusetts 01854 ABSTRACT Substructuring methods allow for the development of system models from component information. Often times however, system response needs to be improved through the modification of one or more component representations. Decoupling the component is necessary in order to accomplish any additional design improvements. Decoupling can be performed different ways. Several approaches are considered for the evaluation of component decoupling from the system. Impedance and Mobility techniques are compared to an alternate force decoupling approach. Several models are studied to better understand the strengths and weaknesses of each of the techniques often employed. Several cases are studied and shown in the paper. INTRODUCTION Frequency Based Substructuring has been used as a valuable tool for many years. This modeling approach using frequency response functions [1- 4] has received much attention in the development of system models. Many researchers have provided alternate approaches for the development of system models. In recent years however, many [5-13] have directed their efforts towards the disassembly or decoupling of system models to identify component characteristics. There are many reasons to obtain component information from a system representation. Often, an individual component may need to be redesigned to meet specific requirements to satisfy system response characteristics. The system response characteristics of the assembled system identify the overall characteristics but do not necessarily directly provide a clear identification of the individual component contributions to the total system response. Being able to identify specific component characteristics that provide the proper system level response is of critical importance when trying to redesign or retrofit components into a system model. Clearly, this information can provide extremely beneficial information if available. For this reason, many have attempted to identify component information from system response characteristics. When system response characteristics along with one of the component response characteristics are known, the remaining component information can be extracted. There are several approaches to finding the unknown component information. Impedance and Mobility modeling approaches have been employed by several researchers [5- 8]. While these approaches lead to the same equation when only response functions at the connections are used, the use of internal DOF in the formulation results in slightly different expressions. Both of these have provided possible mechanisms for obtaining the component characteristics for the unknown component which is part of a system model. An alternate approach was considered in which the constraint forces at the connections are used to estimate the unknown component. In this paper, the Impedance and Mobility approaches are reviewed along with an alternate force constraint approach. These are all compared to each other and strengths and drawbacks are discussed.
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_59, © The Society for Experimental Mechanics, Inc. 2011
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THEORY In general, two components A and B can be used to form a system representation AB. The two components are coupled at connection point (c), that results in a coupling force (fAB ) to tie the components together. An external force (fe) may be applied to either of the components. With this definition, the system is characteristically shown in Figure 1; also note that an equivalent representation of the component may be shown with only one component provided that the system is maintained in dynamic equilibrium using the connection force. Additionally, there may be interior points (i) on either component A or B. For the work presented in this paper, the general system coupling equations are presented first, followed by the three separate decoupling approaches investigated.
Figure 1. General System Description and Nomenclature. General System Modeling Equations Previous decoupling techniques include Impedance and Mobility [5] in which the frequency response functions (FRFs) of the connection DOF on the unknown component B are calculated from the complete known system AB and a known component A. By partitioning the FRF matrices into the connection DOF (c) and internal DOF (i), the known system can be written as,
^u`AB ½ ° c ° ® AB ¾ °¯^u`i °¿
ª> H @AB « cc «> H @AB ¬ ic
> H @ci º» °^f `c ½° AB ® AB ¾ > H @ii »¼ °¯^f `i °¿ AB
AB
(1)
and for the known component A,
^u`A ½ ° c° ® A¾ °¯^u`i °¿
ª> H @A « cc «> H @A ¬ ic
> H @ci º» °^f `c ½° A ® A¾ > H @ii »¼ °¯^f `i °¿ A
A
(2)
The coupling conditions at connection DOF are:
^u`c ^u`c ^u`c AB A B ^f `c ^f `c ^f `c AB
A
B
(3) (4)
and at internal DOFs on component B:
^u`i ^u`i AB B ^f `i ^f `i AB
B
(5) (6)
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Impedance Approach For the Impedance based approach, subtracting the inverse of (2) by the inverse of (1) and introducing the conditions (3), (4), (5) and (6), with some algebraic manipulation one can obtain,
> H @cc B
ª> N @cc º « » «¬> N @cc »¼
ª> H @AB º « cc » «> H @AB » ¬ ic ¼
(7)
with, A 1
A 1
> N @cc > I@cc > H @cc > H @cc > H @ci > H @ic AB A AB A > N @ic > H @ic > H @cc > H @ii > H @ic AB
AB
1
(8)
1
If only connection DOF are measured, then (7) simplifies to,
> H @cc B
A 1
> I@cc > H @cc > H @cc AB
1
> H @cc
AB
(9)
Mobility Approach For the Mobility based approach, subtracting (2) from (1) and introducing the coupling conditions (3), (4), (5) and (6), with some algebraic manipulation one can obtain,
> H @cc B
ª > H @AB cc ¬
§ AB ¨ > H @ci º¼ ¨ ª¬ > I@cc ¨ ©
ª> H @A º > 0@ci º¼ « ccA » «> H @ » ¬ ic ¼
ª> H @AB « cc «> H @AB ¬ ic
> H @ci > H @ci º» ·¸ AB A > H@ii > H@ii »¼ ¸¸¹ AB
A
(10)
If only connection DOF are measured, then (10) simplifies to,
> H @cc > H @cc > I@cc > H @cc > H @cc B
A 1
AB
AB 1
(11)
Both approaches obtain the same solution when only connection DOFs are used, as seen by comparing (9) to (11). Constraint Force Approach Consider a complete known system AB assembled from known subcomponent A and unknown subcomponent B. The response, {x}c, of the system at the connection DOF (c) with a force, {f}e, at some arbitrary external DOF, (e) can be written as,
^x`c
AB
AB > H @ce ^f `e
(12)
An alternate way to write the displacement of system AB would be to use the FRFs of the known component A and assuming a constraint force, {f}c. This will provide the necessary dynamic force that the unknown component B exerts on component A and is given as,
^x`c
AB
ª> H @A ¬ ce
^f ` ½ A > H @cc º¼ °® ABe °¾ °¯^f `c °¿
(13)
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Rewriting (13) to solve for the constraint force gives,
^f `c
AB
A 1
> H @cc ª¬> H @ce ^f `e ^x`c A
AB
º ¼
(14)
From the Frequency Based Substructuring derivation [2], the constraint force between two components can be calculated by,
^f `c
AB
> H @
A
> H @cc B
cc
1
> H@ce ^f `e A
(15)
Rewriting (15) to solve for the unknown component B, AB1
> H @cc > H @ce ^f `e ^f `c B
A
> H @cc A
(16)
This equation holds true for a single connection. For two connections, (15) can be written as,
^f `AB ½ ° c1 ° ® AB ¾ °¯^f `c2 °¿
ª ª > H @A c1c1 «« « «> H @A ¬ ¬ c2c1
A B > H @c1c2 »º ª« > H @c1c1 A B > H @c2c2 »¼ «¬> H@c2c1
> H @c1c2 º» º» B > H@c2c2 »¼ »¼ B
1
ª > H @A º c1ce « » ^f ` «> H @A » e ¬ c2ce ¼
(17)
> @c1c1 was previously calculated with the system having a single connection, one can find the FRF between
Assuming H
B
the two connections can be found by writing (17) into two individual equations and rearranging so that,
> H @c1c2 B
> H @
^f `e > H @c1c1 ^f `c1 > H @c1c2 ^f `c2 > H @c1c1 ^f `c1 ^f `c2 c1ce A
A
AB
A
AB
B
AB
AB1
(18)
Assuming reciprocity,
> H @c2c1 > H @c1c2 B
B
(19)
Rewriting (17) again and using (18) and (19) gives,
> H @c2c2 B
>H@c2ce ^f `e >H@c2c1 ^f `c1 >H@c2c2 ^f `c2 >H@c2c1 ^f `c1 ^f `c2 A
A
AB
A
AB
B
AB
AB1
(20)
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APPLICATION System Description To study the robustness of each technique, a simple mass-spring model was created in MATLAB [14]. This model consists of two individual components A and B combined using rigid connections; extension of these equations to consider flexible connections could also be considered. Table 1 and Table 2 list the physical and modal properties of component A and component B, respectively.
A K1
M1
K2
M2
Table 1. Properties of Component A.
K3
M3
Figure 2. Physical Representation of Component A.
B M4
K4
M5
Table 2. Properties of Component B.
K5
M6
K6
Figure 3. Physical Representation of Component B. FRFs of each component were synthesized from the modal properties listed in Table 1 and Table 2. Using Frequency Based Substructuring [2], the two components were combined using rigid connections. System AB-1 was created using a rigid connection between DOF 3 on component A and DOF 4 on component B, represented by Figure 4. The natural frequencies and damping are listed in Table 3.
A K1
M1
K2
M2
B K3
M3
M4
K4
M5
Table 3. Modal Properties of System AB-1. K5
M6
K6
Figure 4. Physical Representation of System AB-1. System AB-2 was created using rigid connections between DOF 1 on component A to DOF 6 on component B and also between DOF 3 on component A to DOF 4 on component B. Figure 5 displays a representation of this system, and Table 4 lists the natural frequencies and damping of this system.
A K1
M1
K2
M2
B K3
M3
M4
K4
M5
Table 4. Modal Properties of System AB-2. K5
M6
Figure 5. Physical Representation of System AB-2.
K6
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Perturbation of Frequency Response Functions Response analysis was performed in LMS CADA-X Forced Response Simulation module [15] to simulate real measurements. A uniform random force was applied to each of the systems and responses were computed. All FRF manipulation was performed using the LMS CADA-X Matrix Toolbox [16]. Two cases were evaluated. One with FRFs to study the decoupling approaches without any noise on the functions and a second case with noise. A one percent random noise was applied to the force spectrum and frequency response functions were computed for all DOFs. All decoupling approaches were first computed with the applied noise resulting in noisy estimates of the unknown component. All techniques produced similar results when calculated directly with noise, but will not be presented in this paper in order to keep the results brief. These cases will be presented in future work [13] along with experimental studies currently in progress.
Figure 6. Drive Point FRF of Connection DOF 3 on Component A.
Modal parameter estimation was performed to smooth the FRFs, as typically done with measurements. Table 5 lists the frequencies and damping obtained from modal parameter estimation. New FRFs were synthesized from this modal data for use in the decoupling computations. Figures 6, 7 and 8 show the effect of the applied noise and synthesized FRFs from modal parameters.
Figure 7. Drive Point FRF of Connection DOF 3 on System AB-1.
Table 5. Reference Model Modal Characteristics Compared to Model with Noise.
Figure 8. Drive Point FRF of Connection DOF 3 on System AB-2.
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Decoupling of System – 1 Connection The first case considers system AB-1 in which there is a single rigid connection between DOF 3 and DOF 4. Due to the simplicity of the model, effects of truncation are of no concern for any of the cases. Figure 9 shows all techniques produce exact results when pure FRFs are used. Figure 10 shows the effect of noise applied to the data. The Constraint Force approach shows noise scattered throughout the entire FRF due to the noise being present in the constraint force calculation, equation (14). The Impedance and Mobility approaches gave a fairly accurate result except for a spurious peak occurring at approximately 60 Hz. Further investigation to this discrepancy is related to the shift in antiresonances in connection FRFs which are inverted in the calculation. Figure 11 displays an overlay of the FRFs used in the computation and the resultant FRF estimation of Component B. The problematic frequencies are highlighted, which are both located at antiresonances of system AB-1 and component A. (These antiresonances of the perturbed FRFs have a slight shift in frequency from the true FRFs, which is shown to produce these peaks during the manipulation of the FRF equations.) An advantage of both the Impedance and Mobility approaches is the ability to include additional internal DOFs which are present on both the system and known component. The estimation of the unknown component is significantly improved with the use of internal DOF, as shown in Figure 12. Using DOF 1 produced slightly more accurate results for both approaches. Including both DOFs in the computation produces the same results as the results using DOF 1. (Use of additional internal DOF is currently under investigation with larger models to allow for more general selection of DOF; on this limited size model, the use of additional DOF is limited.) For all cases studied thus far, the Mobility with internal DOF was shown to produce more accurate results.
Figure 9. FRF Estimation of Component B using Exact FRFs: Reference (Black), Force Approach (Red), Impedance and Mobility (Blue).
Figure 10. FRF Estimation of Component B using Perturbed FRFs: Reference (Black), Force Approach (Red), Impedance and Mobility (Blue).
Figure 11. Overlay of FRFs used in Calculation and Perturbed Estimation of Component B (Top: Component A, Bottom: System AB-1).
Figure 12. FRF Estimation of Component B using Internal DOF: Reference (Black), Impedance (Red), Mobility (Blue).
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Decoupling of System – 2 Connections The second case studied uses system AB-2, in which component A and B are connected with two rigid connections. As with the first case, all three methods produce exact results when pure FRFs are used, as shown in Figure 13. Impedance and Mobility approaches are much more efficient approaches than Force, as only a single calculation must be performed. As derived in the theory section, the Force approach needs a FRF on the unknown component initially found from the single connection case to find the additional connection FRFs. Figure 14 presents the estimation of the connection drive point FRFs on component B using perturbed FRFs. As previously mentioned, the Constraint Force approach requires the FRF of component B with a single connection; therefore any inaccuracies in this estimation will be amplified in successive calculations. Impedance and Mobility are again significantly improved with the use of an internal DOF, as shown in Figure 15. Mobility approach with internal DOF again produced the most accurate estimations.
Figure 13. FRF Estimation of Component B using Exact FRFs: Reference (Black), Force Approach (Red), Impedance and Mobility (Blue).
Figure 14. FRF Estimation of Component B using Perturbed FRFs: Reference (Black), Force Approach (Red), Impedance and Mobility (Blue).
Figure 15. Estimation of Component B using Internal DOF: Reference (Black), Impedance (Red), Mobility (Blue). OBSERVATIONS Current investigations of the Constraint Force technique were shown to be promising with a single connection. While this technique is not as efficient for multiple connections, current work is focused on computing the unknown component with multiple connections in a single calculation. A single matrix calculation would significantly reduce the noise amplified in successive calculations. Impedance and Mobility appeared to be advantageous when internal DOF are available, although slight shifts in antiresonances due to modal parameter estimation on the noisy FRFs were shown to produce spurious peaks when only connection DOF are used. These spurious peaks are not present when the noisy FRFs are directly used in the decoupling approaches, although the entire FRF computed has a significant amount of noise. Future studies will be focused on applying these techniques to a test fixture in which a variety of test issues will be presented.
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CONCLUSIONS The decoupling problem was presented in which FRFs are of an unknown component are estimated from a known system and component. Impedance and Mobility techniques are investigated and compared to a Constraint Force-based decoupling approach. The effect of noise was studied for all techniques to replicate real measurements. The current Constraint Force approach was shown to amplify noise of the estimated component for multiple connections. Impedance and Mobility approaches were shown to produce spurious peaks in the estimation of the unknown component when only connection DOF FRFs are used. The use of internal DOF significantly improved the estimated FRF for these approaches. REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Otte, D., “Development and Evaluation of Singular Value Analysis Methodologies for Studying Multivariate Noise and Vibration Problems”, Doctorate Dissertation, Catholic University Leuven, Leuven, Belgium, May 1994 Cuppas, K., Sas, P., Hermans, L., “Evaluation of the FRF Based Substructuring and Modal Synthesis Technique Applied to Vehicle FE Data”, Proceedings of the Twenty-Fifth International Seminar on Modal Analysis, Leuven, Belgium, Sept 2000. Piergentili, F., “Rotational Degree of Freedom Estimation of Frequency Response Functions for Substructured Experimental Components”, Master’s Thesis, University of Massachusetts Lowell, Aug 1999. Ren, Y., Beards, C., “On Substructure Synthesis with FRF Data”, Journal of Sound and Vibration, Vol 185, No 5, pp 845-866, Sept 1995. D’Ambrogio, W., Fregolent, A., “Sensitivity of Decoupling Techniques to Uncertainties in the Properties”, Proceedings of ISMA 2008 - International Conference on Noise and Vibration Engineering, Leuven, Belgium, Sept 2008. D’Ambrogio, W., Fregolent, A., “Promises and Pitfalls of Decoupling Procedures”, Proceedings of the Twenty-Sixth International Modal Analysis Conference, Orlando, Florida, Feb 2008. D’Ambrogio, W., Fregolent, A., “Prediction of Substructure Properties using Decoupling Procedures”, Proceedings of EURODYN 2005 – Sixth European Conference on Structural Dynamics, Paris, France, Sept 2005. D’Ambrogio, W., Fregolent, A., “Decoupling Procedures in the General Framework of Frequency Based Substructuring”, Proceedings of the Twenty-Seventh International Modal Analysis Conference, Orlando, Florida, Feb 2009. Sjövall, P., Abrahamsson, T., “Substructure System Identification from Coupled System Test Data”, Mechanical Systems and Signal Processing, Vol. 22, No. 1, pp 15-33, Jan 2008. Voormeeren, S., Rixen, D., “Substructure Decoupling Techniques – a Review and Uncertainty Propagation Analysis”, Proceedings of the Twenty-Seventh International Modal Analysis Conference, Orlando, Florida, Feb 2009. Gray, S., Starkey, J., “Dynamic Substructure Separation Using Physical and Modal Models”, Proceedings of the Sixth International Modal Analysis Conference, Kissimmee, Florida, Feb 1988. D’Ambrogio, W., Fregolent, A., “Decoupling of a Substructure from Modal Data of the Complete Structure”, Proceedings of ISMA 2008 - International Conference on Noise and Vibration Engineering, Leuven, Belgium, Sept 2004. Cloutier, D., “Investigation of Various System Model Decoupling Techniques”, Master’s Thesis, University of Massachusetts Lowell, expected completion Jan 2010 MATLAB R2007a, The MathWorks, Natick, Massachusetts. CADA-X Modal Analysis module, Leuven Measurement Systems, Leuven, Belgium. CADA-X Matrix Toolbox module, Leuven Measurement Systems, Leuven, Belgium.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Input Estimation from Measured Structural Response
Dustin Harvey1, Elizabeth Cross2, Ramón A. Silva1, Chuck Farrar1, Matt Bement1 1)
Los Alamos National Laboratory, MS T001, Engineering Institute, Los Alamos, NM 87544 2)
Dept of Mechanical Engineering, University of Sheffield, Sheffield, S1 3JD, UK
NOMENCLATURE A, B, C
System state-space matrices
e
Model error
J
Cost function
J’
Perturbed cost function
"
Adjoint operator
r
Test function
t
Time vector
T
Final time
u
Input guess
u’
Perturbed system states
x
System states
x’
Perturbed system outputs
y
System outputs
ym
Measurements
ABSTRACT This report will focus on the estimation of unmeasured dynamic inputs to a structure given a numerical model of the structure and measured response acquired at discrete locations. While the estimation of inputs has not received as much attention historically as state estimation, there are many applications where an improved understanding of the immeasurable input to a structure is vital (e.g. validating temporally varying and spatiallyvarying load models for large structures such as buildings and ships). In this paper, the introduction contains a brief summary of previous input estimation studies. Next, an adjoint-based optimization method is used to estimate dynamic inputs to two experimental structures. The technique is evaluated in simulation and with T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_60, © The Society for Experimental Mechanics, Inc. 2011
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experimental data both on a cantilever beam and on a three-story frame structure. The performance and limitations of the adjoint-based input estimation technique are discussed.
1. INTRODUCTION 1.1 Motivation Knowledge of excitation to a structure can be useful for a number of different engineering applications. With a better understanding of the dynamic input to a system, the response of that system to its operational and environmental loading conditions can be more accurately determined. This information can be used to optimize design with the intent of improving system performance for failure mechanisms such as yielding, fatigue, stability or excessive deformation. However, it is not always convenient or possible to measure dynamic inputs to a system. For example, direct measurement of the input may be impractical when the excitation has a complex spatial distribution (e.g. wave loading on a ship hull) or when the structure is very large (e.g. suspension bridge subject to traffic loading). In these instances a method for estimating the inputs becomes useful. The estimation of inputs to a system constitutes an inverse problem, which has been studied for a wide variety of applications in structural dynamics (e.g. experimental modal analysis and finite element model updating). 1.2 Background Numerous techniques based in the frequency or time domain have been used experimentally for input estimation; these include deconvolution methods, Kalman filters, dynamic programming, and gradient based methods. For the most part, research has focused on linear, time-invariant systems, involved only numerical simulations, and ignored the problem of identifying spatially-varying loads. In 1992, Carne et al. [1] developed the Sum of Weighted Averages Technique (SWAT), and applied the technique to experimentally identify the loading timehistory on the nose cone of a weapon system. Later, Carne et al. [2] and Mayes [3] demonstrated the success of two methods of identifying the weighting matrix used with SWAT. In [4], an extended Kalman filter and recursive least-squares estimator were applied to a non-linear, spring-mass-damper system to reconstruct a series of various shape impulses. In simulation, the method performed admirably for a three degree-of-freedom (DOF) system. Nordstrom [5] developed a variation on the Kalman filter which was implemented in simulation on a timevariant system, and on a bridge structure with a moving input, both with excellent results. Each of the previously mentioned techniques comes with its own set of limitations. Deconvolution methods involve an inversion of the frequency response function, which in itself is inherently unstable. The SWAT method only identifies the force applied to an object’s center of mass and, therefore, cannot determine the location of inputs. Kalman filtering requires some knowledge of the expected noise in signals. Additionally, as Kalman filtering is run online, it only uses information from the previous state. In situations where the entire data history is known, a better estimate could be made at each instant combining past and future data. For this study, input estimation
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will be performed using an adjoint-based optimization method. Previous applications for this technique include model predictive control and weather forecasting. The adjoint-based method developed herein has none of the above limitations but can be highly computationally and memory intensive. 1.3 Purpose Because many structural characteristics can be determined from understanding the loading applied to a system, improved input estimations result in a better definition of the system as a whole. Previous input estimation research has shown great success in numerical simulation, but few studies have implemented the techniques on physical structures. In this work, inputs estimated by an adjoint-based optimization method are compared to those measured to evaluate its performance. This is done by implementing the method on two structures; a three-story frame structure and a cantilever beam. 1.4 Outline This report contains an overview of the adjoint-based optimization method for input estimation in section two. A more rigorous derivation of the algorithm can be found in Appendix A. Next, the physical structures and numerical models are presented along with results of time series simulations to validate the models for this application. Section 4 presents results for estimating the input to both structures, followed by a discussion of the success of the technique along with its limitations and difficulties.
2. ADJOINT-BASED OPTIMIZATION To begin the adjoint-based optimization, a simple cost function (based on the error between the predicted outputs and the measured outputs) is constructed; the aim is to find an input which minimizes it. To accomplish this minimization, a fairly straight forward gradient-descent process is followed. As briefly shown in Appendix A, the gradient of the cost function with respect to the input can be calculated with two simulations, regardless of the length of the input or complexity of the structure. The adjoint-based optimization method proves to have a lower computational cost when compared to a finite difference approach for estimating the gradient as only two simulations are required to generate an estimated input. Implementation of the adjoint-based optimization method involves a few preliminary steps and a while loop to perform the iterations on the input guess. First, a model is created in state space and some guess must be made for the input. The required number of iterations is largely dependent on the accuracy of the initial input guess. For the while loop, some criterion for stopping the iterations needs to be calculated along with a threshold setting. Since the true input to the tested structures in laboratory testing can be measured directly, a metric calculated with the true input vector and the current estimate can be used as a stopping criterion. In an application where input estimation is required due to the difficulty of measuring inputs directly, a stop criterion based on the difference
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between two successive input estimates could be used. The adjoint-based optimization while loop itself contains three main steps. First, the current input estimate is used to simulate the states of the system using a numerical integration technique. Next, a test function is solved by reverse time numerical integration to calculate the gradient of the cost function with respect to the current input estimate. Finally, the gradient is used to update the input estimate using any standard gradient descent based optimization method. A line search can be used to calculate the step size for input updating. Figure 1 diagrams the adjoint-based optimization method for input estimation flowing from left to right.
Figure 1. Flow diagram for adjoint-based optimization input estimation
3. STRUCTURES AND MODELING The adjoint-based method was implemented on two structures: a three-story structure and a cantilever beam. As shown in Figure 2, the three-story structure consists of four aluminum columns (17.7 × 2.5 × 0.6 cm) which are connected to the top and bottom of each aluminum plate (30.5 × 30.5 × 2.5 cm) creating a four DOF system. Accelerometers were attached at the center line of each floor on the opposite side to the excitation source to measure the system’s response. Input to the system was applied by a shaker at the base floor.
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Accelerometers
Shakker
Figure 2. Three-story T sttructure T verify the accuracy To a and practicality of o the propose ed modeling approach, a num merical simulations were conducted c o both test structures on s and d compared to o experimenttal data. A fou ur DOF lumpe ed-mass mod del was consttructed for the three-storry structure, as a shown in Figure 3. Mo odal Damping g was added for modes 2, 2 3 and 4, as a well as a additional mass, stiffness and a damping to account fo or the effectss caused by the t shaker an nd rails attach hed to the b base. The ressponse of the e accelerome eter was mod deled with a simple s low pass filter. Using a downhiill simplex a algorithm certtain paramete ers were opttimized, including the basse’s stiffness, mass and damping; d the e damping r ratios for mod des 2, 3 and d 4; the accelerometer filte er bandwidth h; and the co olumn stiffnesss. The meassured and s simulated acccelerations du ue to a chirp in nput (25 to 75 5 Hz) are shown in Figure 4.
Figu ure 3. Four DOF D lumped mass modell for three-sttory structure
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Figure 4. Measured M an nd simulated acceleration a d to a chirp due p input (25 to 75 7 Hz) F Figure 5 show ws the cantile ever beam se et up; a steel L-bracket (10.2 × 10.2 × 0.95 cm) is bolted to an aluminum b base (30.5 × 61 6 × 2.5 cm). An aluminum m beam (45.7 7 × 5.0 × 3.2 cm) c is fastene ed to the L-bracket by four bolts and a aluminum block (7.6 × 5 × 1.25 cm). Response off the structure an e was measurred at the free e end of the beam b with a accelerometer and a CC an CD laser disp placement sen nsor. A shake er was attache ed 5.25 cm fro om the fixed end e of the b beam to provide the input.
Shakker
La aser Sensor
F Figure 5. Can ntilever beam m structure
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F the cantilever beam a finite eleme For ent model wa as constructed d using three e beam elements. Addition nal mass, s stiffness, and damping we ere added to the node co orresponding to the input location to model m the effe ect of the s stinger and shaker. The shaker’s masss, stiffness an nd damping were w again optimized o with h the downhiill simplex a algorithm. Figure 6 shows the simulated d tip displacem ment alongsid de the measu urement from the laser disp placement s sensor for a chirp c input from m 15 to 25 Hzz (across the first bending mode).
Figure 6. 6 Simulated and a measured d tip displacem ment
R RESULTS A Adjoint-based d optimization was used to o estimate various input siignals applied d to the cantilever beam and a threes story structure e. The optim mization routin ne was first implemented using simula ated response data and performed p e extremely welll. The routine e was then im mplemented using measure ed response data d from the e structures, which w also w worked succe essfully. Finally, for the thrree-story stru ucture, the ad djoint-based method m was implemented using an u unknown inpu ut location as s well as fewe er accelerome eter readingss. In this secttion estimated d inputs are compared c g graphically to the actual inp puts as measured by a forcce transducer. 3 Input Esttimation for the 3.1 t Cantileve er Beam T The adjoint-b based optimizzation was first f implemented using simulated s ressponse data.. Figure 7 shows the p performance o the optimizzation routine for a combina of ation signal comprised c of sine s and trian ngle waves. The adjoint r routine is able e to recreate the t input sign nal perfectly. This T simulatio on comprised d of 5000 data a points and took t 2476 itterations to co onverge.
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Figure 7. Siimulated inpu ut estimation of o combinatio on signal N Next, the adjo oint-based op ptimization method m was im mplemented using measu ured response e data. The cantilever b beam was exccited with varrious signals each e measurred with a forcce transducerr. Figure 8 shows the time history of b both the meassured input (b black) and esstimated inputt (red) for a 15-25 Hz chirp input signal. The input signal s was c comprised of 4096 data po oints and too ok 85 iteration ns to converg ge to the spe ecified toleran nce. The adjo oint-based o optimization m method is able e to recreate the input to a reasonable degree of acccuracy. The magnitude m and phasing o the estimatte matches th of he measurem ment well. The e differences can be attrib buted to noisse in the mea asurement d data, unmode eled interactio ons between the t transduce er, stinger, and d shaker, or the t fact that th he beam mod del cannot r recreate high frequency res sponse with only o 6 degree es-of-freedom.
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Figure 8. 8 Input estimation of 15-25 5 Hz Chirp Siignal F Figure 9 show ws the perform mance of the adjoint metho od for an imp pulse chain exxcitation to the e system. The impulse c chain signal was compos sed of two square s wave sections at the beginnin ng and midd dle of the sig gnal. The m measurement t shown from m the force tra ansducer sho ows the shakker’s attempt to respond instantaneously to the v voltage steps in the signal. The adjoint--based metho od was able to recreate the e basic shape e and magnittude of an im mpulse chain n input. Again n, some highe er frequency signal compo onents are evvident in the measuremen m ts that do n appear in the estimate not e. The impulse e chain had 4096 4 data po oints and convverged in 185 5 iterations, more m than tw wice as manyy as the chirp signal.
mation of impu ulse chain sig gnal Figure 9. Input estim
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3 Input Esttimation for the 3.2 t Three-Sto ory Structure e A with the cantilever As c bea am the adjoin nt-based optiimization wass first implem mented on the e three-story structure u using simulate ed response data. Figure 10 shows the e performancce of the adjoint method fo or a combinattion signal im mplemented on simulated d response da ata. This com mbination of sine and triangle wave sign nals was com mprised of 5 5000 data po oints. The rou utine ran thro ough 490 itera ations to reach the predicction which matched m the measured m in nput perfectlyy.
Figure 10. Simulated S inpu ut estimation of combinatio on signal T adjoint-ba The ased method was next imp plemented ussing measured d response to o various inpu ut signals. Fig gures 11a, 1 11b, and 11c compare the time historie es of the estim mated signal and the mea asured input signal s for a 45 Hz sine w wave input. Enlarged E sec ctions of the datasets are e provided to o show the phasing p of th he input estim mate and m measurement t. Figure 11c shows the force f envelop pes of both the t measured d and estima ated inputs. The T force e envelope allo ows for directt comparison n between the magnitude e of the inputt estimate an nd measurem ment. The e envelopes we ere computed using a Hilbe ert transform. T sine wave input shown in Figures 11a, 11b and 11c was com The mprised of 16 6384 data points. The adjo oint-based r routine ran 70 05 iterations to produce the t above esstimation. Tho ough the mag gnitude of the e estimate iss 25% too la arge for mostt of the signal, the phasing was recreate ed correctly.
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5
Figure e 11a. Input esstimation of 45 4 Hz sine wa ave
Figure 11b. Input estimattion of 45 Hz sine wave (enlarged)
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Figure 11c. Input estimattion of 45 Hz sine wave (en nvelope) N Next, Figures 12a, 12b, 12 2c and 12d co ompare input estimates to experimental measuremen nts for a 40-8 80Hz chirp in nput signal. The T chirp sign nal was comprised of 1638 84 data pointss. The adjoint--based routine took 490 ite erations to c converge on this t particularr estimation. Figures F 12a and a 12b show w that the estim mation match hes the measu ured input to o a good de egree for most of the tim me history. Figure 12c hig ghlights the areas a where the magnitude of the e estimation do oes not matcch the meassurement inp put as well. However, th he phasing is estimated correctly throughout the e signal.
Figure 12a. 1 Input estiimation of 40--80 Hz chirp input i
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Figure F 12b. In nput estimatio on of 40-80 Hz H chirp input (enlarged)
Fiigure 12c. Inp put estimation n of 40-80 Hzz chirp input (enlarged) (
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F Figure 12d. In nput estimatio on of 40-80 Hzz chirp input (envelope) ( T add more complexity in To nput estimatio on was attem mpted with fe ewer accelero ometers attacched to the th hree-story s structure. For this study, alll accelerome eters were rem moved excep pt one; the sysstem was the en excited by the same 4 40–80Hz chirrp signal use ed previouslyy. Figures 13 3a, 13b, and d 13c compa are the perfformance of the input e estimation rou utine with mea asured data for f a single acccelerometer placed at the e base of the structure. s
Figure 13a a. Input estim mation of 40-8 80 Hz chirp inp put with base e measuremen nt only
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Fig gure 13b. Inp put estimation of 40-80 Hz chirp input wiith base meassurement onlyy (enlarged)
Fig gure 13c. Input estimation of 40-80 Hz chirp input wiith base meassurement onlyy (enlarged) W a single acceleromete With er on the base e floor, the ad djoint routine took much longer at 7090 iterations to converge. c T The estimate ed inputs ma atch the mea asured input nearly as well w as with all four acce elerometers. Next, an a acceleromete r was placed d only on the third floor off the structure e; the structure was then excited with the same c chirp signal. Figures F 14a, 14b, and 14cc compare the e performancce of the inpu ut estimation routine with measured m d data for this accelerometerr placement.
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F Figure 14a. Input I estimation of 40-80 Hz H chirp inputt with third floo or measurem ment only
Figurre 14b. Input estimation off 40-80 Hz chiirp input with third floor me easurement only (enlarged d)
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Figurre 14c. Input estimation off 40-80 Hz chiirp input with third floor me easurement only (enlarged d) W With the acce elerometer on n the top floor, the estim mation converged after 162 270 iterationss of the adjo oint-based r routine. The input estima ation is clearrly not as acccurate as when w all fourr accelerome eters or just the base a acceleromete r is used. Figure 14c show ws problems with w the estim mated phase leading the measurement. m From the e envelopes in Figure 15, th he peak magn nitude of the estimate using only an accelerometerr on the 3rd floor is 2.5 times larger than t the true e value. In comparison c th he trial using g an accelerrometer at th he base estim mates the m magnitude to within 20% fo or all but the end e of the signal.
Figure 15. 1 Input estim mation of 40-8 80 Hz chirp in nput with singlle accelerome eter measure ements (envellope)
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4. CONCLUSION This preliminary study on implementing adjoint-based optimization techniques for use in estimating inputs to dynamic structures produced promising results. In simulation, the method is able to recreate any continuous input to any model, although for some systems many iterations of the optimization may be needed to reach a perfect estimate. For the cantilever beam structure, the adjoint-based optimization performed as well as can be expected considering the simplistic model used with only 6 degrees-of-freedom. Likewise, input estimation for the threestory structure worked well when accelerometers were placed on every floor. With limited sets of measurements, the estimate was not able to accurately reconstruct the magnitude of the input though the phasing still matched the measurement. With an improved model (particularly for energy dissipation on the three story structure), the adjoint-based optimization method would be expected to perform even better than the results shown here. The routine may also be expected to perform better for fewer response measurements if the gradient descent was changed to a global minimization technique.
REFERENCES [1] Carne T G, Bateman V I and Mayes R L. Force reconstruction using a sum of weighted accelerations technique. 1992. Sandia National Laboratories; pp.291-298. [2] Carne T G, Mayes R L and Bateman V I. Force reconstruction using a sum of weighted accelerations technique- max-flat procedure. 1994. Sandia National Laboratories. pp. 1053-1062. [3] Mayes Randall L, Measurements of lateral launch loads on re-entry vehicles using SWAT. 1994. Sandia National Laboratories, Experimental Structural Dynamics Department. pp. 1063-1068. [4] Ma C-K, Ho C-C, An inverse method for the estimation of input forces acting on non-linear structural systems. 2004. Journal of Sound and Vibration .pp. 953-971 [5] Nodstrom, L J L, A dynamic programming algorithm for input estimation on linear time-variant systems. 2006. Computer Methods in Applied Mechanics and Engineering. pp. 6407-6427
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APPENDIX A Mathematically, the details of the adjoint-optimization approach are as follows. The non-linear finite element model can be represented as
f x, u, t
x
(1)
where x is the vector of system states (usually positions and velocities of all the nodes in the mesh of the structure), u is a vector of inputs at nodes on the structure and t is time. The measured outputs, y, can generally be represented as
y
Cx
where, C is some matrix. Stated another way, the outputs are some linear combination of the states of the system. The model’s error is then defined to be
y ym
e
The goal of the optimization is to select u such that e is minimized. Or, more precisely, we want to minimize the cost function
1 T T e e dt 2 ³0
J Since
e
(2)
Cx y m , after some manipulation, this cost function can be rewritten as T
³
J
0
T
T
x T Qx 2 y m Cx y m y m dt
where Q=CTC. If the input u is perturbed by u c , the perturbed state trajectory is given by the tangent linear equation
x c where "
A t xc B t u c or "x c
Bu c
(3)
d A t and A(t) and B(t) are obtained by linearizing Equation 1 about x and u. The resulting dt
perturbation to J is given by
Jc
T
³
0
x T Qx c y m Cx cdt T
³ x T
0
T
Q y m C xcdt T
(4)
The goal of what follows is simply to re-express J’ as a functional linear in u’. To that end, we integrate Equation 3 against a test function, r.
³
T
0
T
³
r T "xcdt
0
r T x c A t x dt
Using integration by parts, we can rewrite the above equation as T
³
0
r T "x cdt
T
³
0
(" * r ) T x cdt r T x c
t T t 0
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where "
*
d T A t . This is true for any test function, r. If we select r such that dt
"*r
Qx C T y m
(5)
r T 0
where T is the final time, then Equation 4 can be rewritten as
Jc
³ x Q y T
0
T
T
m
C x' dt
³ " r T
T
*
0
x' dt
³
T
0
r T "x'dt
T
³
0
r T Bu cdt
Equation 5 is referred to as the adjoint equation. Thus, we have expressed J’ as a functional linear in u’. The gradient of which with respect to u is then simply
DJ Du
rT B
Therefore, givens some initial guess at u, Equation 1 is solved for x. This x is then used in conjunction with the measured data, ym, to solve Equation 5 in reverse-time since r(T) is known. From r, the gradient of the cost function with respect to the input may be calculated, and used to update u (using any number of standard gradient descent based optimization techniques). Note that solving Equation 5 requires what is known as an adjoint version of the simulation code, which can calculate the linearized A(t) and B(t), as functions of x and u. This step allows for non-linear models to be used with the technique.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Damage Detection with Ambient Vibration Data Using Time Series Modeling Mustafa Gul 1, F. Necati Catbas 2 Department of Civil, Environmental and Construction Engineering University of Central Florida, Orlando 1 2
[email protected]
[email protected]
NOMENCLATURE -1
: Mass matrix
B(q): Polynomials in the delay operator q
: Damping matrix
C(q): Polynomials in the delay operator q
: Stiffness matrix
q : Delay operator
-1
-1
x(t): Output of the system
y(t): Output of the ARMAX model
f(t): Excitation force on the system
u(t): Input to the ARMAX model -1
A(q): Polynomials in the delay operator q
e(t): Error term in the ARMAX model
ABSTRACT In this study, a novel approach using a modified time series analysis methodology is used to detect, locate, and quantify structural changes by using ambient vibration data. Random Decrement (RD) is used to obtain pseudo free response data from the ambient vibration time histories. ARX models (Auto-Regressive models with eXogenous input) are created for different sensor clusters by using the pseudo free response of the structure. The output of each sensor in a cluster is used as an input to the ARX model to predict the output of the reference channel of that sensor cluster. After the ARX models for the healthy structure for each sensor cluster are created, the same models are used for predicting the data from the damaged structure. The difference between the fit ratios is used as damage indicating feature. The methodology is applied to experimental data obtained from steel grid structure tests and it is shown that the approach is successfully used for identification, localization, and quantification of different damage cases for both impact and ambient tests. INTRODUCTION Structural Health Monitoring (SHM) is the research area focusing on condition assessment of different types of structures including aerospace, mechanical and civil structures. Damage detection is arguably one of the most critical components of SHM. A common approach to extract damage sensitive features (or damage features) from SHM data is to use time series analysis [1-11]. AR (Auto Regressive), ARX (Auto Regressive model with eXogenous input) and ARMA (Auto-Regressive Moving Average) models are some of the time series analysis methods employed by different researchers. One of the main advantages of such methodologies is that it requires only the data from the undamaged structure in the training phase (i.e. unsupervised learning) as opposed to supervised learning where data from both undamaged and damaged conditions is required to train the model. These types of methods have been widely studied also since their implementation for an automated SHM system is relatively more feasible compared to other methodologies such as damage detection based on model updating.
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_61, © The Society for Experimental Mechanics, Inc. 2011
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710 In a recent study, the authors proposed a time series methodology where ARX models were used to model the free response of different sensor clusters for damage detection [12]. The authors showed that the method was very successful in identifying, locating and quantifying the damage with different numerical and experimental data [13, 14]. In this paper, the methodology is modified and improved for damage detection with ambient vibration data. The details of the methodology, theoretical background and experimental results are given in the following sections. Objective and Scope The objective of this study is to present a time series analysis methodology to identify, locate and quantify damage by using ambient vibration data. First, the theoretical background of time series modeling and its relation with structural dynamics is briefly discussed. Then the methodology is applied to different experimental datasets from a large-scale laboratory model for damage detection. Results from impact and ambient tests are presented in a comparative fashion to show the effectiveness of the approach. THEORETICAL BACKGROUND The equation of motion for an N Degrees of Freedom (DOF) linear dynamic system can be written as in Eqn. (1). (1) where The vectors
is the stiffness matrix. is the damping matrix and is the mass matrix, are acceleration, velocity and displacement, respectively. The external forcing and
. The same equation can be written in matrix form as shown in Eqn. function on the system is denoted with (2) (t for time is omitted). Furthermore, if the first row of this matrix equation is considered for free response case, the relation is written as shown by Eqn. (3).
(2)
(3)
It is seen from Eqn. (3) that if a model is created to predict the output of the first DOF by using the DOFs connected to it (neighbor DOFs), the change in this model can reveal important information about the change in the properties of that part of the system. Obviously, similar equalities can be written for each row of Eqn. (2) and different models can be created for each equation. Each row of Eqn. (2) can be considered as a sensor cluster with a reference DOF and its neighbor DOFs. The reference DOF for Eqn. (3), for example, is the first DOF and neighbor DOFs are the DOFs that are directly connected to the first DOF. Therefore, it is proposed that different linear time series models can be created to establish different models for each sensor cluster and changes in these models can point the existence, location and severity of the damage. The details of the methodology are explained in the following sections. A general form of a time series model can be written as in Eqn. (4)-(5) and an ARX model is shown in Eqn. (6). (4)
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(5)
(6) where, y(t) is the output of the model, u(t) is the input to the model and e(t) is the error term. The unknown parameters of the model are shown with ai, bi, and di and the model orders are shown with na, nb and nd. As indicated above, Eqn. (3) can be used in the case of free response analysis. Although the free response of a structure can be easily measured for some applications, in most of the civil engineering applications it may not always be practical or easy to collect the free response data. Therefore, the authors implemented the Random Decrement (RD) Technique to obtain the pseudo free response data from the ambient vibration data [15-17]. The basic idea behind the RD is that the random response of a system at the time components, which are the step response due to the initial displacements at the time from initial velocity at time
consists of three , the impulse response
, and a random part due to the load applied to the structure between
and
.
By taking average of time segments, every time the response has an initial displacement bigger than the trigger level given in Eqn. (7), the random part due to random load will eventually die out and become negligible. Additionally, since the sign of the initial velocity can be assumed to vary randomly in time, the resulting initial velocity will be zero. (7) After the pseudo-free response data is obtained by using RD, this data is used to create different ARX models for different sensor clusters. Then damage sensitive features are extracted from these models to detect the damage. In these ARX models, the y(t) term in Eqn. (6) is the acceleration response of the reference channel of a sensor cluster, while the u(t) term is defined with the acceleration responses of all the DOFs in the same cluster. Eqn. (8) st shows an example ARX model to estimate the 1 DOF’s output by using the other DOFs’ outputs for a sensor cluster with k sensors. (8) To explain the methodology schematically, a simple 3-DOF model is used as an example. Figure 1 shows the first sensor cluster for the first reference channel. The cluster includes first and second DOFs since the reference channel is connected only to the second DOF. The input vector u of the ARX model contains the acceleration outputs of first and second DOFs. The output of the first DOF is used as the output of the ARX model as shown in the figure. When the second channel is the reference channel, Figure 2, the sensor cluster includes all three DOFs since they are all connected to the second DOF. Sensor clusters for each reference channels are created with the same concept. After creating the ARX models for the baseline condition, the fit ratios of the baseline ARX model when used with new data is employed as a damage sensitive feature. Further details about the methodology and applications are discussed in Gul and Catbas [18].
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Figure 1. Creating different ARX models for each sensor cluster (first sensor cluster)
Figure 2. Creating different ARX models for each sensor cluster (second sensor cluster) After creating the ARX models for the healthy structure, the same models are used to predict the output of the damaged structure for the same sensor clusters. The difference between the fit ratios of the models is used as the DF. The Fit Ratio (FR) of an ARX model is calculated as given in Eqn. (9).
(9)
where
is the measured output,
is the predicted output,
is the mean of
and
is the norm of
. The DF is calculated by using the difference between the FRs for healthy and damaged cases. EXPERIMENTAL STUDIES AND ANALYSIS RESULTS A steel grid structure was used for the experimental demonstration of the methodology. The physical model has two clear spans with continuous beams across the middle supports. It has two 18 ft girders (S3x5.7 steel section) in the longitudinal direction. The 3 ft transverse beam members are used for lateral stability. The grid is supported by 42 in tall columns (W12x26 steel section). The grid is shown in Figure 3 and more information about the specimen can be found in Burkett [19] and Catbas et al. [20]. The grid is instrumented with 12 accelerometers in vertical direction at each node (all the nodes except N7 to.N14). The accelerometers used for the experiments are ICP/seismic type accelerometers with a 1000 mV/g sensitivity, 0.01 to 1200 Hz frequency range and g of measurement range.
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Figure 3. The steel grid model used for the experimental studies and node numbers Damage Simulations Two different damage cases are simulated to show the effectiveness of the method. The first damage case is a local stiffness loss simulation. In the second damage case, a more severe damage is simulated by fixing the boundaries. Baseline Case (BC0): Before applying any damage, the structure is tested to generate the baseline data so that the data coming from the unknown state can also be compared to the baseline data for damage detection. Damage Case 1 (DC1) - Moment release and plate removal at N3: DC1 simulates a local stiffness loss at N3. The bottom and top gusset plates at node N3 are removed in addition to all bolts at the connection (Figure 4). This is an important damage case since gusset plates are very critical parts of the steel structures. Damage Case 2 (DC2) - Boundary restraint at N7 and N14: The second damage cases is a boundary condition change at two corners. This damage case is created to simulate some unintended rigidity at a support caused by different reasons such as corrosion. The oversized through-bolts were used at N7 and N14 to introduce fixity at these two supports (Figure 4). Although these bolts can create considerable fixity at the supports, it should be noted a perfect fixity cannot be guaranteed.
Figure 4. Damage simulations: plate removal at N3 for DC1 (left), boundary fixity at N7 and N14 for DC2 (right) Analysis Results In the following sections, the analysis results of impact and ambient data are presented in a comparative fashion. For the free response analysis, the free responses that are obtained from the impact tests are used. Five impact data sets were used for each case. In the second part, the ambient vibration analysis results are presented. The ambient vibration data is first processed with RD technique to obtain the five different pseudo free response data sets and then the ARX methodology is applied to these pseudo free response data.
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714 Damage Case 1 (DC1): Figure 5 shows that the DFs for DC1 obtained using impact tests whereas Figure 6 shows the results obtained using the ambient vibration data. As seen from both of the figures, the DFs for N3 are considerably higher than the threshold (see Gul and Catbas [18] for details about calculating the threshold value) and other nodes. This fact is due to the plate removal at this node. It is also observed from the figure that the DFs for N2 are also relatively high since N2 is the closest neighbor of N3. Finally the secondary effect of the damage on N5 and N10 is also seen. The comparison of the impact and ambient data results shows that the methodology is successfully implemented for ambient vibration cases. Therefore, it can be concluded that the methodology is successful at detecting and locating the damage for this experimental damage case for both impact and ambient data. Finally note that the DFs for other nodes are around the threshold showing us that these nodes are not affected significantly from the localized damage.
Figure 5. DFs for DC1 using free response (from impact tests) data
Figure 6. DFs for DC1 using ambient vibration data
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715 Damage Case 2 (DC2): The results for the second damage case, DC2, are presented in Figure 7 in Figure 8. The figures again show that the impact and ambient data analysis results are very similar and both are successful in detecting the damage. It is seen from the figures that the DFs for N6 and N13 are considerably higher than the other nodes. This is because of the fact that they are the closest nodes to the restrained supports (note that N7 and N14 are not instrumented). The DFs for N5 and N12 are also high since they are also affected by the damage. The DFs for the remaining nodes are also slightly higher than the threshold since the structure is changed globally for DC2. Note that the DFs for N6 and N13 are around 40 which show that the damage is more severe than DC1. This information is also very consistent with the severity of the applied damage. Finally, note that the variation of the DFs for ambient vibration data is more than the impact case which is an expected situation due to the random and ambient effects in the data.
Figure 7. DFs for DC2 using free response (from impact tests) data
Figure 8. DFs for DC2 using ambient vibration data
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716 CONCLUDING REMARKS In this study, a time series analysis methodology is presented for damage detection using ambient vibration data. First the pseudo free response data is obtained from the ambient time histories with Random Decrement. Then ARX models are created for each sensor cluster by using the pseudo response data. Once the ARX models for baseline and damaged cases are created, the fit ratios are used as damage indicating features. The methodology is applied to the experimental data of a steel grid structure where different damage cases are simulated experimentally. Results obtained from impact and ambient vibration tests are presented. It is demonstrated the damage can be identified, located and quantified by using this approach even for a relatively complex laboratory structure using both impact and ambient vibration data. ACKNOWLEDGEMENTS The research project described in this paper is supported by Federal Highway Administration (FHWA) Cooperative Agreement Award DTFH61-07-H-00040. The support of both agencies and their engineers is greatly recognized and appreciated. The authors would like to express their profound gratitude to Dr. Hamid Ghasemi of FHWA for his support of the advanced exploratory research program. The opinions, findings, and conclusions expressed in this publication are those of the authors and to not necessarily reflect the views of the sponsoring organizations. REFERENCES 1.
Sohn, H., J.A. Czarnecki, and C.R. Farrar, Structural Health Monitoring Using Statistical Process Control. Journal of Structural Engineering, ASCE, 2000. 126(11): p. pp. 1356-1363.
2.
Sohn, H., et al., Structural Health Monitoring Using Statistical Pattern Recognition Techniques. Journal of Dynamic Systems, Measurement, and Control, ASME, 2001. 123: p. 706-711.
3.
Omenzetter, P. and J.M. Brownjohn, Application of Time Series Analysis for Bridge Monitoring. Smart Material and Structures, 2006. 15: p. pp. 129-138.
4.
Nair, K.K., A.S. Kiremidjian, and H.L. Kincho, Time Series-Based Damage Detection and Localization Algorithm with Application to the ASCE Benchmark Structure. Journal of Sound and Vibration, 2006. 291(1-2): p. pp. 349-368.
5.
Zhang, Q.W., Statistical Damage Identification for Bridges Using Ambient Vibration Data. Computers and Structures, 2007. 85(7-8): p. pp. 476-485
6.
Carden, E.P. and J.M. Brownjohn, ARMA Modelled Time-Series Classification for Structural Health Monitoring of Civil Infrastructure. Mechanical Systems and Signal Processing, 2008. 22(2): p. pp. 295314.
7.
Chang, P.C., A. Flatau, and S.C. Liu, Review Paper: Health Monitoring of Civil Infrastructure. Structural Health Monitoring, 2003. 2(3): p. pp. 257-267
8.
Zheng, H. and A. Mita, Two-stage Damage Diagnosis Based on the Distance between ARMA Models and Pre-whitening Filters. Smart Material and Structures, 2007. 16: p. pp. 1829-1836.
9.
Gul, M. and F.N. Catbas, Statistical Pattern Recognition for Structural Health Monitoring using Time Series Modeling: Theory and Experimental Verifications. Mechanical Systems and Signal Processing, 2009. 23(7): p. pp. 2192-2204.
10.
Lu, Y. and F. Gao, A Novel Time-domain Auto-regressive Model for Structural Damage Diagnosis. Journal of Sound and Vibration, 2005. 283: p. 1031-1049.
11.
Monroig, E. and Y. Fujino. Damage Identification Based on a Local Physical Model for Small Clusters of Wireless Sensors. in 1st Asia-Pacific Workshop on Structural Health Monitoring. 2006. Yokohama, Japan.
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Gul, M. and F.N. Catbas. A New Methodology for Identification, Localization and Quantification of Damage by Using Time Series Modeling. in 26th International Modal Analysis Conference (IMAC XXVI). 2008. Orlando, FL.
13.
Gul, M. and F.N. Catbas, A Novel Time Series Analysis Methodology for Damage Assessment, in The Fifth International Workshop on Advanced Smart Materials and Smart Structures Technology, ANCRISST 2009. 2009: Boston, MA.
14.
Gul, M. and F.N. Catbas. A Modified Time Series Analysis for Identification, Localization, and Quantification of Damage. in 27th International Modal Analysis Conference (IMAC XXVII). 2009. Orlando, FL.
15.
Cole, H.A., On-The-Line Analysis of Random Vibrations. American Institute of Aeronautics and Astronautics, 1968. 68(288).
16.
Asmussen, J.C., Modal Analysis Based on the Random Decrement Technique-Application to Civil Engineering Structures. 1997, University of Aalborg: Aalborg.
17.
Gul, M. and F.N. Catbas, Ambient Vibration Data Analysis for Structural Identification and Global Condition Assessment. Journal of Engineering Mechanics, 2008. 134(8): p. pp. 650-662.
18.
Gul, M. and F.N. Catbas, Structural Health Monitoring and Damage Assessment using a Novel Time Series Analysis Methodology with Sensor Clustering. Journal of Sound and Vibration, 2009. (Under Review).
19.
Burkett, J.L., Benchmark Studies for Structural Health Monitoring using Analytical and Experimental Models, in Department of Civil and Environmental Engineering. 2005, University of Central Florida: Orlando, FL.
20.
Catbas, F.N., M. Gul, and J.L. Burkett, Damage Assessment Using Flexibility and Flexibility-Based Curvature for Structural Health Monitoring. Smart Materials and Structures, 2008. 17(1): p. 015024 (12 pp).
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BookID 214574_ChapID 62_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Wind Load Observer for a 5MW Wind Energy Plant
Maksim Klinkov and Claus-Peter Fritzen University of Siegen, Institute of Mechanics and Control Engineering-Mechatronics, PaulBonatz-Strasse 9-11, 57076 Siegen, Germany Phone: +49(0)271/740-4633, fax: +49 (0)271/740-2769, [email protected] Phone: +49(0)271/740-4621, fax: +49 (0)271/740-2769, [email protected]
Nomenclature A
State transition matrix
B
Input matrix
C
Output matrix
ω(t)
Observer state vector
D
Direct feed through matrix
ˆ ξ(t)
Estimated states
x(t)
State vector
FBetz
Wind force according to Betz theory
u(t)
Input vector
c
Aerodynamic resistance
y(t)
Output vector
ρ
Air density
f ( .)
Nonlinear function field
Vwind
Wind velocity
N
Observer state matrix
Arotor
Rotor area of the wind turbine
L
Observer correction matrix
T
Observer correction matrix
f L ( ξˆ , y )
Nonlinear Lipchitz function
Q
Observer correction matrix
ABSTRACT For off-shore wind energy plants (OWEP) inspection, maintenance and repair is much more difficult and more expensive compared to on-shore plants. Therefore, SHM has large potential in this field of application. Wind load monitoring system is especially important as an SHM unit for forecasting of the remaining life-time of the structure. The knowledge of these loads enables to make a better design of the structure and an assessment of damage after extreme events. In the case of WEP it is not possible to measure the forces e.g. resulting from wind or wave loads directly. Therefore, these forces are determined indirectly from dynamic measurements. The indirect reconstruction addresses the inverse problem in mathematical sense and all its consequences, such as ill-posedness. These are solved with an observer-based concept that originally comes from the control engineering area. For the purpose of design, test and prototype implementation of the SHM system for the OWEP project was granted by the German Ministry of Economics together with several research groups and companies. In this contribution an estimation of the wind load for the real 5MW onshore wind energy plant (WEP) in Bremerhaven, Germany is presented which was done within a national project. Wind load reconstruction is carried out online and implemented as a unit into the prototype SHM system.
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INTRODUCTION Many methods for estimation of the external forces were already developed, see e.g. [1] or [2], a comprehensive overview can be found in [3], as well as a discussion of their strong and weak sides [4]. The force reconstruction leads to an ill-posed inverse problem [5]. In addition one needs to have a model (analytical, numerical FEM, identified) and the dynamic responses of the structure (acceleration, strain, velocity) which are contaminated by noise. Typically all known approaches try to overcome the ill-posedness by converting the ill-posed problem into a well-posed one, using different regularization techniques either in frequency or in time domain, see e.g.[1, 6]. If this inversion can be done, the system itself becomes its own “virtual force sensor” which would solve the problem of the load history estimation. A state and input observer is a time domain approach, which originally was invented for control engineering purposes [7]. It allows simultaneous reconstruction of the inputs and states (velocities, positions) of a linear or nonlinear time invariant system. Additionally it is possible to use the results of the observer for the estimation of other quantities like strain or stress at any location of interest where no measurement equipment is applied. A schematic representation of the observer usage for WEP is shown in Figure 1.
Figure 1. Online states (positions, velocities) and inputs (external forces, moments) estimation scheme. The principle is based on the construction of the observer for a general first order non-linear state space system:
x (t) = Ax(t) + Bu(t) + f (( x(t),u(t)), y(t)) y(t) = Cx(t) + Du(t)
(1)
where x(t), u(t) and y(t) are the states, unknown inputs and measured outputs, respectively. Matrices A, B, C and D of the state space model are real, constant and of appropriate dimensions. f(.)=fL +fU is a real nonlinear vector function split into a smooth function fL which satisfies the Lipschitz condition and an unknown nonlinearity fU. In the next step the state space model in Eq.1 is converted into more compact form by combining of the states together with the inputs as follows:
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⎡ x(t)⎤ ξ (t) = ⎢ ⎥, ⎣ u(t)⎦
Eξ(t) = Mξ (t) + f ( ξ (t), y(t)) y(t) = Hξ (t)
(2)
where, the matrices E, M, H constructed in such way that the equality holds. The state space model is obtained from the finite element model of the WEP which is than transformed from nodal coordinates into the modal ones. The modal model considers only the limited number of modes that can be hypothetically exited in case of WEP up to 50Hz. This transformation significantly reduces the complexity of the considered load estimation problem especially calculation of the observer [7] which has following form:
ω (t) = Nω(t) + Ly(t) + T f L ( ξˆ (t), y(t)) ξˆ (t) = ω(t) + Qy(t)
⎡ x(t)⎤ ⎥ ⎣ u(t) ⎦
ξ (t) = ⎢
(3)
An appropriate set of observer matrices N, L, T and Q has to be found so that the error between the estimated and the real inputs and states will converge to zero as the time evolves. The states of the observer include both: the original states of the initial model as well as the required unknown loads. It is possible to find these matrices with the help of a linear matrix inequality (LMI) technique [8]. There are still two assumptions in this method that should be satisfied: the D matrix must have full column rank (acceleration sensors should be located at the position of the force whereas this can be released in case of modal models); the number of sensors has to be bigger than or equal to the number of unknown inputs plus the number of nonlinear terms. Detailed step by step mathematical derivation can be found in [9]. Once the observer matrices are calculated, the inputs together with the modal states can be reconstructed simultaneously online. The ability of online reconstruction and inherited noise robustness make the observer to be very attractive for external load estimation. LOAD RECONSTRUCTION ON THE WIND ENERGY PLANT M5000-2 Within the IMO-Wind project 167 sensors were installed on the M5000-2 onshore WEP in Bremerhaven. The M5000-2 is a 5 MW wind turbine with a tripod tower structure as shown in Figure 2 (left). Equipment installation and data collection has been done by the Federal Institute for Material Research and Testing (BAM VII.2). External load reconstruction was performed offline as a sub-part of the overall SHM package for future offshore WEPs.
Figure 2. Multibrid 5MW wind energy plant in Bremerhaven, Germany (left). Load estimation scheme M5000-2 (right)
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The scheme of the data acquisition and its processing is similar to the laboratory experiment as in [10] and shown in Figure 2 (right), whereas here the data is collected and sent via internet every hour to our lab at the University of Siegen where it goes through the SHM and load reconstruction packages automatically. Finally, the evaluated data flows back to BAM VII.2 server. For the observer design an updated FE model has been used and two sensor types (two 2-D accelerometers and two strain gauges) were needed. Load estimation was then done for the section where the nacelle is connected to the tower. This section is chosen intentionally and includes all possible forces that the tower should subtend (external wind forces; static nacelle orientation, blades dynamics; mass unbalance; etc.). In Figure 3 one of the working states of the WEP M5000-2 together with operational conditions (wind velocity, pitch angle, etc.) is illustrated. On the upper right subplot of Figure 3 the WEP is shown from the top, where the black thick lines represent the nacelle with the rotor. Strain and acceleration sensors are fixed to the tower and have their own orientation in space which is displayed on the same subplot respectively. The nacelle is always changing its orientation in space this in its turn produces difficulties for the calculation of the observer because it is based on the modal state space model which has fixed orientation. Therefore measurement transformation was performed to overcome this problem.
Figure 3. Operational state M5000-2: nacelle orientation with respect to the wind together with strain gauges coordinates (top left and right); wind velocity and pitch of the rotor blades (bottom left and right) The above operational conditions are very close to the ideal ones for the WEP M5000-2 see data sheet [11]: a) wind comes almost orthogonal to the rotor area, with a nearly constant wind speed of approximately 12 m/s; b) the blades have a relatively small pitch angle. They allow using the Betz theory for validation of the estimated load. According to Betz, the wind load which is applied to the rotor area can be calculated as: 1 2 A FBetz = cρVwind rotor 2
(5)
where, c is an aerodynamic resistance coefficient, Vwind is the wind speed, ρ is the air density and A is a rotor area accordingly. The wind is assumed to blow orthogonal to the rotor area. The estimated and Betz forces are depicted in Figure 4 for a time duration of 25 minutes and the above mentioned operational conditions.
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Figure 4. Estimated external force for M5000-2
Figure 4 shows a certain difference between estimated and Betz force especially in the region where the wind velocity is increasing. This is due to Betz theory; see Eq. 3, where the Betz force follows the wind velocity behavior for a fixed coefficient c without consideration of pitch angles (see for comparison the wind velocity in Figure 3 in the time interval 0-1500 seconds, wind velocity increases as a time evolves). As a result the Betz force cannot be used solely for the validation, instead one need also to take into consideration the pitch angle of the rotor blades. A closer look on Figure 4 shows that the estimated force is very close to the Betz force in the regions where the pitch is close to zero (~100 and ~1300 seconds) and becomes smaller as rotor blades are pitched (see pitch and wind velocity in Figure 3 in time interval 0-1500 seconds). Magnification of Figure 4 demonstrates that the estimated force has similar slower dynamics as the Betz force but also includes higher frequencies which obviously are the components of all interaction forces between the tower, nacelle and rotor blades. Observations of the estimated load spectrum in Figure 5 left shows that the external load lies in a low frequency range up to 25 Hz. Moreover a closer look to the spectrum in Figure 5 right demonstrates that the stall-effect (blade passage) and its multiples play a very important role in the dynamic excitation of the WEP. The blade passage frequencies are plotted with orange lines on both plots. The green lines represent the first and the second natural frequencies of the tower which are also present in the estimates load spectra. On the other hand the spectrum of the wind velocity that has typical logarithmic decay is shown in Figure 6.
Figure 5. Estimated load spectrum [0 - 15] Hz range (left); [0 - 3.5] Hz range (right)
Comparison of both figures 5 and 6 lead to an important issue that the load which the tower of the WEP subtends is not consisting solely of the wind load but rather includes loads/forces that come from an interaction of other components of the WEP.
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Figure 6. Wind velocity spectrum
By means of the estimated force it is possible to calculate shear forces at any location of interest, as an example the shear force at the connection of nacelle and tower for the same operational conditions as in Figure 3 and time interval of 1500 seconds is shown in Figure 7. An additional important issue of the force estimation is its usage for the future structural design.
Figure 7. Estimated shear forces at nacelle-tower cross section for 11.8 m/s wind speed and turbulence intensity 20.76%.
At the moment the most important components of the WEP are designed with the help of commercially available software packages which allows the calculation of the wind load and considers different parameters such as: transients in wind speed, direction, various turbulence spectra. In Figure 8 both shear forces at the nacelle tower connection from the commercial software package (CSP) are plotted together with calculated Betz force for 12.2 m/s wind speed and 19.79% turbulence intensity. By comparing Figures 7 and 8 one can notice that the shear forces from CSP are much higher for almost the same operational conditions, and do not include all interaction forces of the WEP. The main disagreement between estimated and modeled forces is noticeable in the Y directional shear force; this is due to the different angle of attack of the wind itself, see Figure 3 (right top) and the nacelle orientations. In the CSP the nacelle is positioned into the main wind direction which is 240° which also means that the shear force in the X-direction reaches its maximum (see Figure 3 top left), whereas the in case of force estimation the nacelle had only 224.5°. Modeling results might be sufficient for the present design strategy but it generally leads to structure overdimensioning, and might be improved in future by the records of the estimated dynamic forces from real on field structure. Of cause there is still a need of validation of the reconstructed forces by means of real
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measured shear forces for its further reliable usage. This will be done in the near future by the BAM VII.2, which is aiming to use strain gauges for that.
Figure 8. Shear forces at nacelle-tower cross section by GH Bladed for 12.2 m/s wind speed and turbulence intensity of 19.79%.
CONCLUSION The proposed observer produces online reconstruction of the time-history of the external loads. These results were obtained on the basis of a beam element FEM. The observer estimates realistic forces with respect to the Betz loads. Further investigations of the observer properties should be made together with validations of the obtained result that will be accomplished as a next step within the IMO-Wind project in cooperation with the partners Offshore Wind Technology GmbH (OWT) and BAM VII.2. Additionally, external moments that play a central role in the life time and fatigue prediction of the tower structure will be estimated and validated in the same way as the external forces. During the project the authors made an observation that the WEP is experiencing a compete bench of forces that are arising throughout the WEP operation and need to be considered for the life time prognosis. Among them are loads from pitching, yaw control, stall and start of the rotor. The model improvement by usage of shell elements in FE modeling will also lead to the improvement of the force estimation. One of the perspectives of the observer usage can be a monitoring of the rotor blade loads that can assure an optimal pitch angle for a prolonged operation. This is extremely important in case of offshore WEP where the loss of one rotor blade due to the failure caused by unrecognized initial damage and overload operational conditions will lead to reduction of operational availability during the high wind and wave season. This concept can be also extended to all critical component of WEP. As an outcome of the IMO-Wind project the SHM prototype software package for the WEP was released, which includes: sensor functionality and reliability check, damage detection for the tower for changing environmental conditions, load monitoring with compensation of changes of environmental and operational conditions.
ACKNOWLEDMENT The authors are grateful to the German Ministry of Economics for the financial support of the IMO-WIND project by funding of BMWi (grant no. 16INO327), to the Federal Institute for Material Research and Testing (BAM VII-2) and to the company AREVA-Multibrid for providing the measurement data.
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REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
Nordström. L.J.L., Input estimation in structural dynamics, Chalmers University of Technology Sweden: Göteborg, Sweden. 2005. Doyle J.F. and Adams R., Multiple force identification for complex structures Experimental Mechanics, 42 ((1)): p. 25 -36. 2002. Stevens K., Force identification problems - an overview. Proceedings of SEM, Spring Meeting, Houston: p. 838-844. 1987. Klinkov M. and Fritzen C.-P., An Updated Comparison of the Force Reconstruction Methods. Key Engineering Materials, 347: p. 461-466. 2007. Jacquelin E., Bennani A., and Hamelin P., Force reconstruction and regularization of a deconvolution problem. Journal of Sound and Vibration, 265: p. 81-107. 2003. Steltzner. A.D. and Kammer. D.C. Input force estimation using an inverse structural filter. in Proceedings of the 17th International Modal Analysis Conference: Proc. SPIE. March 1999. Ha Q. P. and Trinh H., State and input simultaneous estimation for a class of nonlinear systems. Automatica, 40: p. 1779-1785. 2004. Boyd S., et al., Linear Matrix Inequalities in Systems and Control Theory, Philadelphia: SIAM books. 1994. Klinkov M. and Fritzen C.-P., Online estimation of external loads from dynamic measurements, in International Conference on Noise and Vibration Engineering, Sas P. and De Munck M., Editors: Leuven, Belgium. p. 3957-3968. 2006. Klinkov. M. and Fritzen. C. P. Online wind load estimation for the offshore wind energy plants. Stanford University: Stanford CA. September 11-13, 2007. Areva-Multibrid. M5000 Technical data. Available from: http://www.multibrid.com/index.php?id=9&L=1Areva-Multibrid
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
An Integrated SHM Approach for Offshore Wind Energy Plants Claus-Peter Fritzen, Peter Kraemer and Maksim Klinkov University of Siegen, Department of Mechanical Engineering, Paul-Bonatz-Str. 9-11, 57068 Siegen, Germany, [email protected], [email protected], [email protected]
ABSTRACT Global warming, the limitation of combustible resources and lack of public acceptance of nuclear power plants including the problem to find appropriate nuclear waste deposits, has pushed renewable sources of energy towards the top of the power generation agenda. This has helped to promote wind power as one of the most cost effective of the renewable technologies. Since the energy gain from off-shore power plants (OWEP) is higher as onshore, many offshore wind parks worldwide are projected. The other side of the medal is that the costs for inspections and maintenance offshore are much higher than for onshore plants. Especially under harsh condition on the sea, classical inspection methods are not practicable. For this reason it makes sense to develop monitoring systems in order to reduce the number of inspections, to identify damage in an early phase and to forecast the remaining life-time of OWEPs. The first part of the paper gives a short overview on the importance of SHM systems for OWEPs. In the second part some suitable methods for monitoring different parts of OWEPs are mentioned. Finally, methods for online force identification sensor fault identification and damage detection/localization accompanied on field tests of a 5 MW plant will be presented. INTRODUCTION: SHORT OVERVIEW OF MONITORING SYSTEMS FOR OWEP A large OWEP represents a big technical challenge with respect to design, operation and maintenance. It is a matter of fact that the downtime of the plant for extended repair and maintenance (harsh condition on the sea) resulting from undiscovered damage can lead to big economical losses. Costs models [1,2] for OWEPs show that operation and maintenance is a considerable factor in the total cost of a plant [3]. It has to be noted that, as wind farms are sited further offshore, vessel time will increase for each visit due to greater transport time and potentially higher risk of weather downtime. To reduce these costs, there is a need to develop a monitoring system which provides the operator necessary and reliable information about the plant’s state and allows the initiation of maintenance actions. Not only the blades and gearbox, but also the tower and foundation have to be monitored, because the knowledge about the structural loads resulting from wind and water waves and the resulting material stresses is not yet fully understood. For a better understanding of the monitoring of OWEPs the relevant parts are displayed in Figure 1. Here the foundation is based on a tripod construction, whereas this depends on the design philosophy of the producers. Other possible foundations are: monopile or jacket constructions. A very good overview on different constructions of WEP or OWEP and their dynamical behavior is given in [4]. SHM systems for different parts of OWEPs are in development since many years. The most monitored components today are the machine components and the blades. The gearbox, bearings, etc. are online controlled with methods derived from condition based monitoring (CBM). Since a lot more experience exist in CBM than in SHM, many companies and research institutes worldwide offer their services for monitoring machine parts, whereas this domain is dominated by the producers of the machine components. A big challenge remains for the monitoring of slowly rotating machine parts. Also the signal transmission between the sensors and the monitored parts is often long and the separation of the signals coming from many components simultaneously is therefore difficult. Monitoring of these parts is mostly done with accelerometers. There is a challenge to develop sensors for slow rotating machinery.
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_63, © The Society for Experimental Mechanics, Inc. 2011
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Figure 1. Parts of OWEP. Example based on Multibrid M5000-2 offshore prototype The most problematic part of the machine is the planetary gear with its complicated kinematics. A total damage of planetary gear follows by its exchange/repair on the sea is associated with immense costs. The monitoring of the rotor blades is still in development and only few companies offer online monitoring on the field. The applicability of a big palette of methods on rotor blades is tested today. E.g. Mc Gugan uses acoustic emission techniques to identify damages in blades [5], Lehmann et. al. [6] applies guided waves and acoustic emission techniques for active and passive monitoring systems. Volkmer uses changes in the frequency spectrum to identify damages or mass increasing by ice-foundation [7]. Also the impedance method was tested on blades [8]. In [9] four different techniques for monitoring of wind turbine blades are compared: transmittance function, resonant comparison, operational deflection shape and wave propagation. The use of optical fiber sensors lately opened new ways for monitoring the blades of OWEPs. First attempts can be found in [10]. An overview of some damage identification methods for wind turbines can be found in [11]. One of the most neglected but expensive part of the energy plant from an SHM point of view is the tower and the foundation. Only few companies and research institutes spend time and money to investigate this, although the impact of the loads from waves or from extreme wind gusts, the ground erosion, etc. are not well-known at the moment. SHM systems for tower and foundation are developed for e.g. by Rolfes et al. [12]. They use the proportionality between the strain and the vibration velocity at certain places on the OWEP for damage detection. Nichols [13] used an attractor-based approach for damage detection of offshore structures under ambient excitation. In [14-16] a combination of the Stochastic Subspace Fault Detection Method and pattern recognition techniques was proposed for damage detection of OWEP. Another often neglected but essential aspect of permanently monitored structures is that the functionality of the monitoring system depends not only on the damage identification method but also on the sensor network installed on the structure. In the ideal case the sensors should have a longer life-time than the structure they have to monitor. In practice, especially under harsh conditions, the sensors themselves have a limited life expectation. The probability that sensor faults occur during an OWEP life time of 20 years is very high. In particular the functionality of strain gauges on sea conditions is guaranteed only for few years. Errors from malfunctioning sensors affect the damage identification result. In some cases the effect of a faulty sensor on the damage identification algorithm can be greater than the effect of a severe damage of the structure.
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When a sensor fault occurs, the damage identification algorithm gives a false alarm and the structure will be incorrectly classified as damaged. For this reason, it is necessary to identify the faulty sensors and to exclude them from the damage identification algorithm. For long-time monitoring it is important to integrate sensor fault detection algorithms in the SHM system configuration [3,14,16]. For high levels of the SHM hierarchy like damage prognosis, load identification techniques for different parts of OWEP are necessary. Today many research institutes and companies study different possibilities to identify loads caused by wind and waves on OWEP, e.g. [17-20]. Also guidelines for load monitoring of OWEPs are proposed [21,22]. Kohlmeier et. al. [23] study especially the wave loads on the foundation and tower. Fritzen and Klinkov [24] proposed a robust observer for online load identification. A holistic monitoring of OWEP was performed within IMO-WIND project [25,26], (Integrated Monitoring and Evaluation System for Off-shore Wind Energy Plants) sponsored by the German Ministry of Economics. The consortium consists of six companies and two research establishments: Multibrid (producer of 5MW OWEP), Offshore Wind Technology (design, modeling and calculation), μ-Sen (condition monitoring), IGUS (SHM of blades), Infokom (networks and communications), GermanLoyd Wind (guidelines), Federal Institute for Materials Research and Testing, BAM (measurements and evaluations methods) and the University of Siegen (force identification, SHM of tower, foundation, machine parts and blades including sensor fault detection). CONDITIONS FOR MONITORING OF OWEP To perform online monitoring of OWEPs it is very important to know the influences of different operational and environmental conditions on the dynamical behavior of the plant, whereas the wind velocity and direction, the temperature, the position of the nacelle, the rotation velocity of the blades, the atmospheric conditions (rain) are changing permanently. Sometimes also the boundary conditions are changing (e.g. by ground erosion). For example, the wind velocity, the wind direction and the height of waves affect the excitation level of the blades and structure (e.g. at low wind velocity, only a few numbers of modes are excited). It is also known from bridges that a high wind velocity can cause a change in the eigenfrequencies [27]. The change in the temperature can affect dynamical behavior. Since the structure of the OWEP is not perfectly symmetric along the tower axis (caused by transformers, pumps, etc. which are placed along the tower), the distribution of mass/mass moment of inertia changes due to the orientation of the nacelle, and with them also the dynamics of the system. Since the methods for damage identification and prognosis proposed in the project IMO-WIND deal at the moment mostly with the structure and the foundation, mainly these items will be considered in this paper. DAMAGE IDENTIFICATION OF THE STRUCTURAL PARTS The concept for the permanent SHM of OWEP structure presented in this paper consists of three steps: 1) damage detection under changing operational conditions; 2) sensor fault detection; 3) damage localization. The concept is presented in Figure 2. Damage detection Methods for damage detection under changing operational conditions were developed by Sohn et al. using a combination of AR-ARX (AR models with exogenous inputs) models with Non Linear Principal Component Analysis (NLPCA) [32]. Kullaa applies missing data analysis [33] or factor analysis [34] and non-linear factor analysis [35] to eliminate the environmental effects from damage sensitive features. Yan et. al [36] propose a local PCA for structural damage diagnosis under changing environmental conditions. A very good review of the methods for compensation of environmental conditions can be found in [27]. In this contribution an algorithm for damage detection based on the SSFD method developed in [37] and, alternatively, a method based on the multivariate AR modeling [38] is used. Both methods are designed for output-only systems under the assumption that the unknown excitation is a stationary white noise signal. The basic idea of these methods consists of extracting some features θ n from the incoming measured data and comparing them with the features θ 0 of reference data. E.g. for SSFD method the residual errors ζ n are calculated from the left singular vector of the baseline Hankelmatrix H (θ 0 ) arranged column-wise in the matrix K and the Hankel matrices themselves from the incoming data (e.g. acceleration measurements).
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Figure 2. Co oncept for Stru uctural Damag ge Identificatio on of OWEP
ζ n = vec( K ( θ 0 )H αβ ( θ n )) with 1 ≤ n ≤ N D
(1)
In this equation ND define es the total number n of da atasets. Therrefore, if dam mage occurs, the residual error will in ncrease beca ause the resp ponse spaces spanned by the damaged d system will be different from f those sp panned by the baseline system. A scalar s damag ge indicator DI can be defined d as DI D n := ζ nT Σ −1ζ n , where Σ is the c covariance ma atrix of the residuals of the e undamaged structure. T These method ds turn out to o be very sen nsitive to strucctural damage es but also to o changes in the environm mental and o operational co onditions of the e system. Consequently th hey were adap pted to identifyy damage of OWEP O by cla assification o measured data of d w.r.t. the e means of measured m inpu ut parameterss representing g the environm mental and operational o c conditions of the t OWEP likke temperature e, wind velociity, position off the nacelle, height of wavves, etc, [15].. For each
(m )
c class m a reference datasett θ 0 is chossen. The dam mage indicatorss are calculate ed for each class separately from the in ncoming data (i data in clas ss m) compare ed with the co orresponding class c referencce. A big challeng ge of the clas ssification is to o avoid the “ccurse of dimensionality” (e.g. by grid cla assification) and a to find o only a few nu umbers of rep presentative classes c and references fo or the calcula ation of dama age indicatorss. For this p purpose, classification tech hniques like k-means or Expectation-M E Maximization (EM) [39] are e used. To re educe the n number of cla assification pa arameters, a NLPCA can be applied to o the measurred input para ameters. Thiss adapted m method for pe ermanent SH HM of OWEP P works in tw wo phases: offfline (training g) and online e (monitoring). A good training of the e damage dettection system m during the offline phase e (assume: structure is no ot damaged) is a good p premise for a well working damage detection syystem. In the e online pha ase the data are assigne ed to the c corresponding g class. Whe en the incoming online da ata doesn’t co orrespond to one of the existing e classses a new c class is generrated. Sensor fault detection S d Id dentification and a reconstru uction of faultyy sensors havve been studiied e.g. by Du unia et al. [40 0] and by Kercchen et al. [4 41] applying the PCA, Ma attern et al. [42] using ne eural networkks. Worden [4 43] applies a combination n of autoa associative ne eural network ks (AANN) an nd ARX mode els while Kullaa [44] usess the missing data analysis. Further p possibilities to o identify and reconstruct signals s from faulty f sensorss were develo oped in [45,4 46] by means of mutual in nformation an nd Kalman filtters.
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If a change in the system occurs (which is not caused by changing ambient excitation or operational conditions) it will be checked if a sensor fault is responsible for this change or not. The detection of sensor abnormalities is based on the concept of mutual information (MI) between two signals X and Y from two different sensors. This requires that two different channels measure some amount of redundant information. MI measures the information of one channel X that is shared by another channel Y.
MI ( X ,Y ) =
p( x , y )
∫∫ p(x, y ) log p x (x) p y ( y ) dxdy
(2)
where p( x , y ) is the joint probability density and p x , p y are the marginal densities of X and Y. From Eq. 2 we can see, that if X and Y are independent, then p( x , y ) = p x ( x ) p y ( y ) and thus MI becomes zero.
The MI for all possible combinations of sensor outputs y m and y l (except m=l) is computed which leads to a MImatrix. The basic idea is that the MI changes when an individual sensor fault f m is present, e.g. in the m'th channel ( ~ y m = y m + f m ). This fault appears only in the m-th channel. Thus we should expect that all combinations with index m should show a reduction of the mutual information after failure of sensor m. This allows us to localize the defect sensor. Damage localization
Figure 3. Damage localization algorithm [47, 48] When the change in the system is not caused by changing ambient excitation, changing operational conditions or sensor faults, it will be assumed that a structural damage has occurred which will be localized in the third step of the SHM system. For damage localization a model based algorithm based on the inverse eigensensitivity is applied to the structure, for details see [47] and [48]. The basic idea of the technique is to compare the measured mode shapes and the eigenfrequencies of the OWEP (which requires a modal analysis of structures under ambient excitation) with the mode shapes and the eigenfrequencies from an analytical model of the undamaged OWEP for a specific operational condition. When damage occurs (loss of stiffness by cracks or loose bolts on the tower) the measured mode shapes and eigenfrequencies deviate from the reference data. The position of damage can be localized with the help of the model. It is important that the modal analysis is applied to measured data “collected” under the same excitation and operational conditions. The main components of this algorithm are displayed in Figure 3.
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O ONLINE LOAD IDENTIFICA ATION Reconstruction R n of external lo oads/forces iss an importantt part of the ovverall OWEP SHM S system, especially be ecause the e external forcess/loads resulting from wind and waves ca an not be mea asured directlyy. Knowledge e of these load ds enables u to make an us n assessmentt of damage after a extreme events like sttorms and upd dated forecassts of the rema aining lifetime of the OW WEP structure e, and to imprrove the signifficant items du uring the desiign phase for the future con nstruction. T OWEP co The omponents lik ke tower, roto or blades as well w as the trripod foundation are the pa arts where the external fo orces estimation are of grea at interest. L Load history id dentification has h been stud died extensive ely for the lasst two decade es. Generally all force reco onstruction m methods that where w propos sed by engine eers can be summarized s in n three main groups g [49]: 1) 1 Deterministtic method (based on the model and measured m sign nals); 2) Stoch hastic method (statistical models); 3) Artiificial intelligen nce-based m methods (neurral networks); commonly they can be rep presented as in Figure 4. T force reco The onstruction pro oblem in all th hese groups solved by using indirect mea asurement tecchniques, which include th he transforma ations of related measured d quantities such s as acce eleration, velocity, position or strain see e Figure 4 (right). These transformations generally lead to a so-ca alled inverse problem or de econvolution problem p of the e following c convolution (D Duhamel-Integ gral): t
y( t ) = ∫ H ( t − τ )F ( τ )dτ
(3)
0
where the sysstem propertie w es H(t) and re esponses y(t) are known while excitation ns F(t) are unkknown. It is well w known th hat inverse prroblems are ‘ill-posed’ in th he mathematiccal sense, tha at is one of 1) the existence e, 2) the uniqu ueness, or 3 the stabilityy of solution iss violated [50, 51]. If this invversion can be done, the system 3) s itself becomes b its ow wn “virtual fo orce sensor” which w would solve s the prob blem of the lo oad history esstimation. A va ariety of methods were elab borated to o overcome the e above menttioned difficultties [52-53]. Stevens S [54] gave an exccellent overvie ew of this top pic, which in ncludes some e earlier studies on inverse e analysis of external forcces as well ass Inoue et al. [51] compile ed a good s summary for im mpact force esstimation tech hniques.
Figure 4. OWEP O (left), lo oad reconstrucction problem (right). Some of these S e methods are e based on the e frequency or o impulse resp ponse function ns or use regu ularization and d dynamic p programming in the time do omain, which was proposed by Trujillo and a Busby [55 5] and applied d by Doyle [56 6, 57] and N Nordström [58 8]. Others use e an integration of measured acceleration n signals [52] or can only determine d the sum of all fo orces and mo oments applied to the centrre of mass (su um of weighte ed acceleration n technique SWAT) S [59]. Most M of the a authors assum me the prior kn nowledge of th he force location and requirre first to recorrd the system responses an nd then do th he force histo ory reconstrucction. The reg gularization techniques or a time shift off the collected d measureme ent data in n non-collocated d case [60] (where ( senso ors and loadss positions are a not identical) are used d to overcom me the illp posedness of the inverse problem. The ere are other researchers who treated the same pro oblem from th he control e engineering po oint of view [6 61] by using ob bserver prope erties that are able to estima ate the states and unknown n inputs or e external disturrbances simulttaneously. O One of the promising p tec chniques whicch was teste ed on the OW WEP model is a simultan neous state and input e estimator (SS SIE). The SSIE is a time domain apprroach, which originally wa as invented for f control en ngineering p purposes [62]]. It allows sim multaneous reconstruction n of the inputs and states (velocities, positions) p of a linear or n nonlinear time e invariant sys stem. The ma ain principle is based on th he constructio on of the obse erver for a ge eneral first o order nonlinea ar state space e system:
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x& (t) = Ax(t) + Bu(t) + f (( x , u ), y ) y(t) = Cx(t) + Du(t)
(4)
where x(t), u(t) and y(t) are the states, unknown inputs and measured outputs, respectively. Matrices A, B, C and D are real, constant and of appropriate dimensions. f(.) is a real nonlinear vector function. The proposed observer T
[62,63] estimates a combined state and input variable ξ (t ) = ⎡ x (t )T u(t )T ⎤ . For the observer design an ⎣ ⎦ appropriate set of observer matrices should be found, so that the error between the estimated inputs and states and the real one will converge to zero as the time evolves. It is possible to find these matrices with the help of a linear matrix inequality (LMI) technique. Two assumptions in this method have to be satisfied: the D matrix must have full column rank and the number of sensors must be greater or equal to the number of unknown inputs plus the number of unknown nonlinear terms. Once the observer matrices are calculated, the inputs and states can be reconstructed simultaneously online. The ability of online reconstruction and inherited noise robustness make the SSIE very attractive for external load estimation. RESULTS OF THE INTEGRATED SHM SYSTEM ON OWEP M5000-2 The SHM approaches were applied on measured time data from the prototype OWEP M5000-2 of the company AREVA-Multibrid in Bremerhaven, Germany (see Figure 5, left). The sensor signals are provided by eight accelerometers positioned on the tower as shown in Figure 5, right. The signals were measured simultaneously with a sample rate of 50Hz. The measurement time for one data set was ten minutes. In order to perform long time monitoring 5758 data sets were considered. The first 2016 data sets were continuously measured in November 2007 (first 1008) and March 2008 (next 1008). The remaining 3742 data sets were measured between March and September 2009 (one data set in one hour).
Figure 5. OWEP M5000-2 (left); sensor positions (right) Since the SHM approaches uses the premise that the excitation of the structure is unknown but normally distributed (output-only system with white noise excitation) it is avoided to apply the algorithms on data measured by strong transient excitation. In order to exclude such undesired data sets from the analysis, at first a multivariate outlier technique based on Mahalanobis-distance with thresholds provided by means of the F-distribution are applied on the measured data [64]. This leads to the fact that only 2669 data sets can be evaluated.
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The integrated SHM system evaluates the measured data online via internet and gives the operator information about the health state of the sensors and plant. Additional information about the loads acting on the structure is also extracted from the data. The scheme of the online monitoring system is shown in Figure 6.
Figure 6. Integrated SHM System for OWEP M5000-2 A part of the damage detection results are showed in Figure 7. Figure 7 (left) displays the damage indicators after the training phase. Different colors marked the indicators belongs to different classes of EOC. In Figure 7 (right) the damage indicators of one class obtained during the training phase are compared with indicators from the online phase. It can be seen that the indicators from the online data do not exceed the indicator from the training data, so no alarm will be initialized. Since no damage occurred on the monitored OWEP (M5000-2) until today, the possibility to identify small damages by changing EOC will be exemplified by means of simulations. For this purpose a FEM-model of M5000-2 was used in order to generate “measurement” data sets. The model was stochastically excited at the top of the tower by forces resulting from real measured wind velocities. A further periodical excitation of the tower, the effect of the blade passage was also considered. The simulated positions of the nacelle and the wind directions correspond to the measured one. The system response was “measured” in form of “accelerations” at the degree of freedoms corresponds to the real sensor positions (see also Figure 5, right). The measurement duration and the sample rate corresponds to the real measurement data.
Figure 7. Classes with damage indicators (left) and damage indicators in one class (right) The total number of simulated data sets was 1109. The damage was introduced in the last 101 data sets. The introduced “damage” is a progressive stiffness reduction (Young’s-modulus) of 1, 5 and 10% of element 59, see Figure 8, left. These damages cannot uniquely related to the shift of the frequency spectra. Also changes of stable
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poles obtained by means of operational modal analysis are masked by the effects of changing of environmental and operational conditions.
Figure 8. Simulated damage on Element 59 (left) and results of damage detection for different damage extensions and number of classes (right) The damage detection method gives good results using a relatively high number of training data sets and classes. Figure 8 (right) shows the results using 800 training data sets. Here no false alarms occur and also small damage of 1% stiffness reduction could be well identified. A significant reduction of the number of training data sets can cause few false alarms sporadically. This “deficit” can be referred to the high sensitivity of the NSFD algorithm w.r.t. each system change. This effect is a well-known behavior of non model-based algorithms for damage identification: “if an algorithm is sensitive to the damage so it is also sensitive to changing of EOC” [65]. In our case the reduction of the numbers of false alarms can be achieved by the reduction of the number of references (classes). But this apparent improvement reduces also the sensitivity of the method to detect small damages. Since no faults of the channels 65-72 (see Figure 5, right) occur during the measurements, the signal of sensor 3 (channel 67) was artificially substituted in the last 200 data sets by a noise signal with the same statistical properties as the original signal. The results of MI approach shows clearly sensor 3 as faulty in the last 200 data, see Figure 9. For a better exemplification of the results the change of the MI-matrix in the data set 2539 is displayed in Figure 9, right. It can be seen that the MI value between the sensor three and all the other sensors changes significantly.
Figure 9. Results of the sensor fault detection algorithm (left: long time monitoring; right: MI change for measurement no. 2539) For validation of the damage localization method by absence of a real damage on the M5000-2 the FE-model in Figure 5, right is used. The modal data used for model-updating and as residuals for localization algorithm is provided in an automatic way by mean of an operational modal analysis using vector autoregressive models [16].
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Due to the sparse sensor network on the tower only four bending modes of the structure could be well identified. So the inverse eigensensitivity method was applied by using only these four modes of the structure determined at the four sensor locations in Figure 5. In order to investigate the possibility to identify damages in the tower, the local stiffness of the tower was reduced by 5%, see Figure 10, left. The results in Figure 10 (right) show the correct identified damage position. More complicated seems to identify damages under the water surface, where no sensors are installed and the structure is very stiff. Therefore a stiffness reduction of 5% at the hot spot of the tripod was simulated (Figure 11, left). This damage was also correctly identified, as shown in Figure 11, right.
Figure 10: Damage of the tower (left: damage simulation; right: results of localization)
Figure 11: Damage of the tripod (left: damage simulation; right: results of localization)
The above mentioned methods for SHM were successfully applied on measured time data from a laboratory structure (scaled model of a OWEP) or on the WEP M5000-1, see [3,14,16]. Two sensor types (two 2-D accelerometers and two strain gauges see Figure 12 left) were needed for the estimation of the external loads which are acting on the tower of the WEP. These data was then fed into the observer, based on a reduced modal model which itself was constructed from an FE model (see Figure 12 right).
Figure 12. Load estimation for the prototype WEP
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The initial FE model was improved using model-updating techniques, converted into modal coordinates and finally transformed into state space notation. The thick red spots on the model (see Figure 12 right) represent the locations of the accelerometer (top) and strain gauge (bottom), respectively. Load estimation was then done for the section where the nacelle is connected to the tower. This section is chosen intentionally and includes all possible forces that the tower should subtend (external wind forces; static nacelle orientation, blades dynamics; mass unbalance; etc.). For the qualitative validation of the estimated load one of the working states (see Figure 13 right) of the WEP M50002 that was close to the ideal operational conditions has been chosen see data sheet [66]: a) wind comes almost orthogonal to the rotor area, with a nearly constant wind speed of approximately 12 m/s; b) the blades have a relatively small pitch angle. They allow using the Betz theory for validation of the estimated load. According to Betz the wind load which is applied to the rotor area can be calculated as: FBetz =
1 2 c ρVwind A 2
(5)
where, c is an aerodynamic coefficient, Vwind is the wind speed, ρ is the air density and A is a rotor area accordingly. The wind is assumed to blow orthogonal to the rotor area. The estimated and Betz forces are depicted in Figure 13 for a time duration of 25 minutes and the above mentioned operational conditions
Figure 13. Estimated external force for M5000-2 (left), operational conditions (right) Figure 13 shows a certain difference between estimated and Betz force especially in the region where the wind velocity is increasing. This is due to Betz theory; see Eq. 5, where the Betz force follows the wind velocity behavior for a fixed coefficient c without consideration of pitch angles (see for comparison the wind velocity in Figure 13 right in the time interval 0-1500 seconds, wind velocity increases as a time evolves). As a result the Betz force cannot be used solely for the validation, instead one needs also to take into consideration the pitch angle of the rotor blades. A closer look on Figure 13 left shows that the estimated force is very close to the Betz force in the regions where the pitch is close to zero (~100 and ~1300 seconds) and becomes smaller as rotor blades are pitched (see pitch and wind velocity in Figure 13 right in time interval 0-1500 seconds). CONCLUSIONS For the last decades practice has proved that the wind energy industry is one of the leading and promising among the renewable energy manufactures. Additionally the industrial trend moves to a bigger construction of WEP's and their global shift to offshore regions due to higher wind speeds. Experience show that the cost of operation and maintenance raise dramatically with the shift into offshore zone compared to onshore WEP's. Especially, vessel transportation of the damaged components/spare parts or inspection teams is cost demanding and challenging in case of stormy weather.
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Therefore, SHM and also CBM systems with the ability of online remote forecasting of critical failure situations of components is an indispensable integral part of new WEP concepts. The integration of the SHM system into the WEP will not only save cost but also provide information for future design improvement of these items. Presently, research teams and companies attempt to develop such monitoring systems or independent monitoring components for gear and bearings, blades and the supporting structure. The IMO-WIND project forming a network of WEP manufactures and research institutes illustrates the importance of multidisciplinary teams to develop SHM system for the offshore WEPs and to be successful with system integration. ACKNOWLEDGEMENT The authors are grateful to the German Ministry of Economics and the companies Multibrid, OWT, μ-Sen, IGUS, Infokom and GL Wind for the financial support of the IMO-WIND project by funding of BMWi (grant no. 16INO327). We are also grateful to the Federal Institute for Materials Research and Testing (BAM VII-2) and to the company AREVA-Multibrid for providing the measurement data. REFERENCES
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Delft University Wind Energy Research Institute (Duwind), 2001, “Concerted Action on Offshore Wind Energy in Europe”, Report Ref.: Duwind 2001.006. Offshore Design Engineering (ODE), 2007, “Study of the costs of offshore wind generation”, A Report to the Renewables Advisory Board (RAB) & DTI, URN NUMBER 07/779. Fritzen C-P, Kraemer P, Klinkov M., 2008 “Structural Health Monitoring of Offshore Wind Energy Plants”, 4th European Workshop on Structural Health Monitoring, Cracow, Poland: 3-21. Gasch R., Twele J., 2002, „Wind Power Plants – Fundamentals, Design, Construction and Operation“, James & James. McGugan M., Sorensen B. F., 2007, “Fundamentals for Remote Condition Monitoring of Offshore Wind Turbine Blades”, Proc. 6nd Intl. Workshop on SHM, Stanford: 1913-1919. Lehmann M., Büter A., Frankenstein B., Schubert F., Brunner B., 2006, “Monitoring System for Delamination Detection-Qualification of Structural Health Monitoring (SHM) Systems”, Conference on Damage in Composite Material CDCM 2006, Stuttgart. Volkmer P., Kühl A., Müller F., Scholbach D., Volkmer D., 2006, “Continuous Natural Frequency monitoring of Rotor Blades for Detection of Damages, Ice-foundation and Dynamik Overloads”, Deutsche Windenergiekongress, Bremen, Germany. Pitchford W, 2007, “Impedance-Based Structural Health Monitoring of Wind Turbine Blades”, Master Thesis Virginia Polytechnic Institute and State University. Ghoshal A., Sundaresan M. J., Schulz M. J., Pai P. F., 2000, “Structural health monitoring techniques for wind turbine blades”, J. of Wind Eng. and Industrial Aerodynamics, 85: 309-324. Guemes J. A. et. al., 1998, ” Strain and damage monitoring of wind turbine blades by piezoelectrics and fiber optic sensors”, Proc. of the ECCM-8, 3,: 357–64. Ciang C.C., Lee J.-R., Bang H.-J., 2008, “Structural health monitoring for a wind turbine system: a review of damage detection methods”, J. of Measurement Science and Technology,19: doi:10.1088/09570233/19/12/122001. Rolfes R, S. Zerbst , G. Haake, J. Reetz and P. Lynch, 2007, “Integral SHM-System for Offshore Wind Turbines using Smart Wireless Sensors” 6th Intl. Workshop on SHM, Stanford: 1889-1896. Nichols, J. M., 2003, “Structural health monitoring of offshore structures using ambient excitation”, Applied Ocean Research, 25: 101-114. Kraemer P., Fritzen C.-P., 2007, “Concept for Structural Health Monitoring of Offshore Wind Energy Plants”, Proc. 6nd Intl. Workshop on SHM, Stanford: 1881-1888. Moll J., Kraemer P., Fritzen C.-P., 2008, “Compensation of Environmental Influences for Damage Detection using Classification Techniques”, 4th European Workshop on Structural Health Monitoring, Cracow: 1080-1087. Kraemer P., Fritzen C.-P., 2008, “Damage Identification of Structural Components of Offshore Wind Energy Plants”, Deutsche Windenergie Konferenz, DEWEK 2008 Bremen, Germany. Liersch J., Gasch R., 1996, “Calculation of Fatique Loads by Digital Simulation of Wind Turbines which are controlled by Commercial Controllers in Real Time”, Proc. EUWEC'96, Gothenburg.
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18 Rettenmeier A., M. Kühn, T. Kramkowski, F. Koch, 2006, “Load and Power Performance Measurements of the Multibrid M5000 Prototype”, DEWEK, Bremen, Germany. 19 Reuter A., N. Cosack, 2004, “On-line load tracking using standard turbine visualisation data”, EWEC, London. 20 Fischer T., Kühn M., 2006, “Control Requirements for Load Mitigation of Aerodynamic and Hydrodynamic Loads of Offshore Wind Turbines”, DEWEK, Bremen, Germany. 21 Söker H., C. Illig, N. Cosack, J. Kröning, M. Damaschke, 2006, “A Guide to Design Load Validation”, DEWEK, Bremen, Germany. 22 Argyriadis K., Gill L., Schwartz S.,” Wave Load Prediction Methods in Offshore Wind Turbine Modelling and their Influence on Fatigue Load Analysis”, available on: http://www.gl-group.com. 23 Kohlmeier M., Mittendorf K.., Kossel T., Habbar A., Zielke W., 2007, “Wave Load Prediction Methods in Offshore Wind Turbine Modelling and their Influence on Fatigue Load Analysis”, European Offshore Wind Conference & Exhibition, Berlin. 24 Fritzen C.-P., Klinkov M., 2007, “Online Wind Load Estimation for Offshore Wind Energy Plants”, Proc. 6th International Workshop on SHM, Stanford: 1905-1912. 25 Rohrmann R., Rücker W., Said S., 2005, “IMO-WIND An integrated monitoring system for offshore wind energy turbines”, 6th Intl. Conf. on Structural Dynamics (on CD-ROM), Paris. 26 Rohrmann R. G., Rücker W., Thöns S., 2007, “Integrated monitoring Systems for Offshore Wind Turbines”, Proc. 6nd Intl. Workshop on SHM, Stanford: 1897-1904. 27 Sohn H., 2007, “Effects of environmental and operational variability on structural health monitoring”, Phil. Trans. R. Soc. A, 365: 539-560. 28 Oppenheim, A.V., R.W. Schafer, 1998, “Discrete-Time Signal Processing”, 2nd ed., Prentice-Hall. 29 Kolerus J., 2000, „Zustandsüberwachung von Maschinen“, 3. Auflage, Expert-Verlag. 30 Antoni J., Randall R.B., 2004, “The Spectral Kurtosis: a useful tool for characterising nonstationary signals”, Mechanical Systems and Signal Processing, Vol. 20: 286-307. 31 Antoni, J., 2007, “Cyclic spectral analysis of rolling-element bearing signals: Facts and fictions”, Journal of Sound and Vibration, Vol. 304: 497-529. 32 Sohn, H., K. Worden and C. R. Farrar, 2002, “Statistical Damage Classification Under Changing Environmental and Operational Conditions,” Journal of Intelligent Material Systems and Structures, 13: 561-574. 33 Kullaa, J., 2005, “Damage Detection under a Varying Environment using the Missing Data Concept”, Proc. 5th International Workshop on SHM: 565-573. 34 Kullaa, J., 2002, “Elimination of Environmental Influences from Damage-Sensitive Features in a Structural Health Monitoring System”, Proc. of 1st European Workshop on SHM, Paris: 742–749. 35 Lämsa V., Kullaa J., 2007,“Nonlinear Factor Analysis in Structural Health Monitoring to Remove Environmental Effects“, Proc. 6nd Intl. Workshop on SHM, Stanford: 1092-1099. 36 Yan, A.-M., G. Kerschen, P. De Boe and J.-C. Golinval, 2005, “Structural damage diagnosis under varying environmental conditions - part 2”, Mechanical Systems and Processing, 19: 865-880. 37 Basseville M., M. Abdelghani and A. Benveniste, 2000, “Subspace-based fault detection algorithms for vibration monitoring”, Automatica, 36, 2000: 101-109. 38 Neumaier A. and T. Schneider, 2001, “Estimation of Parameters and Eigenmodes of Multivariate Autoregressive Models”, ACM Transactions on Mathematical Software, 27 (1): 27-57. 39 Nabney I. T., 2002, “NETLAB-Algoritms for Patterrn Recognition”,Springer. 40 Dunia, R., S. J. Qin, T. F. Edgar and T. J. McAvoy, 1996, “Identification of faulty sensors using principal component analysis”, AiChe Journal, 42 (10): 2797-2812. 41 Kerschen G., P. De Boe, J.-C. Golinval and K. Worden, 2004 “Sensor validation for on-line vibration monitoring”, Proc. 2nd Europ. Workshop on SHM, Munich, Germany: 819-827. 42 Mattern D. L., L. C. Jaw, T.-H. Guo, R. Graham and W. McCoy, 1998, “Using Neural Network for Sensor Validation”, 34th Joint Propulsion Conference, Cleveland, Ohio, USA. 43 Worden K., 2003, “Sensor validation and correction using auto-associative neural networks and principal component analysis”, Proc. IMAC XXI, Orlando, USA. 44 Kullaa J., 2007, “Sensor Fault Identification and Correction in Vibration-Based Multichannel Structural Health Monitoring”, Proc. 6nd Intl. Workshop on SHM, Stanford: 606-613. 45 Kraemer P., Fritzen C.-P., 2007, “Sensor Fault Identification Using Autoregressive Models and the Mutual Information Concept”, DAMAS 2007, Torino, Key Engineering Materials, 347: 387-392. 46 Kraemer P., Fritzen C.-P., 2008, “Sensor Fault Detection and Signal Reconstruction using Mutual Information and Kalman Filters”, International Conference on Noise and Vibration Engineering, Leuven, Belgium: 3267-3282. 47 Fritzen C.-P., Bohle K.,, 2001, “Model-Based Damage Identification from Change of Modal Data – A Comparison of Different Methods”, Proc. 2nd Intl. Workshop on SHM, Standford: 849-859.
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48 Balageas D., C.-P. Fritzen and A. Güemes, 2006, “Structural Health Monitoring”, ISTE Ltd, London, 2006. 49 Uhl T., 2007, “The inverse identification problem and its technical applications”, Archive of Applied Mechanics, 77: 325-337. 50 Jacquelin E., Bennani A., Hamelin P., 2003, “Force reconstruction: analysis and regularization of a deconvolution problem”, Journal of Sound and Vibration, 265: 81-107. 51 Inoue H., Harrigan J.J., Reid S.R., 2001, “Review of inverse analysis for indirect measurement of impact force”, Appl. Mech. Rev. 54(6), November (2001): 503-524. 52 Elliott K., Buehrle R., James G., 2005, “Space shuttle transportation loads diagnostics”, Proceedings of the 23rd International Modal Analysis Conference, Orlando, Florida. 53 Lee M.L., Chiu W.K., Koss L.L., 2005, “A numerical Study into the reconstruction of impact force on railway tracklike structures”, Sage Publications, 4(1): 19-45. 54 Stevens K., 1987, “Force identification problems - an overview, Proceedings of SEM”, Spring Meeting, Houston:. 838-844. 55 Trujillo D. M., Busby H. R., 1997, “Practical inverse engineering”, CRC Press, London. 56 Adams R., Doyle J.F., 2002, “Multiple force identification for complex structures”, Experimental Mechanics, 42(1): 25-36. 57 Doyle J.F.. 1984, “An experimental method for determining the dynamic contact law”, Experimental Mechanics, 24(4): 265-270. 58 Nordström L. J. L., 2005, “Input estimation in structural dynamics”, PhD thesis, Chalmers University of Technology Sweden, Göteborg, Sweden. 59 Genaro G. G., Rade D. A., 1998, “Input force identification in time domain”, Proceedings of the 16th International Modal Analysis Conference: 124-129. 60 Kammer D. C., Steltzner A. D., 2001, “Structural identification of Mir using inverse system dynamics and Mir/Shuttle docking data”, Journal of Vibration and Acoustics 123(2): 230-237. 61 Söfker D., Ahrens J., Ulbrich H., Krajcin I., 2003, „Modellgestützte Schätzung von Kontakt-Kräften und Verschiebungen an rotierenden Wellen“, Schwingungen in rotierenden Maschinen V. Verlag Vieweg. 62 Ha Q. P., Nguyen A. D., Trinh H., 2004, “Simultaneous state and input estimation with application to a two-link robotic system”, Proc. ASCC of the 5th Asian Control Conference: 322-328. 63 Klinkov M., Fritzen C.-P., 2006, ”Online estimation of external loads from dynamic measurements”, Intl. Conf. on Noise and Vibration Engineering, ISMA, Leuven, Belgium, on CD. 64 Rencher A.C., 2002, “Methods of Multivariate Analysis”, John Wiley & Sons, Inc. 65 Worden K., Farrar C.R., Manson G. and Park G., 2005, “Fundamental Axioms of Structural health Monitoring”, In: Proc 4th International Workshop Structural Health Monitoring, Stanford University: 26-41. 66 Areva-Multibrid MTd., 2008, http://wwwmultibridcom/indexphp?id=9&L=1.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Preliminary Validation of a Complex Aerospace Structure Michael Arviso, D. Gregory Tipton & Patrick S. Hunter Sandia National Laboratories1 P.O. Box 5800 Albuquerque, NM 87185
ABSTRACT A series of modal tests were performed on a complex aerospace structure, consisting of a shell structure with joints and discrete payloads, in order to validate a finite element model of the structure. Modal tests have been performed on individual assemblies followed by model updating using the measured modal data. The final configuration has placed all assemblies together as a complete unit which includes a multitude of joints and interfaces. Frequency response functions (FRFs) were chosen as the validation metric. INTRODUCTION The purpose of this work is to ascertain the validity of a finite element model by determining if it produces an accurate representation of the dynamic characteristics of the complex aerospace structure. Model validation is performed with the use of frequency response functions (FRF’s) as the independent validation metric. Test and model FRF’s were compared to determine the validity of the model. Several modal tests were performed with a variety of excitation inputs (locations, directions, and magnitude) to gather frequency response functions and mode shape information in order to update the model. These tests were performed in stages to aid in model correlation. Testing was conducted at multiple assembly levels starting with individual components and culminating in the complete structure. The final configuration includes a multitude of welds, rivets, brackets, bonds, and joints with uncertain material properties and joint stiffnesses. Data of interest ranged from 100 to 1000 Hz which enveloped the test environment for the structure. The complex aerospace structure has been dynamically tested with concurrent model updating for a three year period. In the first year a model was developed extremely quickly using new design through analysis tools that needed experimental data to be used to perform the validation. During this phase, the major steps included correcting modeling errors such as oversimplifications and connectivity issues. Initially the basic structure without any brackets or payloads was examined and tested in order to improve correlation and to isolate modeling issues. A multitude of material parameters were calibrated during this phase. Modal test data during this phase was used to compare modal frequencies and shapes to help improve the model. These tests were extremely helpful in bringing the model to better agreement with the structure. As we know, models are susceptible to errors, oversimplifications, incorrect assumptions, and unknown parameters that may be corrected only through a series of testing and calibration. In the second year, focus was on investigating the uncertainty in the model. There were modeling uncertainties associated with unknown parameter values as well as variability of manufacturing (part to part) parameters. Experimental uncertainty primarily included assembly variability, as only one structure is available for testing. A natural question was presented, which asks, as more uncertainty is added to the model, is it really easier to validate the model? The answer is yes, however the uncertainty used in validating the model must be carried
1
Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_64, © The Society for Experimental Mechanics, Inc. 2011
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forward when performing subsequent calculations. In doing so, a model will have been created that is not as useful, in part due to the high uncertainty [5]. Through testing it was shown that uncertainty included in the model was plausible [5]. The main question was, “Is it reasonable to include uncertainty in analytical models at all?” Well, real structures do have variability; therefore any model of a real structure should also include variability. Uncertainty investigated in the model included two material thicknesses, one adhesive modulus, and three joint stiffness properties.
MODEL The aerospace structure consists of an exterior shell surrounding interior bracing, which supports several brackets holding payloads. Figure 1 shows a simplified illustration of the structure. The bracing is attached to the exterior shell using a combination of rivets, bonds, and welds, which are not explicitly modeled. The brackets are attached to the bracing using bolted joints, which are represented in the model using one-dimensional spring elements in each degree of freedom. The payloads are bolted to the brackets, which are also represented in the model with springs. Forces were input into the structure in directions axial, normal, and tangential to the exterior shell and at one of the mounting feet.
Figure 1: Simplified Illustration of Structure nd
The finite element model of the structure is composed of 2 order elements, a mix of hexes, quads and beams resulting in 5.6 million degrees of freedom. The modes and FRFs of the structure were calculated using Salinas, a massively parallel structural dynamics code developed at Sandia National Laboratories, [1].
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743 TESTING Focus of the final year of testing rests with two main configurations, one being the final configuration. The first test was conducted on a portion of the structure with its associated payload, which was used as an intermediate step leading up to the final test. The second test was performed on the full structure with all the payloads and brackets in place. FRFs were recorded for both configurations and used as calibration data for the full system model. Modal parameters, including the frequencies, damping, and shapes were extracted and compared with the model predictions. The first test was conducted on a partial assembly of the structure using modal hammer impacts as excitation [2]. These inputs produced very clean FRF’s. Varying levels of amplitude produced nearly identical FRF’s; indicating that nonlinearity of the system was not present at these levels. Since the structure is basically axisymmetric, the shell ovaling modes show up as two lobed and three lobed modes. Four lobed ovaling frequencies were above the band of interest. Approximately 10 modes were extracted from the measured FRF’s, up to 1000 Hz, using the SMAC algorithm [3]. The final configuration test was performed on the full structure with all the payloads and brackets in place. This test configuration used excitation inputs from both modal hammers and a single modal shaker. The modal shaker used a continuous random input with Hanning windows applied to the data. To minimize the effect of the Hanning window on the measured damping, long time blocks were utilized. The input location and direction are more accurately controlled when a modal shaker is used instead of a modal hammer. This makes the FRFs obtained using the modal shaker input more suitable for use in the validation efforts to remove input uncertainty. Data acquisition systems were set to measure in the 0 to 2000 Hz range. Filters were used with a cutoff frequency of 3000 Hz. Hanning windows were used in conjunction with the shaker while no window was used with the modal hammer inputs. Mode shapes were measured with a total of 204 accelerometers arranged throughout the payloads and brackets along with several rows on the shell. A total of 30 modes were extracted and the damping ratios varied between ¼ and 2 percent. Driving point FRFs with inputs normal to the shell are overlaid and shown in Figure 2. Magnitude of the FRF is plotted from 100 to 1000 Hz, which is the frequency range of interest for this structure. As seen in the figure there are definitely two bands of modes that are present. The first band is seen in the 400 to 500 Hz region and is attributed to shell modes with some payload interaction. The second band of modes is shown in the 600 to 700 Hz region and is attributed to payloads and associated structure interacting with the shell.
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Figure 2: Driving point FRF for inputs normal to shell in two different locations
Driving point FRFs with inputs axial to the shell are overlaid and shown in Figure 3. As seen in the figure, there is only one band of modes that are present. This band is seen in the 400 to 500 Hz region and is attributed to shell modes with some payload interaction. The second band of modes that was presented before is not excited with the input axial to the shell.
Figure 3: Driving point FRF for inputs axial to shell in two different locations
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745 Driving point FRFs with inputs tangential to the shell are overlaid and shown in Figure 4. As seen in the figure there is definitely two bands of modes that are present once again as in Figure 2. The first band is seen in the 400 to 500 Hz region and is attributed to shell modes with some payload interaction. The second band of modes is shown in the 600 to 700 Hz region and is attributed to payloads and associated structure interacting with the shell.
Figure 4: Driving point FRF for inputs tangential to shell in two different locations
Driving point FRF with shaker input near one of the mounting feet are overlaid and shown in Figure 5 from two different load levels produced by the shaker. Again, magnitude of the FRF is plotted from 100 to 1000 Hz, which is the frequency range of interest for this structure. In this figure the two distinct bands of modes are not present as seen in the previous figures. The close match between these two data sets shows that nonlinearity is negligible, at least at these levels.
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Figure 5: Driving point FRF for inputs near one of the mounting feet with shaker at two different amplitude levels
MODEL UPDATING A preliminary finite element model would be free of errors in the ideal world; however this is not the case. In order to resolve errors, there must be some sort of test or test data available to compare against in order to update the model. In our previous work, there have been ongoing efforts and a number of separate procedures used to update the model in the following areas: material properties, material thicknesses, meshing, structural omissions, mass verifications, connectivity issues, mode verification, and animation to verify or calibrate the model. Efforts in early tests have shown all of the previously mentioned issues that were investigated led to greater insight into the test structure thus producing a more refined model of the aerospace structure. It is known that models are prone to have uncertainties in all these areas that may only be corrected through testing and model calibration as shown in Table 1 with the before and after analysis frequencies and test modes [4]. Note the largest error goes from 35% to 3.5% after calibration. The next step is evaluation of the calibrated model to ascertain the model’s validity.
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Mode Description
Test Frequency (Hz)
Analysis Frequency Before (Hz)
Error Before
Analysis Frequency After (Hz)
Error After
2,0 Ovaling / Payloads In Phase with Shell 2,0 Ovaling / Payloads In Phase with Shell 2,0 Ovaling / Payloads Out of Phase with Shell 2,0 Ovaling / Payloads Out of Phase with Shell Payloads Axial One Payload Rocking 3,0 Ovaling / All Payloads Out of Phase with Shell 3,0 Ovaling / 2 Payloads Out of Phase with Shell 3,0 Ovaling / 2 Payloads Out of Phase with Shell 3,0 Ovaling / 2 Payloads Out of Phase with Shell 3,0 Ovaling / 2 Payloads Out of Phase with Shell
259 271 299 321 348 527 644 678 696 702 722
173 175 257 280 236 406 618 634 657 698 717
-33% -35% -14% -13% -32% -23% -4% -6% -6% -1% -1%
261 273 301 322 356 535 644 672 694 678 739
0.7% 0.6% 0.6% 0.4% 2.2% 1.5% 0.0% -0.9% -0.3% -3.5% 2.3%
Table 1: Analysis Mode Updating – Subassembly of Main Structure
Uncertainty quantification can be applied in modeling and testing. Model verification including solution verification, uncertainty quantification, and two validation steps were used to compare the experimental and model results. Level of uncertainty does affect how the model is validated, but it also affects how the uncertainty is carried forward in subsequent predictions. The model was validated in both the full frequency range enveloped by the test parameters as well as the operational frequency range of interest [5]. Figure 6 depicts FRFs from a chosen validation point on the structure with experimental (blue) overlaid upon the analytical (red) FRFs with associated analytical uncertainty. Analytical FRF’s were generated for a variety of material thicknesses, adhesive modulus, and bolt stiffness parameters. Adding additional uncertainty, such as changing the material thicknesses, adhesive modulus, or bolt stiffnesses, to the analytical model only serves to remove the distinct peaks at the higher frequencies making it difficult to depict any modes at the higher end while the lower peak around 350 Hz is quite apparent. Experimental data showed variability in the 5.5% range while the variability of the analytical data approached the 10 to 15% range. The experimental variability only included assembly uncertainty, since only one structure is available for testing.
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10
|H(f)|
10
10
10
10
3
2
1
0
-1
10
2
10
3
Frequency (Hz) Figure 6: Validation Point 1, All Model Results (red) and Experimental Results (blue)
Shown below in Figure 7 is the comparison of the mean calibrated analytical model enveloped with the experimental results. Again, we are only analyzing data from one structure, so is the model valid? With only one structure, there is no means to find true variability of the structure. Because there is unit to unit variability, uncertainty must be included in the model. Even though the mean value realization is a good match to our experimental data, we cannot use this realization for all future analyses because the model was validated with uncertainty. In order to use the mean model realization, this one model would have to be validated to the experimental data, and as stated before, this is only one copy of the structure.
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10
|H(f)|
10
10
10
10
3
2
1
0
-1
10
2
10
3
Frequency (Hz) Figure 7: Validation Point 1, Calibrated Model Results (red) and Experimental Results (blue)
The final year of the project has placed the complex aerospace structure into the final assembly configuration for testing. As both the experimental and the analytical team move forward to the final assembly there are a multitude of joints, brackets, and payloads that must be instrumented to determine what interaction all the assemblies will have as a complete unit. Preliminary modes and descriptions as predicted by the analytical team will be presented in Table 2. As can be seen, a great deal of work needs to be performed in updating the final configuration in an effort to validate the finite element model. There are several modes that were not excited during the testing. Continuing efforts will be focused on updating the model and several additional tests are in order to perform the validation series on this structure.
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Mode Description Payloads axial Out of Phase Main Payload top rocking; Payloads axial In Phase Payloads axial In Phase Main Payload top rocking Axial – Main top & bottom Payloads Out of Phase Bending – Main bottom Payload rocking Bending – Main bottom Payload rocking 2,1 Ovaling – Payloads Lateral Out of Phase Small Payload & Bottom bracing Axial 2,1 Ovaling – Payloads Lateral In Phase Payloads and Small Payload Lateral In Phase Payload 1 Lateral – Small Payload Axial Torsion 2,2 Ovaling – Payload Lateral Out of Phase 2,2 Ovaling – Payload 1 Lateral – Small Payload Axial 2,2 Ovaling – Payload 1 Axial 2,3 Ovaling – Small Payload Axial 2,3 Ovaling Top bracing rocking
322
Preliminary Analysis Frequency (Hz) 323
-0.3%
438
325
-25.8%
331 459 448 430 436 510 609 551 558 771 638 522
326 327 415 462 467 533 550 561 578 599 618 625
-1.5% -28.8% -7.4% 7.4% 7.1% 4.5% -9.7% 1.8% 3.6% -22.3% -3.1% 19.7%
643
642
0.2%
675 700 891 728
674 675 697 767
-0.1% -3.6% -21.8% 5.4%
Test Frequency (Hz)
Error
Table 2: Preliminary Analysis Frequencies vs. Test Frequencies– Complete Assembly
DISCUSSION The third year focused on combining all the individual assemblies into a final configuration that may be used as a single representation of a field of identically produced complex aerospace structures. The final configuration serves as a means to collect experimental data which may be used to verify or calibrate the analytical model that will be separate from a test that is used to acquire validation data to rate or verify the finite element model. Data collected in the final year still resides as preliminary data that is to be used in the validation efforts. Validation metrics are still being considered and defined to ascertain the validity of the latest finite element model. Validation of a complex aerospace structure is by no means a finished project. There are still many ideas or approaches that may be considered to perform additional verification and validation experiments. Additionally, the focus of this series of experiments is to consider the range from 100 to 1000 Hz. As the model and structure move forward in areas of study, other areas of interest may arise. Some emphasis must be given to the testing circumstances in which only one test unit is available and is to serve as a representation of the entire field of complex aerospace structures. With this in mind, it is understood that there exists unit to unit variability as well as fabrication uncertainty.
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751 CONCLUSION Performing the instrumentation, data acquisition, testing, and analysis for both final configurations was by no means an easy task. The FRFs were collected and used as calibration data for the full system model. Modal parameters, including the frequencies, damping, and shapes were extracted and compared with the model predictions. A great deal of work exists in order to correlate the model to the experimental data as seen in Table 2, and the next step would be to agree to the type of test that needs to be performed to serve as the final validation test, which may be used to ascertain the validity of the model. There also exists the assumption of linearity that is used for modal analysis. While inputs are being driven at low levels this assumption is quite valid, but as inputs begin to approach higher levels the assumptions are no longer valid. Real environmental loads may drive nonlinearities that are not captured by this analysis. With this said, there still exists a large area of uncertainty which was not captured by this analysis. REFERENCES [1] Bhardwaj, Manoj; Pierson, Kendall; Reese, Garth; Walsh, Tim; Day, David; Alvin, Ken; Peery, James; Farhat, Charbel and Lesoinne, Michel, "Salinas: A Scalable Software for High-Performance Structural and Solid Mechanics Simulations", Supercomputing 2002. Baltimore, MD. Nov 2002. [2] Carne, T.G. and Stasiunas, E.C., “Lessons Learned in Modal Testing-Part 3: Transient Excitation for Modal Testing, More than just Hammer Impacts,” Experimental Techniques, vol.30, No.3, May/June 2006, pp 69-79. th [3] Hensley, D.P., and Mayes, R.L., “Extending SMAC to Multiple References,” Proceedings of the 24 International Modal Analysis Conference, pp.220-230, February 2006. [4] Rice, A. E., Carne, T. G., Kelton, D. W., "Model Validation of a Complex Aerospace Structure," Proceedings of IMAC XXVI Conference, SEM, Orlando FL, 2008 [5] Rice, A. E., Arviso, M., Paez, T. L., Carne, T. G., Hunter, P. S., "Uncertainty Quantification in Model Validation of a Complex Aerospace Structure," Proceedings of IMAC XXVII Conference, SEM, Orlando FL, 2009
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Modeling the Elastic Support Properties of Bernoulli-Euler Beams
T.A.N. Silva1,2, N.M.M. Maia2 1
Department of Mechanical Engineering, Instituto Superior de Engenharia de Lisboa Rua Conselheiro Emídio Navarro, 1, 1959-007 Lisboa, Portugal E-mail: [email protected] 2
Department of Mechanical Engineering, Instituto Superior Técnico Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal E-mail: [email protected]
ABSTRACT Considering the transverse vibration problem of a machine rotor, the authors deal with the free vibration of elastically restrained Bernoulli-Euler beams. Based upon Rayleigh’s quotient, an iterative strategy is developed to identify the approximated torsional stiffness coefficients, which allows to bringing together experimental results obtained through impact tests and the ones of the theoretical model. The proposed algorithm treats the vibration of continuous beams taking into account different stiffness coefficients at the left end side and intermediate supports and the effect of attached mass with inertia at the free beam tip, not just on the energetic terms of the Rayleigh’s quotient but also on the mode shapes, considering the shape functions defined in branches. A number of loading cases are studied and examples are given to illustrate the validity of the model and the accuracy of the obtained natural frequencies. KEYWORDS: Transverse vibration of beams; Elastic supports; Torsional stiffness coefficients.
1
INTRODUCTION
The study of beam-like components that present cross section variations along the length and carry concentrated masses and/or springs is often addressed by means of approximated numerical methods, like Rayleigh’s quotient. The accuracy of such an approach depends on the chosen shape function, according to Rayleigh’s theorem [1]. Figure 1 shows the physical system under consideration, a rotor with a disc at the right end side. The objective of the present work is to develop an accurate model of the system in order to replicate the experimental natural frequencies in lateral bending. While the rotor itself presents no problem and can easily be studied as a Bernoulli-Euler beam with a concentrated mass, the identification of the adequate torsional and linear stiffness properties associated to the end supports remains a challenge. In a similar context, De Rosa et al. [2] suppose the beam elastically restrained against rotation and translation at both ends, so that it is possible to study all the common boundary conditions. Those authors show that trigonometric functions work slightly better than the static deflections and highlight the accuracy of Rayleigh’s quotient to the true frequencies. References [3-6] present exact solutions for the frequency equation of a Bernoulli-Euler beam restricting the stiffness coefficients, in order to reproduce some particular cases, and accounting for the rotation inertia of attached discs and their eccentricity. Similar problems are treated in [7, 8], with intermediate supports. Wu and Chen [9] studied the bending vibrations of wedge Bernoulli-Euler beams with any number of point masses, verifying that the mass distribution on the beam affects the dynamical behavior more than the mass addition itself. The works of Biondi and Caddemi [10, 11] treat beams with
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_65, © The Society for Experimental Mechanics, Inc. 2011
753
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Figure 1: System under study (MFS, SpectraQuest, Inc.).
discontinuities in the curvature or in the slope functions, allowing the adoption of different materials or different cross sections in the model. Elishakoff and Pentaras [12] worked on the free vibration of non-homogeneous Bernoulli-Euler beams using a polynomial approach for the mode shape of the natural frequency of interest. The developed formulation does not require numerical tackle and provides a useful tool to the design phase. The above referred papers deal with slender beams, verifying l / D > 10 . As a consequence, the inertial rotation energy of an infinitesimal element dx is negligible compared to its translational energy, justifying the applicability of the Bernoulli-Euler beam theory. Sometimes, when this principle is violated, the problem has to be treated using Timoshenko beam theory [13-15]. In the present work one shall use Rayleigh’s quotient, with shape functions that have into account the torsional flexibility of the supports, while assuming infinite rigidity along the vertical direction (figure 2). The algorithm will also include the effect of the mass and torsional inertia of the attached disc. Such an approach aims at a more precise model that reproduces better the real dynamic behavior of the system, rather than assuming simply supported or perfectly clamped ends. After showing that the simply supported case is not the best way to model the physical system, the updating procedure for identifying the value of the torsional stiffness of both supports will be explained.
z , w( x, t )
kt2
kt1
m, J
x a
l
Figure 2: Model of a beam with elastic torsional supports and a rigid disc at x = l.
2
THEORETICAL BACKGROUND
To obtain an analytical model that reproduces the physical system (figure 1) one considers the Rayleigh’s quotient, using the Bernoulli-Euler beam theory to define the shape functions. On a first attempt one tries the simply supported (SS) case. However, the results for this ideal case differ from the experimental ones, especially for the second natural frequency. Therefore, one proposes to build a model considering that the supports of the beam present different torsional stiffnesses kti (elastic supports - ES), whose values are iterated until the error between experimental and numerical results satisfies a given tolerance τ.
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Regarding Rayleigh’s method, the energy dissipation is neglected and therefore the Principle of Conservation of Energy holds: Δ (T + V ) = 0
⇒
T m ax = V m ax
(1)
where T and V are the kinetic and potential energies, respectively. Through the quantification of the work performed by the elastic forces, in relation to the equilibrium position of the beam, the potential energy is
V=
1 l M ( x , t ) dθ 2 ∫0
(2)
where M ( x , t ) is the bending moment given by M ( x , t ) = E ( x ) I ( x ) ∂ 2 w ( x , t ) ∂ x 2 , l is the length of the beam and θ is the rotation angle given by θ ( x, t ) = ∂ w( x, t ) ∂ x . E( x) is the Young modulus and I ( x) the second moment of area of the beam; w( x, t ) represents the lateral displacement of the beam in relation to its equilibrium position. Under undamped free vibration conditions, the time variation of w( x, t ) can be shown to be harmonic and therefore the displacement response is given by the harmonic variation of a shape function φ ( x) :
w ( x , t ) = φ ( x ) sin(ω t + ϕ )
(3)
As w ( x , t ) max = φ ( x ) , it follows that 2
Vmax =
⎛ d 2φ ( x ) ⎞ 1 l E ( x ) I ( x ) ⎜ ⎟ dx 2 2 ∫0 ⎝ dx ⎠
(4)
On the other hand, the kinetic energy is given by
T=
1 l w& ( x, t )2 d m ∫ 0 2
(5)
From (3), w& ( x , t ) max = ω φ ( x ) . As d m = ρ ( x ) A( x ) dx , where ρ ( x) is the density of the material and A( x) the cross sectional area of the beam, one has
Tmax = ω 2 2.1
1 l ρ ( x ) A( x) φ 2 ( x) d x 2 ∫0
(6)
Rayleigh’s quotient
Rayleigh’s quotient, R , results from the application of (1) and therefore, for a beam with no extra masses and springs, equating (4) and (6) leads to
R = ω2
∫ =
l 0
∫
E ( x) I ( x) (φ ′′( x) ) d x 2
l 0
ρ ( x) A( x) φ 2 ( x) d x
(7)
where prime denotes differentiation with respect to the spatial coordinate x. Rayleigh’s quotient has several interesting properties from the numerical point of view. One of them is the upper bound approximation for the natural frequency value, as long as one provides a shape function φ ( x) close enough to the true mode shape, respecting at least the geometric boundary conditions of the problem. As a first order variation on φ ( x) corresponds to a second order variation on ω 2 , Rayleigh’s quotient has a stationary value (a
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minimum in this case) in the neighborhood of the true mode shape. This is verified not only for the fundamental natural frequency, but for all of them, i.e., a good approximation for the nth shape function φ n ( x ) leads to a good approximation for the nth natural frequency ω n . Taking into account the possible existence of concentrated parameters along the length of the beam, such as springs and masses (and corresponding inertias), one can rewrite (4) and (6). Therefore, the total maximum value for the potential energy will be given by
Vmax = Vb + ∑Vki + ∑Vkt i
j
(8)
j
where Vb is the potential energy of the beam itself, given by (4), and
Vki =
2 1 1 k i φ n2 ( xi ) , Vkt = k t j φ ′n ( x ) x= x j j 2 2
(9)
are the potential energies of each local translational and torsional stiffnesses k i and k t , respectively. j The maximum kinetic energy will be given by
Tmax = Tb + ∑ Tmr + ∑ TJ s r
(10)
s
where Tb is the kinetic energy of the beam itself, given by (6), and
Tmr =
2 1 2 1 ω mr φ n2 ( xr ) , TJ s = ω 2 J s φ ′n ( x ) x = x s 2 2
(11)
are the kinetic energies of each local mass m r and inertia J s , respectively. Hence, Rayleigh’s quotient applied to a general case, will be defined as
R = ωn2 =
∫
l 0
∫
E ( x) I ( x) (φn′′( x) ) d x + ∑ ki φn2 ( xi ) + ∑ kt j φ ′n ( x ) 2
i
l 0
2
j
x=x j
ρ ( x ) A( x ) φn2 ( x ) d x + ∑ mr φn2 ( xr ) + ∑ J s φ ′n ( x ) x = x 2
r
s
(12)
s
Note that in our case study, the beam is uniform and the translational springs have infinite stiffnesses. Thus,
EI ∫ (φ ′′n ( x ) ) d x + ∑ kt j φ ′n ( x ) 2
l
R = ωn2 =
0
x= x j
ρ A∫ φ 2n ( x ) d x + ∑ mr φn2 ( xr ) + ∑ J s φ ′n ( x) x = x 2
l
0
2.2
j
2
r
s
(13)
s
Shape functions
In Rayleigh’s quotient the selection of the shape function may have a significant impact on the solution of the problem. The question is how to choose such an appropriate shape function. This should at least verify the geometric boundary conditions, although more accurate results are expected if it verifies the natural boundary conditions too. Moreover, it is also known that, in general, trigonometric functions provide better results than polynomial functions or functions based upon static deflections. Certainly a good choice would be to take the shape function that results from the solution of the equilibrium equation of the Bernoulli-Euler beam with uniform material and cross section in free vibration:
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∂ 4 w( x, t ) ∂ 2 w( x, t ) c + =0 ∂x 4 ∂t 2 2
(14)
where c 2 = ( EI ) ( ρ A) and w( x, t ) = φ ( x) T (t ) ; the shape function φ ( x) is given by
φ ( x ) = C 1 sin( β x ) + C 2 cos( β x ) + C 3 sinh( β x ) + C 4 cosh( β x ) where ‘C’ are constants and β is related to the natural frequency through ωn = ( βn l )
2
(15)
( EI ) (ρ Al4 ) . Applying the
set of boundary conditions to (15) leads to an eigenproblem, whose eigenvalues are β n l (n = 1, ... ∞ ) and whose eigenvectors are constituted by the constants ‘C’. From the eigenvalues, the natural frequencies are calculated and, from the eigenvectors, the mode shapes are defined as a function of the eigenvalues, as
φ n ( x ) = C ( n ) sin( β n x ) + C ( n ) cos( β n x ) + C ( n ) sinh( β n x ) + C ( n ) cosh( β n x ) 1
2
3
4
(16)
For the problem under study (figures 1 and 2), several sets of boundary conditions were assumed in order to reproduce the physical system conditions. As one considers an intermediate support at x=a, one must complement the boundary conditions with the correspondent continuity equations. As a result, φ ( x) should be defined as a piecewise function, as follows:
⎧⎪{φ n ( x )}1 for x ∈ [ 0, a ] ⎪⎩{φ n ( x )}2 for x ∈ [ a , l ]
φn ( x) = ⎨
(17)
with
{φ n ( x )}1 = C ( n ) sin( β n x ) + C ( n ) cos( β n x ) + C ( n ) sinh( β n x ) + C ( n ) cosh( β n x )
(18)
{φ n ( x )}2
(19)
1
2
3
4
= C 5( n ) sin( β n x ) + C 6( n ) cos( β n x ) + C 7( n ) sinh( β n x ) + C 8( n ) cosh( β n x )
In the next sub-sections the considered sets of boundary conditions and continuity equations are described. 2.2.1
Simply supported beam (SS)
To reproduce this ideal case one assumes ki = ∞ ∧ kt j = 0 for both beam supports, placed at x = 0 ∧ x = a , which leads us to the following set of boundary conditions:
∂ 2 w( x, t ) w(0, t ) = ∂x 2 ∂ 2 w( x , t ) ∂x 2
= x=l
=0
(20)
x =0
∂ 3 w( x , t ) =0 ∂x 3 x = l
(21)
and continuity equations:
{φn ( x)}1 x =a = {φn ( x)}2 x =a = 0
(22)
{φ ( x)}
= {φn' ( x)}
(23)
{φ ( x)}
= {φn'' ( x)}
(24)
' n '' n
1 x =a
1 x =a
2 x =a
2 x =a
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2.2.2
Beam on elastic supports (ES)
Considering a more realistic approach, the proposed model consists of a solution with a beam on elastic supports (ES) carrying a disc at its free end. The expression “elastic support” designates a support where k i = ∞ ∧ k t > 0 j
with the boundary conditions given by:
w (0, t ) = 0
∂ 2 w( x, t ) ∂x 2
EI
∂ 2 w( x , t ) ∂x 2
EI EI
∂ 3 w( x, t ) ∂x 3
(25)
= kt1
∂w( x, t ) ∂x x =0
(26)
= −Js
∂ 3 w( x , t ) ∂x ∂t 2
(27)
x =0
x =l
= mr x=l
∂ 2 w( x, t ) ∂t 2
x=l
(28) x =l
and continuity equations:
{φn ( x)}1 x =a = {φn ( x)}2 x =a = 0
{φ ( x)} ' n
{φ ( x)} '' n
1 x=a
1 x =a
(29)
= {φn' ( x)}
(30)
2 x =a
− kt2 {φn' ( x)}
1 x= a
= {φn'' ( x)}
(31)
2 x= a
Note that for this case we consider two configurations:
3
•
Beam on elastic supports (ES1) with k t = k t ; 1 2
•
Beam on elastic supports (ES2) with k t ≠ k t . 1 2
NUMERICAL PROCEDURE
With the purpose of identifying the elastic properties of the beam supports, a numerical procedure has been developed, bearing in mind the reconciliation between the numerical results for the first natural frequency and the experimental ones ( ω 1 ). To start this updating procedure, one assumes as a first approximation the solutions EXP
for a simply supported case (SS), β 1 l
SS
and φ1 ( x )
SS
. From (13) and for iteration p = 1, the torsional stiffness
coefficients are given by:
kt1
(1)
= k t2
(1)
=
(
ω 21 EXP ρ A∫ φ 1 ( x) SS d x + m {φ1 ( x)}2 l
2
0
2 SS @ x = l
+ J {φ1 ( x)}2
2 SS @ x = l
) − EI ∫ φ ′′( x) l
0
1
2 SS
{φ1 ( x)}1 SS @ x =0 + {φ1 ( x)}1 SS @ x = a 2
2
In an iterative scheme, one shall use the solutions for a beam on elastic supports, β 1 l calculating, when required, different stiffness coefficients k t1 and k t2 , as follows:
ES
dx (32)
and φ 1 ( x )
ES
,
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Step 1. with β 1 l
and φ 1 ( x )
2
as function of k t ( p ) and k t ( p ) , compute k t ( p + 1 ) : 1 2 1
⎝
2
0
{φ (x)} ' 1
Step 2. similarly, with β 1 l
ES
and φ 1 ( x )
ES
2
⎝
{φ (x)} ' 1
EXP
− ω1
ES1 the procedure is limited to one step, where k t 1 4
as function of k t
0
(
(33)
1 ES @ x = 0
2
This process is repeated until the criterion ω1
⎞ − ⎛ EI l φ ′′( x) 2 d x +k ( p ) φ ' ( x) 2 { 1 }1 ES @ x=a ⎞⎟⎠ ⎟ ⎜ ∫0 1 ES t2 ES @ x = l ⎠ ⎝
2
2
ω 21 EXP ⎛⎜ ρ A∫ φ 1 ( x) ES d x + m{φ1 ( x)}2 ES @ x =l + J {φ1' ( x)}2 l
kt2 ( p +1) =
ES
ω 21 EXP ⎛⎜ ρ A∫ φ 1 ( x) ES d x + m{φ1 ( x)}2 ES @ x =l + J {φ1' ( x)}2 l
kt1 ( p +1) =
ES
( p +1) 1
and k t
( p) 2
, compute k t ( p + 1) : 2
⎞ − ⎛ EI l φ ′′( x) 2 d x +k ( p +1) φ ' ( x) 2 { 1 }1 ES @ x=0 ⎞⎟⎠ ⎟ ⎜ ∫0 1 ES t1 ES @ x = l ⎠ ⎝
2
2
(34)
1 ES @ x = a
ES
( p + 1)
)
2
< τ is verified,
= k t2
( p + 1)
τ being a given tolerance. For the case
is obtained as described in (32).
EXPERIMENTAL APPLICATION
As case study setup one uses the Machinery Fault Simulator of figure 1, with A ≈ 1.27 × 10−4 m2 , E = 73 GPa , I ≈ 1.28 × 10 −9 m 4 and ρ ≈ 2.77 × 10 3 kgm −3 . From this setup, our study is focused on the modeling of the beam with an intermediate support in a certain position a ∈ [ 0, l ] . The disc one uses is placed at x = l and has
m ≈ 1.6 kg and J ≈ 1.5 × 10 −3 kg.m 2 . The modal data, namely the natural frequencies were obtained by impact tests on the test equipment. Those results were used as inputs, under the designation ω n
EXP
, on the numerical
model with the aim of updating the value of the torsional stiffnesses k t , besides allowing the comparison of j results. All the experiments were performed with the objective of building a straightforward tool for mechanical design. 5
RESULTS AND DISCUSSION
Table 1 shows the results obtained for the first 2 natural frequencies, for the ideal case (SS) as well as for the two considered configurations of the elastically restrained beam (ES1 and ES2). The experimental values of the natural frequencies presented in table 1 are average values obtained from at least three impact tests. As it can be observed, the experimental value for the fundamental frequency does not differ very significantly from the ideal case. However, the experimental value for the second natural frequency is quite different from the obtained for the SS case, justifying the application of a model with elastic supports. The calculation of k t and k t has been made for the cases mentioned in section 2.2.2, ES1 and ES2, following 1
2
the iterative procedure explained in section 3, with the aim of matching the value of the first natural frequency. In order to validate the results obtained for the stiffnesses, one has used them to predict the value of the second natural frequency. Also in table 1, the summary of the obtained results, predictions and relative errors is showed. Therefore, one has for each ES case, the value of the measured natural frequencies, the results of k t and k t 1
2
(after a certain number of iterations p), the correspondent predicted values for the second natural frequency and the resulting relative error.
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Figure 3 shows the evolution of k t and k t values and the resulting value of the fundamental frequency, during the iterative procedure for each ES case. 1
2
Table 1: Comparison between the theoretical values of the natural frequencies for each case study – SS, ES1 and ES2 – and the experimental results.
ω1 EXP [Hz] 26.023
ω2
EXP
[Hz]
199.636
Case #
n.º of Iter. ( p end )
SS
-
ES1
7
ES2
6
ω 1 # [Hz]
ω 2 # [Hz]
(Error [%])
(Error [%])
24.957 (4.1) 26.023 (0.0) 26.023 (0.0)
72.570 (63.6) 441.739 (121.3) 200.530 (0.4)
kt1 [Nmrad-1]
k t2 [Nmrad-1]
-
-
234.172
234.172
259.483
248.296
(b)
(a) Figure 3: Evolution of k t (a) and ω 1 j
#
(b) for the ES case.
With respect to the prediction of the second natural frequency value one should emphasize that the relative error is considerably lower for the result of the ES2 case, even negligible. This fact highlights how important is to consider different stiffnesses coefficients at each support of the beam. Another significant consideration is the inclusion of mass and inertia elements not just in the energetic terms of the Rayleigh’s quotient but also in the mode shapes. Figure 4 intends to illustrate the influence of those elements in the resulting first 4 mode shapes, where solid lines represent the mode shapes obtained for a beam on elastic supports with the free end (case not mentioned in section 2.2) in opposition to the ones obtained for the ES3 case (dashed lines).
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Figure 4: Mode shape deviation due to considering a disc at x=l: with disc (dashed line) and without disc (solid line).
6
CONCLUSIONS
In order to predict the dynamic behavior of a shaft mounted on elastic supports, with a disc at the free end, a theoretical approach has been developed. From this study, some conclusions can be drawn: - Modeling a beam on elastic supports is more accurate than assuming it as simply supported. Moreover, assuming that those stiffnesses are different from each other also leads to better results; - The inclusion of mass and inertia of the disc in the mode shapes has a significant effect on the natural frequencies values; - It has been proven that after calculating the torsional stiffnesses coefficients based on the experimental value of the first natural frequency it is possible to predict with sufficient accuracy the value of the second natural frequency.
ACKNOWLEDGMENTS The authors would like to acknowledge the support of their colleague António Roque. Part of this work was supported by FCT, under the project POCI 2010 and the PhD grant SFRH/BD/44696/2008.
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REFERENCES [1]
Meirovitch, L., Fundamentals of vibrations, McGraw-Hill, Boston, 2001, ISBN 0-07-118174-1.
[2]
DeRosa, M., Franciosi, C. and Maurizi, M., On the dynamic behaviour of slender beams with elastic ends carrying a concentrated mass, Computers & Structures, Vol. 58, No. 6, pp. 1145-1159, 1996.
[3]
DeRosa, M. and Auciello, N., Free vibrations of tapered beams with flexible ends, Computers & Structures, Vol. 60, No. 2, pp. 197-202, 1996.
[4]
Auciello, N., Transverse vibrations of a linearly tapered cantilever beam with tip mass of rotary inertia and eccentricity, Journal of Sound and Vibration, Vol. 194, No. 1, pp. 25-34, 1996.
[5]
Nallim, L. G. and Grossi, R. O., A general algorithm for the study of the dynamical behaviour of beams, Applied Acoustics, Vol. 57, No. 4, pp. 345-356, 1999.
[6]
Maiz, S., Bambill, D. V., Rossit, C. A. and Laura, P., Transverse vibration of Bernoulli-Euler beams carrying point masses and taking into account their rotatory inertia: Exact solution, Journal of Sound and Vibration, Vol. 303, No. 3-5, pp. 895-908, 2007.
[7]
Grossi, R. and Albarracn, C., Eigenfrequencies of generally restrained beams, Journal of Applied Mathematics, Vol. 2003, No. 10, pp. 503-516, 2003.
[8]
Albarracin, C., Zannier, L. and Grossi, R., Some observations in the dynamics of beams with intermediate supports, Journal of Sound and Vibration, Vol. 271, No. 1-2, pp. 475-480, 2004.
[9]
Wu, J.-S. and Chen, D.-W., Bending vibrations of wedge beams with any number of point masses, Journal of Sound and Vibration, Vol. 262, No. 5, pp. 1073-1090, 2003.
[10] Biondi, B. and Caddemi, S., Closed form solutions of Euler-Bernoulli beams with singularities, International Journal of Solids and Structures, Vol. 42, No. 9-10, pp. 3027-3044, 2005. [11] Biondi, B. and Caddemi, S., Euler-Bernoulli beams with multiple singularities in the flexural stiffness, European Journal of Mechanics - A/Solids, Vol. 26, No. 7, pp. 789-809, 2007. [12] Elishakoff, I. and Pentaras, D., Apparently the first closed–from solution of inhomogeneous elastically restrained vibrating beams, Journal of Sound and Vibration, Vol. 298, No. 1-2, pp. 439-445, 2006. [13] Lee, S. Y. and Lin, S. M., Vibration of elastically restrained non-uniform Timoshenko beams, Journal of Sound and Vibration, Vol. 183, No. 3, pp. 403-415, 1995. [14] Posiadała, Free vibrations of uniform Timoshenko beams with attachments, Journal of Sound and Vibration, Vol. 204, No. 2, pp. 359-369, 1997. [15] Lin, S. C. and Hsiao, K. M., Vibration analysis of a rotating Timoshenko beam, Journal of Sound and Vibration, Vol. 240, No. 2, pp. 303-322, 2001.
BookID 214574_ChapID 66_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Noise Reduction of Continuously Variable Transmission (CVT) for Automobile Nobuyuki OKUBO, Hiroki AIKAWA and Takeshi TOI CAMAL, Department of Precision Mechanics, Chuo University 1-13-27 Kasuga, Bunkyo-ku, Tokyo, 112-8551, Japan E-mail:[email protected]
Yasuhide HIRABAYASHI, Akihisa TSURUTA AISIN AW
Abstract: The continuously variable transmission, CVT for automobile, which consists of the primary and secondary pulley connected by the steel belt with many thin plate elements inserted in line, is becoming popular because of efficient fuel consumption, smooth variation and acceleration.
But due to engagement and
disengagement between the pulley and element, a distinctive noise may be generated according to the revolution. In this paper such distinctive noise at specific frequency is confirmed using a test rig of CVT and also the Operational Deflection Shape, ODS of the belt at very limited points is measured due to difficult access of non- contact sensor in the test rig. Then the FE model of the CVT is created and updated to match the FRF calculated and measured. Based on the FE model and measured ODS at very limited points, the excitation force can be assumed during the rotation that causes the ODS and generates the noise. Finally alternative element design is proposed to reduce the distinctive noise and verified by the actual measurement. 1. Introduction Recently due to the global warming phenomenon, the reduction of Co2 and the fuel consumption of automobile become very important concern and thus the continuously variable transmission, CVT is installed in many automobiles because of its advantages. However the structure of CVT is different from the ordinary AV and therefore it may generate the distinctive noise caused by the CVT belt. In common cases in order to take counter measures to solve the problem, the transmission case is reinforced or the noise absorbing or insulating materials is used to cover the body. In this paper the thin plate elements inserted in line of the CVT belt are focused and modified its shape to improve the noise. First based the FE model of the CVT, the excitation force acted on the belt during rotation is identified. Then the alternative element design is proposed to utilize the FE model with the identified excitation force and verified by the actual measurements.
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_66, © The Society for Experimental Mechanics, Inc. 2011
763
BookID 214574_ChapID 66_Proof# 1 - 23/04/2011
764 2. Dynamic response of CVT during rotation CVT consists of the primary pulley, the secondary pulley and the steel belt with about 400 thin plate elements. Fig.1 shows the test rig in this study. The primary pulley is driven by the motor and the secondary pulley is followed through the belt and reduced the speed by the brake. The rotation speed available in the rig is stationary and ranged from 0 to 200 rpm. In general the distinctive vibration and noise are generated at specific order component that the contact circumference of the pulley is divided by the thickness of the element. In this study the thickness of the element is uniformly 1.5 mm and therefore the 117th order should be focused. Also the rotation speed is set maximum 200 rpm and then the vibration and noise at basic frequency 200/60x117=390 Hz should be discussed. First the distinctive noise is confirmed to measure the acoustic pressure at 3 cm apart from the primary and secondary pulley. Fig.2 shows the result where 390 Hz noise is revealed as the distinctive noise component. Then the vibration response of the element of belt during rotation is measured in normal direction with non-contact gap sensor at 5 element interval. As a result the operational deflection shape, ODS at 200 rpm namely 390 Hz is obtained as shown in Fig.3 which indicates that the deflection is large at disengagement part. 3. Excitation force identification 3.1 Force identification method In order to propose the alternative element design to improve the distinctive noise, not only the model of the CVT but also the excitation force during rotation should be identified. The ODS during rotation can be written as the linear superposition of each mode shape using the modal coordinate as follows,
{x} = [φ ]{ξ }
(1)
φ is the mode shape and ξ is the modal coordinate namely the modal weight of each mode shape. Further ξ can be expressed as,
where X is the ODS,
1
ξr = where m r is the r th modal mass,
mr
− ω + 2 jς r ω r ω + ω 2
2 r
{φr }T {F }
(2)
ω is the angular frequency, ω r is the r th natural angular frequency, ς r is
the r th modal damping ratio and φ r is the r th mode shape component at the excitation point. It is necessary to guarantee the accuracy of force identification because the measurable points of CVT are
BookID 214574_ChapID 66_Proof# 1 - 23/04/2011
765 very limited to the upper part of disengagement of element at the secondary pulley due to difficulty to access. Therefore by using a simple plate model the accuracy of force identification is verified. 3.2 Verification of force identification Since the ODS of the CVT at 200 rpm is the linear superposition of several mode shapes, the specific frequency of the simple model at which the excitation force is identified is set not to coincide one of the natural frequency but involve several mode shapes. First the model testing is applied to the plate as shown in Fig.4(a), where an exciter with a load cell is attached and the acceleration pick up is used to measure the response. Fig.5 shows the typical FRF and many mode shapes are observed. Then 220 Hz is chosen as the specific frequency to include several mode shapes. In order to discuss the accuracy of force identification with respects to the number and location of measurement points, two cases are dealt with, 12 points (b) and 4 points (c) as shown in Fig.4. Fig.6 shows the comparison of measured and calculated force at 220 Hz that indicates 90 % accuracy in case of 12 points while 50 % in case of 4 points. 4. Modal analysis and ODS prediction of CVT 4.1 Modal analysis An exciter is used to excite the CVT at the secondary pulley and LDV to measure the response of the element in normal direction. Fig.7 shows the typical FRF and 127, 297, 384, 400, 441 and 485 Hz mode shape are considered to contribute the ODS at 200 rpm namely 390 Hz that does not coincide any mode. According to the equation (1), the modal coordinate that is the modal weight can be calculated by inverse manner knowing the ODS and mode shapes and shown in Fig.8, which proves the necessity to take into account several mode shapes. 4.2 ODS prediction In order to propose the alternative element design to improve the distinctive noise, try and error approach wastes much time and cost and thus the ODS prediction based on numerical simulation such as FE analysis provides an efficient one. As shown in Fig.9 the CVT is so modeled where the belt is expressed as solid with uniform cross section corresponding to the element shape, the connections between the element and the belt and between the belt and the pulley are idealized by springs and the pulley is supported by springs at both shaft ends. The model updating is utilized to tune the spring constants to match the measured and calculated natural frequencies and mode shapes. Fig.10 shows the typical mode shape pair between measured and calculated to confirm a good agreement. Not like the resonance problem that only one mode shape plays the role and consequently the force identification to know where and how the excitation force acts is not necessary, the excitation force should
BookID 214574_ChapID 66_Proof# 1 - 23/04/2011
766 be identified where and how. Therefore the excitation force to generate the ODS is assumed at the engagement part of primary pulley, two points in x, y and z direction as shown in Fig.11. Base on the identified force using the equation (2) and the original FE model before element modification, the ODS under rotation can be predicted as shown in Fig.12 compared with measured one and a good agreement is observed. 5. Prediction of improvement by element modification 5.1 ODS prediction For alternative modification design of element shape, there is a constraint not to change the performance of the CVT and therefore the contact surface between the belt and the element and that between the element and the pulley are kept the same, on the contrary the top and the bottom shape may be modified. Because of easy manufacturing, both are slightly cut as shown in Fig.13 hereafter denoted as Modification 1. Then the ODS of the belt at 200 rpm against the same excitation force identified above is predicted and shown in Fig.14, where the Modification 1 results larger ODS and thus the noise is concluded to increase. 5.2 Verification for Modification 1 element In order to verify the prediction of Modification 1 element, actually modified elements are installed in the belt and the CVT is operated under the same condition. The distinctive noise of Modification 1 element at 390 Hz is measured as shown in Fig.15 compared with the original element,and it is observed 2 dB(A) increase as predicted. 5.3 Noise improvement The FE model with identified excitation force is confirmed to be able to predict acceptable ODS and the Modification 1 is found the opposite way to improve the noise. As a result Modification 2 element where the top part is enlarged about 1 mm as shown in Fig.16 is proposed. The predicted ODS of Modification 2 element is shown in Fig.17 and it is observed maximum displacement is suppressed and consequently noise reduction can be expected. 6. Conclusions 1) The distinctive noise of CVT under rotating condition is confirmed at specific frequency. 2) The excitation force can be identified to a certain degree even if the number and location of measurement point are limited and several mode shapes are contributed on the ODS. 3) A proper FE model of the CVT and excitation force under rotating condition are obtained for ODS prediction. 4) Based on such ODS prediction, the shape of element in the belt is dealt with to improve the noise.
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767 References [1] K.Tsukuda,et al,”Toyota’s New Belt-drive Continuously Variable Transaxle for 1.3-liter FWD cars”,SAE Technical Paper 2006-01-1305. [2] H.Tani,et al,”A Study on the Behavior of a Metal V-belt for CVTs”,CVT.HYBRID 2007 Yokohama. [3] H.Shimizu, et al,”Delelopment of 3D Numerical Simulation of Dynamic Behavior of Metal V-belt”,JSAE No.8-99(in Japanese). [4] M.Hodate, et al,”Develop of a Method for Analyzing CVT Casing Radiated Noise Induced by Belt Excitation Forces”,2005 SAE International 2005-01-1460.
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768
Z
Primary Pulley Motor
X
Y Input point Response point (a) Setup for measurement of FRF
Brake Secondary Pulley Fig. 1 Test rig of CVT
(b) 12 point Fig. 4
Setup of experiment (simple model)
Primary Secondary
10dB(A) 390Hz
10000
Frequency Hz
450
220Hz
0.1
Fig. 2 SPL at 200rpm
343Hz 392Hz 458Hz 497Hz 642Hz 745Hz 150Hz 172Hz
FRF m/s2/N
SPL dB(A) 350
(c) 4 point
0
Frequency Hz Fig. 5
20 40
1
FRF of simple model
0.8
Secondary Pulley
Force N
30
10
1024
50 60 70
0
Fig. 3 Operational deflection shape at 200rpm
Fig. 6
Measurement
12 point
4 point
Comparison of measured and calculated force
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769 127Hz 297Hz 384Hz 400Hz 441Hz
FRF m/s2/N FRF
485Hz
EMA 0
Frequency Hz Fig. 7
1024
EMA:384Hz FEM:390Hz
FRF of CVT
Fig. 10
Mode shape pair
W eight m
3e-7 Pulley
A
Element
A
0
-1.5e-7 297 384 400 441 Natural Natur al frequency Hz
127 Fig. 8
485
(a) Over view
Modal weight of CVT at 200rpm
Fig. 11
(b) Section A-A
Assumed excitation force position
Displacement m
0.5e-8m
Experiment Prediction 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 1
Fig. 9
Position
FE analysis model Fig. 12
Comparison of ODS
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770
Fig. 13
Displacement m
0.5e-8m
Structural modification1
Original Structural modification2
0.5e-7m
Displacement m
75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 1
Fig. 17
Original Structural modification1 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 1
Position
SPL dB(A)
Fig. 14
Comparison of ODS
2dB(A)
390Hz
10dB(A)
Original Structural modification1 350 Fig. 15
Frequency Hz
450
Comparison of SPL at Primary Pulley
Fig. 16
Structural modification2
Position
Comparison of ODS
BookID 214574_ChapID 67_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Smirnov Stationarity Criterion Applied to Rocket Engine Test Data Analysis
Stéphane Muller, Astrium Space Transportation, 66, Route de Verneuil 78133 les Mureaux Cedex France Yves Mauriot, ONERA, BP72 - 29, avenue de la division Leclerc – 92322 Chatillon Cedex France
Nomenclature FFT NPSP PSD RAT RMS
Fast Fourier Transformation Net Positive Suction Pressure Power Spectral Density Reverse Arrangements Test Root Mean Square
ABSTRACT The problem of the stationarity of test measurements used to identify a transfer function by PSD calculations is addressed. Several stationarity criteria are compared with a special focus on the Smirnov test. A methodology based on this test is then proposed and experimented on the identification of a turbopump transfer function.
I. INTRODUCTION The use of the well known FFT, for example for PSD calculations, is widely spread and is so commonly used that some basic constraints such as the stationarity condition are sometimes misjudged or underestimated by engineers; this leads of course to biased results. A typical situation was experienced by the authors during the identification of a rocket pump transfer function using fire test data. Due to the partial cavitation of the pump, the frequency content of the pressure transducers measurements was rather dense and fluctuating (see Figure 1): the question of the stationarity of the signal had therefore to be addressed and an extensive analysis of this aspect appeared to be mandatory before any use of FFT. As a first step, some engineering judgment criteria were used to separate the stationary and unstationary whiles of the signal. But it appeared rapidly more fruitful to test and use mathematically justified stationarity tests in order to reach a reliable status agreed by all about the correct use of the data. Another possibility would have been to use specific time-frequency techniques adapted to unstationary signals, but as a first approach we preferred not to push forward in this direction due to seemingly random fluctuations of the frequency content of the signals to be analysed. The paper presents the comparison of the efficiency of different non-parametric tests with a special focus on the Smirnov test and its application on the experimental computation of a transfer function. This work was sponsored by the European Space Agency (ESA) in the frame of the ACEP contract (Ariane Consolidation and Evolution Programme) and technically managed by the French Space Agency (CNES).
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_67, © The Society for Experimental Mechanics, Inc. 2011
771
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772
Figure 1 – Recorded data in the frequency & time domain
II. A SHORT SURVEY OF STATIONARITY TESTS We assigned to a stationarity test to be: non-parametric i.e. that didn't require any hypothesis on the nature of the distribution law of the signal, adapted to the detection of the main sources of unstationarity we were facing and that could bias the FFT treatment, i.e. amplitude, phase and frequency variations on small time scales. The Runs Test This classical test is based on the counting of runs defined as a succession of identical symbols. The total number of runs gives an indication about the random character. For example, we can define a characteristic of the signal on a short and fixed length time interval by the positive or negative response to the question “Is the RMS amplitude of the signal on this interval superior to the mean RMS amplitude evaluated on the whole duration of the signal ?”. If the signal is random, the probabilities of both responses are equal. Thus in case of a low number of runs or on the contrary of a number of runs close to the population of analysed intervals (see Figure 2), we can infer that the signal is likely to be influenced by an underlying factor and thus can not be considered as being random.
PPPPPPPPPPPPPPNNNNNN : 1 run NPPNPNPNPNPNPNPNPNNP : 18 runs Figure 2 - Examples of non-random series detected by the Runs Test Although the runs test is sometimes used as stationarity test (see [1] and [2]), such applications appeared to us not very convincing, the runs test being basically suited to determine the randomness of a process (see [3] and [4]). The Reverse Arrangements Test (RAT) This test consists in counting in a data sequence ui (i=1 to N) the number of times ui > uj for all pairs (i,j) with i < j (number of reverse arrangements).
(i=1 to N)
BookID 214574_ChapID 67_Proof# 1 - 23/04/2011
773 In the case of N independent observations, the number of reverse arrangements is a random variable with a normal distribution defined by:
µ=
N ( N − 1) 2 N (2 N + 5) ( N − 1) ,σ = 4 72
(1)
When one applies it to the power of a signal, this kind of test enables to detect underlying trends and then unstationarity in the signal. This test was used successfully in the cardiology field to identify short “stationary” vs. “unstationary” lapses (see [5] and [6]). The Smirnov test This non-parametric test is especially interesting as it is based on the direct analysis of distribution laws and may thus be considered as being linked to the notion of strong stationarity. This test is an adaptation of the Kolmogorov test to the comparison of two experimental distributions. In our situation, these two distributions will typically characterize the signal for two successive time intervals. The statistical indicator K is defined the following way for two experimental cumulative distribution functions
Fn (x)
and
Fn' ( x)
corresponding to
two samples (x1….xn) and (x’1….x’n) of the same size n :
K = max x Fn ( x) − Fn' ( x) 1 if xi ≤ x 1 n where Fn ( x) = ∑ δ xi ≤ x with δ xi ≤ x = n i =1 0 elsewhere
(2)
K defines so a distance between the two distributions evaluated through the biggest interval. For an important number n of data, the probability of the indicator to exceed a given threshold expresses through (see [7] and [8]):
n→ ∞
P( K >
c 2 n
∞
) → α (c) = 2∑ (−1) r −1 e −2 r c
2 2
(3)
r =1
For a limited number of data, the distribution law of the indicator can be tabulated. We used such tables for the data processing. Let us remark that the use of the Smirnov test is licit for the treatment of experimental (finite size) samples obtained from measurements, in so far as the distributions represented by these experimental density functions are continuous, as stated by W.W. Daniel [9]. This is true, as the values we are dealing with (pressure measurements for example in our case) are located in intervals of real numbers. The Smirnov test has been used for the analysis of the stationarity of sound signals [4] and to evaluate the typical stationarity duration within a recorded signal [10]. Constraint of use It must be pointed out that both RAT & Smirnov tests must be used with independent random data as input: the raw sampled recorded signal is thus not fitted for such kind of test. The classical way to proceed and that we used consists in performing as pre-treatment a power computation of the signal on sliding intervals which duration is of the order of magnitude of the highest period of interest; the limitation of the duration is aimed at limiting the averaging effect that could hide the unstationarity. Moreover, the objective of detecting important phase or frequency shifts requires some specific additional treatments that we will mention later on.
III. APPLICATION OF RAT AND SMIRNOV TESTS ON TEST RECORDS Both RAT and Smirnov tests were used to analyse pressure records of a rocket cavitating pump, the goal being to select the stationarity intervals in which an inlet vs. outlet transfer function could be identified.The objective was to select the set of 2 s intervals in which both inlet and outlet signals were stationary. This duration was imposed by the level of frequency accuracy required during the further Fourier Treatment.
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774 RAT tests The RAT test was applied considering 10 intervals of 2 s and an arbitrary Type I error threshold of 5 %. The signal was band-filtered in the frequency range of interest. Figure 3 shows a typical example of the selected part of the signal.
Figure 3 – Application of the RAT test on a measurement The application of the RAT enabled thus to select more accurately valid data intervals than by our preliminary approach. Smirnov tests With the Smirnov tests we adopted a recursive procedure to evaluate the duration of stationarity of the signal in the course of time. We thus did not take a fixed value of 2 seconds as in the RAT tests but considered population of 5 to 20 RMS computed on the minimal intervals achievable with regard to the lower frequency limit, that is 0.2 s. The stationarity was pronounced when the Smirnov test performed on each of these subdivisions was positive. The same Type I error threshold was used. The Figure 4 illustrates the results obtained on an interval selected by the RAT test (dotted section of the Figure 3). Each white point represents the result of the test performance, a black point corresponds to a rejection of an interval with respect to a Type I threshold of 5 %. Let us recall that the type-I error rate corresponds, in our situation, to the probability of rejection by the test in case of a signal interval which is, in fact, stationary.
Figure 4 – Application of the Smirnov test on a measurement We defined as success criterion a series of accepted tests corresponding to decreasing interval durations. That means that whatever the scale considered, the signal can be considered as being stationary. This methodology applied to the tests pointed out some stationary parts of the signal (beginning respectively at
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775 470s and 520s) for the scales of 16 s to 2 s intervals. For these parts of the signal, the result was thus coherent with RAT results. However, one has to notice that the cascade approach developed here is much more selective than the single RAT test. Indeed, this approach showed us that in our application case corresponding to a significantly fluctuating phenomenon, we had to limit our search for stationarity lapses to time intervals not larger than 2 s, in order to finally get a sufficiently wide statistics of identification results. Enhanced Smirnov test The Smirnov test in previously defined form is limited by the constraint of the frequency lower limit (5 Hz) that leads to minimum tested intervals of 2 s. It is thus impossible to analyse the stationarity of the signal at a smaller scale. To increase our confidence on the results, we improved the technique by comparing distributions of triplets instead of pairs. This more elaborated form of the Smirnov test has been described by Birnbaum & Hall [11] which evaluated the exact distribution law of the statistical indicator. This indicator writes:
K 3 = sup x , i , j Fn( i ) ( x) − Fn( j ) ( x) where Fn(i ) ( x) =
1 if x k ≤ x 1 n (i ) δ xk ≤ x with δ x(ki )≤ x = ∑ n k =1 0 else
(4)
Still with the objective of increasing the number of samples to be tested, we lowered the duration of the intervals in which the RMS is computed (0.2 to 0.1 s). This is of course an infringement as the duration of 0.2 s was linked with the frequency range we’re focusing. Nevertheless, tests performed on synthetic sine signals (perfectly stationary) in the frequency range of 5.5 to 9.5 Hz showed statistical indicator always lower than 0.1. The bias introduced was thus minor in comparison with the gain brought by the increase of samples.
IV. CALIBRATION OF THE SMIRNOV TEST FOR THE ESTIMATION OF A TRANSFER FUNCTION ESTIMATION Amplitude shift detection ability In order to elect accurately stationary lapses of the signal, or more exactly stationary lapses of the power of the signal, we had to tune the Smirnov test. In our application, the goal was to determine a frequency transfer function, i.e. a gain and a modulus. That is why we performed the tuning with regard to these two aspects. The ability of the Smirnov test to detect an amplitude variation was evaluated through test cases performed on
( ) 5
a simple function with a variable modulus exp a t . The objective was to obtain a tuning leading to the rejection of high amplitude variations. The stationarity analysis we applied was based on a subdivision in 3 parts of the time window. The size of the considered samples was 10, which leads to a Type-I error rate ( α Smirnov ) between 1 % and 72 % and values of the K3 statistical indicator ranging from 0.8 to 0.4 (see [11]). The Table 1 indicates the maximal amplitude variation rate compatible with Smirnov test acceptance, with regard to a Type I error rate. The application of the test case demonstrates logically that high Type-I error rates lead to a more selective. The Smirnov test proved thus to be a very efficient tool to eliminate important amplitude variations between intervals.
α Smirnov (%) FT (t + 3s ) / FT (t )max
1
13
72
5.12
1.91
1.18
Table 1 – Tuning of the Smirnov test vs. amplitude shift Frequency shift detection ability The frequency shift detection was one of the major issues of our work, as the variations of the frequency in the recorded data were very quick (see Figure 1). The selection of zones of low frequency variations was performed by the way of a cascading Smirnov test applied on thin band-pass signal. The Figure 5 shows the result of a Smirnov analysis with interval pairs performed on a filtered signal with a decreasing band-pass (3-12 Hz to 5-9 Hz). The procedure can be used in coordination with the FFT procedure, the targeted frequency resolution of the FFT being used to tune the width of the band-pass filter. When the test was applied to our signal filtered on a narrower bandwidth, the number of rejection cases increased from 9 to 29, so the selectivity of the test increases. We interpreted this behaviour as a consequence of the fluctuations of spectral content of the analysed signal. The final selected interval at 520 s is identified by the blue triangle in the Figure 5. Thus in spite of the fact that the Smirnov test applied to the power of the signal do not permit to
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776 identify instationarity due to frequency fluctuations, its use on a signal which has preliminarily been filtered on a narrow bandwidth permitted to overcome this weakness.
N : negative test result P : positive test result Figure 5 – Influence of the filtering on the Smirnov test results Phase Shift The last source of bias in the establishment of a transfer function is a strong evolution of the phase shift. This type of evolution is undetectable by any test performed with an RMS as input: the Smirnov as well as RAT are inadequate to pointing out such type of behaviour. For the detection of this kind of phenomenon, we used therefore a classical interspectra technique that proved through calibration test with synthetic signals to be sensitive to phase shifts.
V. TEST CASE : A TRANSFER FUNCTION IDENTIFICATION The previously presented method based on the Smirnov test was used to perform a transfer function estimation and was crosschecked with : a methodology based on the single RAT test application on the RMS of the signal, a pure ‘engineering judgement methodology’ based on the selection of filtered signal lapses showing stable phase and modulus behaviour.
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777 The gain and phase results of the obtained transfer function are plotted against the NPSP, which is proportional to σ +1, σ being the cavitation number (see Figure 6). 1
270
0,8
Gain's phase (deg.)
Adimensionned gain modulus
0,9
0,7 0,6 0,5 0,4 0,3
225
180
135
0,2 0,1 0 0,92
0,94
0,96
0,98
'Engineering Judgement Approach'
1 NPSP(bar)
1,02
RAT Methodology
1,04
1,06
1,08
90 0,92
0,97
1,02
1,07
NPSP (bar) Smirnov Methodology
'Engineering Judgement Approach'
RAT Methodology
Smirnov Methodology
Figure 6 – Pump transfer function identification Independently of the various approaches for the transfer function identification, leading to different determinations of gain modulus and phase, it is interesting to notice that the cloud of points corresponding to the Smirnov methodology covers a large domain of cavitation number and incorporates both results obtained by the RAT and by the engineering judgement methodology. The crosscheck was thus positive and enabled us to strengthen the confidence in the validity of the results.
VI. CONCLUSION Facing very disturbed test signals, the question of the stationarity and more globally the validity of the identification of a transfer function based on FFT calculations was addressed. A survey of some classical stationary tests led to select the Smirnov test. After a calibration phase the use of this type of algorithm proved to be efficient to select 'usable' part of the signal answering all the conditions to perform FFT analysis. The obtained transfer function strengthened the confidence in the results obtained by more empirical method. The authors recommend to envisage the use of this technique in any case stationarity of a random signal is to be checked before applying standard spectral analysis tools.
References [1]
Gerald F. Harris, Susan A. Riedel, Donald Matesi, and Peter Smith. Standing Postural Stability Assessment and Signal Stationarity in Children with Cerebral Palsy. IEEE Transactions on Rehabilitation Engineering, Vol. 1, n° 1, March 1993
[2]
Jack P. Landolt and Manning J. Correia. Neuromathematical Concepts of Point Process Theory. IEEE Transactions on Biomedical Engineering, Vol. BME-25, n° 1, January 1978
[3]
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M.A. Maiianas, M. Guillkn, J.A. Fig, J. Morera and P. Caminal. Analysis of Stationarity and Statistical nd Changes in Myographic Signals from Respiratory Muscles. Proceedings of the 22 Annual EMBS International Conference, July 23-28, 2000, Chicago IL.
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M. H. DeGroot and M. J. Schervish. Probability and Statistics. Addison Wesley
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W.J. Conover. Practical Nonparametric Statistics. Wiley & Sons
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Wayne W. Daniel. Applied Nonparametric Statistics, 1978, p.274
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Z.W. Birnbaum and R.A. Hall.Small Sample Distributions for Multi-Sample Statistics of the Smirnov Type. Ann. Math. Statist. Volume 31, Number 3 (1960), 710-720.
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BookID 214574_ChapID 69_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Determination of Dynamically Equivalent FE Models of Structures from Experimental Data
Taylan KARAAöAÇLI1, Erdinç N. YILDIZ1 and H. Nevzat ÖZGÜVEN2
1
TÜBøTAK-SAGE, The Scientific and Technological Research Council of Turkey-Defense Industries Research and Development Institute, 06261 Ankara, Turkey 2
Middle East Technical University, Department of Mechanical Engineering 06531 Ankara, Turkey
ABSTRACT In various applications it is important to determine dynamically equivalent spatial finite element (FE) model of complex structures. For instance, obtaining the FE model of an existing aerospace structure is a major requirement for reliable aeroelastic analysis. In such applications a reliable FE model may not be always available, and when this is so a dynamically equivalent FE model derived from modal test will be very useful. This paper presents a noble method to determine spatial FE model of a structure by using experimentally measured modal data along with the connectivity information of measurement points. The method is based on the mass and stiffness orthogonality equations written using experimentally determined mode shapes and natural frequencies. These equations are solved for geometric and material properties constituting global spatial mass and stiffness matrices of an initial FE model. Starting from this initial FE model, mass and stiffness orthogonality equations are updated iteratively employing experimentally obtained natural frequencies and corresponding eigenvectors from the FE model. Iterations are continued until eigensolution of the updated FE model closely correlates with experimentally measured modal data. A simulated case study on GARTEUR scaled aircraft model is presented in order to demonstrate the applicability of the method. NOMENCLATURE
A
>Ak @ >Am @ E G >I @ I1 I2 I12 J
Cross sectional area Coefficient matrix of structural identification equations derived from stiffness orthogonality Coefficient matrix of structural identification equations derived from mass orthogonality Elastic modulus Shear modulus Identity matrix Second moment of area about y axis in local coordinates of a beam element Second moment of area about z axis in local coordinates of a beam element Product moment of area Polar moment of area
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_69, © The Society for Experimental Mechanics, Inc. 2011
785
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>K @ >ke @
Stiffness matrix Element stiffness matrix
L
Length of a beam element Mass matrix
m N n p >T @ >Te @
Number of degrees of freedom of the finite element model Number of experimentally obtained normal modes Total number of degrees of freedom of the FE model Number of primary coordinates
>M @ >me @
>U @ >V @
>U @ >V @
Transformation matrix used in Guyan’s Expansion Coordinate transformation matrix for a finite element matrix Left singular matrix Right singular matrix
*
Conjugate transpose of left singular matrix
*
Conjugate transpose of right singular matrix
>6@ >) @
>) @ t x
>) x @ ^I ` >O @ U
Element mass matrix
Matrix whose diagonal elements are singular values Mass normalized modal matrix Mass normalized experimental modal matrix Mass normalized experimental modal matrix expanded to the size of FE model Eigenvector Diagonal matrix composed of the squares of undamped natural frequencies Density
1. INTRODUCTION Dynamic characteristics of aerospace structures are to be known in order to predict their aeroelastic behavior. For instance, flutter characteristic of an aircraft structure can be studied by using its dynamically equivalent FE model. Flutter analysis is used to determine the safe flight envelop of an aircraft. Accuracy of the FE model has a major effect on the reliability of the flutter analysis and it can be achieved by correlating FE model of the aircraft structure with its experimentally measured modal data. There has been extensive research on correlation of dynamically equivalent FE model of a structure with experimental modal data. Usually, first a FE model is obtained from CAD data and then it is corrected/improved by using experimental data. Methods used to correct FE models of structures are generally called model updating procedures. Studies in the field of model updating can be classified in two groups: direct and indirect model updating techniques. One of the important examples of early direct updating procedures, namely the Method of Lagrange Multipliers, has been used by Baruch and Bar Itzhack [1] to correct stiffness matrices of structures, and also by Berman and Nagy [2] to correct mass and stiffness matrices of structures. Another important direct updating method has been proposed by Sidhu and Ewins [3] to determine the error of mass and stiffness matrices using analytical and experimental modal data. Those studies have been followed by various contributions [4-5] in the field of model updating during last twenty years. A recent contribution made by Carvalho et al. [6] in this area is distinguished among many others. The method has several advantages: Firstly, it does not need any model reduction or expansion, and secondly it is capable of preventing the appearance of spurious modes in the frequency range of interest. Although direct methods prove to be useful in exact matching of mathematically obtained and experimentally determined modal data, there exist some disadvantages that put the work on the secondarily
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searched results side. One such disadvantage is that the original coordinate connectivity of the FE model is lost and resulting mass and stiffness matrices become fully populated. When connectivity in the stiffness matrix is lost, off-diagonal stiffness elements appear for the degrees of freedom among which no direct connection is present. Loss of connectivity information in model updating of an aircraft structure makes it almost impossible to adjust geometric and material properties of standard finite elements such as beams and shells to obtain spatial matrices dictated by direct updating methods. Even if changes may be applied by mass and spring elements, the response of the structure under static loading is degraded which makes aeroelastic studies such as divergence analysis impossible. Loss of connectivity may also degrade semi-definiteness of stiffness matrix and may cause rigid body modes of the aircraft structure to be lost. In case of applying indirect model updating procedures, adjustments in the FE model are made on individual elements in an iterative manner instead of adjustments in the whole system matrices as in the case of direct updating techniques. One of the most widely used indirect methods is the Inverse Eigen Sensitivity Method which makes use of sensitivies of experimentally measured modal data with respect to updating parameters of the FE model. Sensitivity matrix is determined by introducing perturbation to structural parameters of the FE model and analyzing the amount of change in the modal response. Although indirect methods such as Inverse Eigen Sensitivity Method are widely used in model updating of the FE model of structures [7-12] they have certain disadvantages. One such disadvantage is that FE counterparts of the experimentally measured normal modes must appear in the initial FE model to guarantee convergence. Actually this is a general requirement for any of the state-of-the art model updating techniques (direct or indirect) and dictates construction of a detailed initial FE model of the structure. Determination of an accurate initial FE model of an aircraft structure demands considerable time and effort if it is not accomplished by its original design team. Even if such an initial FE model is obtained, the application of the indirect model updating method requires considerable time and experience. The method presented in this paper is developed for structures such as aircraft structures that can be modeled by beam elements. However, the method can further be extended for structures that are modeled by using other type of elements as well. The method starts with an ‘empty’ FE model of which geometric and material properties are not assigned but only connectivity information is available. Connectivity is achieved by connecting measurement degree of freedoms (dofs) with beam elements of known lengths. Initial estimates of geometric and material properties required for the FE model are determined by mass and stiffness orthogonality equations derived from experimentally measured modal data. Sufficient number of modal data is required so that the number of equations obtained is more than the number of unknowns. Alternatively, the number of unknowns should be reduced below the number of equations, by grouping similar elements with the assumption that elements within the same group have the same geometric and material properties. By this way an initial FE model of the structure with eigenvectors corresponding to experimentally determined mode shapes is obtained. Then, the eigenvalues and eigenvectors of the initial FE model are calculated. Eigenvectors of the FE model corresponding to experimental mode shapes are used along with experimentally obtained natural frequencies in order to reconstruct mass and stiffness orthogonality equations. By using constrained least square solution of the orthogonality equations, updated geometric and material properties are calculated. In each iteration, solutions are constrained by lower and upper bounds to avoid divergence problem. Updated properties are used to construct the updated FE model whose eigenvectors are to be used in the next iteration. Iterations are continued until eigensolution of the updated FE model closely correlates with experimentally measured modal data. The application of the method developed is illustrated on a GARTEUR scaled aircraft structure and it proves to be a promising method that deserves to be tested on real aircraft structures. 2. THEORY The theory is presented for structures which can be modeled by Euler-Bernoulli beam elements. Although the extension of the method for structures which are modeled by using other types of elements is not trivial, the same approach can be used. 2.1. Construction of FE Model from Modal Data Starting with Empty Matrices and Connectivity Information The development of the method has started by asking the following questions: Starting only with the knowledge of experimentally measured modal data and of measurement points, is it possible to obtain dynamically equivalent spatial FE model of a real aircraft structure? And also, to what extend the complexity, i.e. number of dofs of the
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FE model, may be reduced by still closely correlating the calculated modes with experimentally measured modal data. These questions are the manifestation of the desire of eliminating several difficulties arising in the application of currently available model updating techniques. In current methods, it is necessary to start with a pretty accurate initial FE model. In order to eliminate the time and effort required for the development of a detailed initial FE model, it is chosen to start with an empty FE model. This model is constructed by connecting measurement points with beam elements without assignment of any geometric or material properties, except beam lengths. The initial estimates of properties are then determined by using mass and stiffness orthogonality of experimentally obtained incomplete normal modes. In indirect model updating procedures, dofs of FE models are usually at least an order of magnitude larger than measurement dofs. In this method it is suggested to take measurement points as the nodal points of the FE model and connect them with beam elements. In case of an aircraft structure, the skeleton of the wings will consist of beam like structures such as ribs and spars. Adjustment of connectivity of the FE model so that its look will be similar to the skeleton of the aircraft of interest may help in more accurate and physically meaningful initial estimates of geometric and material properties. 2.2. Expansion of Experimentally Measured Normal Modes After the construction of a FE element mesh model, the next step is the determination of initial estimates of geometric and material properties. The necessary equations to solve for initial estimates of properties are derived from mass and stiffness orthogonality relations of experimentally determined normal modes. Consider the following mass and stiffness orthogonality equations:
>) @T >M @>) @ >I @
(1)
>) @T >K @>) @ >O @
(2)
For a n dof FE model, the mass normalized modal matrix will be a nxn square matrix. However, experimental modal matrix corresponding to that FE model will be incomplete, since there will be no measurement for rotational dofs, and also measurement points will be less than the total nodes in the FE model. Moreover, experimental modal matrix will be truncated as we are only interested in modes within the frequency range of interest. Therefore the experimental modal matrix will be highly incomplete. However, as it will be clear in the following section, experimentally measured normal modes have to be expanded to the size of the eigenvectors of the FE model to be able to estimate the geometric and material parameters. Guyan’s Expansion [13] is a simple and reliable technique to be used for expansion of experimentally determined normal modes. In Guyan’s Expansion, the transformation matrix between primary and slave coordinates is determined from stiffness matrix of the FE model. Since for the initial FE model we have mesh information only, it seems there is no stiffness matrix to be used. However, the critical question is the following: to obtain transformation matrix, is it really necessary to use the stiffness matrix of the true FE model or a stiffness matrix of some other FE model with the same mesh can be used? Let us consider two different FE models of the same aircraft structure with the same connectivity, i.e. the same number of elements, number of nodes and lengths for each element, but different cross sectional and material properties for individual elements. Then the stiffness matrices will be different; but if translational dofs of nodal points of two structures are forced to have the same displacement pattern such as that given by an experimental mode shape, it is expected that unconstrained dofs such as rotational and slave dofs in one FE model will be close to the corresponding dofs in the other model. Based on this approximation, beam elements of empty FE model of the aircraft structure will be assigned arbitrary geometric and material properties to obtain a stiffness matrix, from which the transformation matrix required in Guyan’s Expansion Method will be calculated.
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According to Guyan’s Expansion Method global stiffness matrix is partitioned to separate elements corresponding to primary and slave coordinates as follows:
ª K ss «K ¬ ps
>K @
K sp º K pp »¼
(3)
Then the transformation matrix can be obtained as:
> @
ª >K ss @1 K sp º « » [I ] ¬ ¼
>T @
(4)
Finally, experimentally measured mass normalized normal modes can be expanded into FE structure size by using the transformation matrix given in Eq. (4) as follows:
>) x @nxN >T @nxp >) tx @pxN
(5)
2.3. Derivation of Structural Identification Equations from Mass and Stiffness Orthogonality Relations Once experimental normal modes are expanded into FE model size, the next step is the derivation of structural identification equations from mass and stiffness orthogonality relations of normal modes. Consider a n dof FE model with k elements. Global mass and stiffness matrices can be expanded in terms of element mass and stiffness matrices as follows: k
>K @nxn ¦ [ke ]nxn
(6)
e 1
k
>M @nxn ¦ [me ]nxn
(7)
e 1
> @
> @
where k e and me are element stiffness and mass matrices;
>K @
> @
and M
are global stiffness and mass
matrices, respectively. Actual size of element matrices is mxm where m is the number of dofs of an element, but during assembly process they should be used in sparse forms (in the size of nxn ) as can be seen in Eqs. (6) and (7). Substituting Eqs. (6) and (7) in Eqs. (1) and (2) yields k
>) @ ¦ >m e @>) @ >I @ T
e 1
(8)
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k
>) @ ¦ >k e @>) @ >O @ T
(9)
e 1
Eqs. (8) and (9) can be further expanded as follows:
1 if ® ¯0 if
^I ` ^>m @ >m @ >m @`^I ` r T
s
1
2
k
^I ` ^>k @ >k @ >k @`^I ` r T
s
1
2
k
r
s
(10)
rzs
O r if r s ® ¯ 0 if r z s
(11)
from which it is possible to write
^I ` >m @^I ` ^I ` >m @^I ` ^I ` >m @^I ` r T
r T
s
r T
s
1
s
2
k
^I ` >k @^I ` ^I ` >k @^I ` ^I ` >k @^I ` r T
r T
s
r T
s
1
s
2
k
1 if ® ¯0 if
r
s
(12)
rzs
Or if r s ® ¯ 0 if r z s
(13)
Since the individual element matrices in Eqs. (12) and (13) are sparse, they can be condensed as follows:
^I ` >m @ ^I `
^I ` >m @ ^I `
(14)
^I ` >k @ ^I `
^I ` >k @ ^I `
(15)
r T 1 xn
r T 1 xn
where
e nxn
e mxm
and
s
e mxm
r T
s
>m @
e mxm
e 1 xm
nx1
>k @
and
^I `
e mxm
r
nx1
>k @
e mxm
coordinates as follows:
e mx1
s
e 1 xm
nx1
the rth eigenvector
>m @
r T
s
e nxn
e mxm
e mx1
are local element matrices in global coordinates.
^I ` r
e mx1
represents the part of
corresponding to the dofs of the element of interest.
are obtained by transformation of element matrices in local coordinates to global
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>m @
>Te @T >mel @>Te @
>k @
>Te @T >kel @>Te @
e mxm
e mxm
where
>m @
l e mxm
and
(16)
(17)
>k @
l e mxm
> @
are element matrices in local coordinates, and Te is the transformation matrix for
the specific element, which transforms matrices from local (element) coordinates to global coordinates.
>m @
l e mxm
Replacing
>k @
l e mxm
and
into right hand side of Eqs. (14) and (15) we have:
^I ` >m @^I ` ^I ` >T @ >m @>T @ ^I ` e
(18)
^I ` >k @^I ` ^I ` >T @ >k @>T @ ^I `
(19)
r T
e
e
r T
e
e
s
r T
e
e
T
s
r T
e
e
T
s
l e
e
e
s
l e
e
e
e
Before substituting Eqs. (18) and (19) into Eqs. (12) and (13) one last arrangement is necessary as shown below:
^I ` >T @ >m @>T @ ^I `
§¨ >T @ I r ·¸ ml §¨ >T @ I s ·¸ © e e ¹ e© e e ¹
^I ` >T @ >m @>T @ ^I `
§¨ >T @ I r ·¸ ml §¨ >T @ I s ·¸ © e e ¹ e© e e ¹
r T
e
T
e
r T
e
l e
T
l e
e
s
e
e
s
e
e
^`> @ ^`
(20)
^`> @ ^`
(21)
T
T
Right hand sides of Eqs. (20) and (21) can be put in more compact forms with the following definition:
^I ` rl
e
^`
§¨ >T @ I r ·¸ © e e¹
(22)
By using the above definition in Eqs. (20) and (21), and substituting them into Eqs. (12) and (13) we obtain:
^I ` >m @^I ` ^I ` >m @^I ` ^I ` >m @^I ` rl T
1
l 1
sl
rl T
1
2
l 2
rl
rl T
2
k
l k
sl k
1 if ® ¯0 if
r
s
rzs
(23)
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^I ` >k @^I ` ^I ` >k @^I ` ^I ` >k @^I ` rl T
1
l 1
sl
rl T
1
2
l 2
sl
rl T
2
k
l k
sl
k
Or if r s ® ¯ 0 if r z s
For 3-D Euler-Bernoulli beam elements, size of element matrices
(24)
>m @ and >k @ in local coordinates is 12x12 e
e
and parametric expressions for matrix elements are too lengthy to be presented in this paper. For that reason, element matrices of a simplified 2-D Euler-Bernoulli beam model without axial dof (see Figure 1) are given here just to explain the application of the method. However, the formulation is obtained for 3-D Euler-Bernoulli beam elements, and this formulation is used in the case study presented in this paper. Two-noded beam element shown in Figure 1 has 4 dofs and its element mass and stiffness matrices are given as follows:
>k @ l e
>m @ l e
6 L 12 6 L º ª 12 « 6L 4 L2 6 L 2 L2 »» EI « L3 « 12 L 6 L 12 6 L » « » 2 L2 6 L 4 L2 ¼ ¬ 6L 22 L ª 156 « 4 L2 UAL « 22 L 13L 420 « 54 L « 2 ¬ 13L 3L
On the other hand,
^I ` rl
e
^I ` rl
e
54 13L º 13L 3L2 »» 156 22 L » » 22 L 4 L2 ¼
T
(26)
vector given in Eqs. (23) and (24) is as follows:
^Q 1 T1 Q 2 T 2 `T
where Q and
(25)
(27)
represent translational and rotational dofs at nodes, respectively.
Figure 1. 2-D Euler-Bernoulli beam model without axial dof
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Substituting Eqs. (25), (26) and (27) into Eqs. (23) and (24), the following equations are obtained:
a1rs ( UA)1 a2rs ( UA) 2 akrs ( UA) k
1 if ® ¯0 if
b1rs ( EI )1 b2rs ( EI ) 2 bkrs ( EI ) k
O r if r s ® ¯ 0 if r z s
rs
r s rzs
(28)
(29)
rs
where a e and be are known constants calculated from mesh properties and experimental mode shapes. It can easily be seen that coefficients
aers and bers can be calculated by using experimental normal modes and
matrices given in Eqs. (25) and (26). It is important to note that experimental normal modes should be mass normalized, which can be calculated by using experimentally obtained modal constant matrices [14]. When 3-D Euler Bernoulli beam elements are used, Eq. (29) takes the following form:
b1rs ( EI1 )1 c1rs ( EI 2 )1 d1rs ( EI12 )1 e1rs GJ 1 f1rs ( EA)1 b2rs ( EI1 ) 2 ... f 2rs ( EA) 2 element 2
element 1
bkrs ( EI1 ) k f krs ( EA) k element k
(30)
Or if r s ® ¯ 0 if r z s
Equations (28) and (30) can be put into the following compact forms:
>Am @^xm ` ^bm `
(31)
>Ak @^xk ` ^bk ` (32) rs
rs
rs
rs
where [ Am ] and [ Ak ] are coefficient matrices that consist of a e and be , c e … f e
terms. ^x m ` and ^x k ` are
unknown vectors which consist of products of geometric and material properties as given in Eqs. (28) and (30). Eqs. (31) and (32) will be referred to as structural identification equations. The number of individual equations in matrix equations (31) and (32) is determined by the number of experimentally determined modes. If N experimental modes have been extracted in the frequency range of interest, N ( N 1) / 2 number of independent equations can be written for each of Eqs. (31) and (32). If there are more unknowns than the number of equations because of the limited number of reliable mode shapes experimentally obtained in the frequency range of interest, the number of unknowns can be reduced by grouping similar beam elements with the assumption that elements within the same group have the same geometric and material properties. Then the initial estimates for the unknown system parameters can be calculated as follows:
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^xm `
>A @^b `
(33)
^xk `
>A @^b `
(34)
m
m
k
k
> @
> @
where Am and Ak
> @
show pseudo inverses of Am and
>Ak @ respectively.
Studies have shown that experimental errors and errors introduced during expansion of experimental modes may give rise to ill conditioned coefficient matrices [ Am ] and [ Ak ] . Another source of ill conditioning is the order of magnitude difference between the numerical values of the unknown parameters. Avoiding the problems due to ill conditioned coefficient matrices will be addressed in the next section. 2.4. Singular Value Decomposition (SVD) Analysis of Coefficient Matrices SVD analysis has shown that major singular values and corresponding left and right singular vectors of ideal coefficient matrices are still preserved in actual matrices. Nevertheless experimental errors, errors due to expansion of normal modes and unbalanced order of magnitude between unknowns introduce uncommon singular values and corresponding singular vectors in the actual matrices. Consider the following SVD of the actual coefficient matrices:
>Am @ >U m @>¦ m @>V m @*
(35)
> A k @ >U k @>¦ k @>V k @*
(36)
Pseudo inverses of coefficient matrices turn out to be:
>A @ >V @>¦ @>U @ m
m
*
m
(37)
m
>A @ >V @>¦ @>U @ k
k
*
k
k
(38) where
>¦ @ and >¦ @are pseudo inverses of >¦ @ and >¦ @ , respectively, and they are obtained by replacing m
k
m
k
every non-zero singular value by its reciprocal. Therefore, reciprocals of erroneous singular values of small magnitudes become dominant terms of the pseudo inverse of coefficient matrices. Consequently, Eqs. (33) and (34) may give rise to erroneous solutions. To avoid this, lower singular values of coefficient matrices have to be eliminated systematically. 2.5. Iterative Solution Procedure After determining initial estimates of products of geometric and material parameters by solving equations (33) and (34), mesh only FE model is completed by using elastic and inertial estimates obtained. Since initial estimates are least square solutions, eigensolution of the initial FE model will not perfectly correlate with experimental modes.
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To improve correlation the following algorithm is applied: Step 1: Solve eigenvalues and eigenvectors of the initial FE model in order to obtain modal data corresponding to experimentally measured ones. Step 2: Reconstruct coefficient matrices of Eqs. (31) and (32) by using eigenvector counterparts of experimental normal modes to reduce ill conditioning. Step 3: Solve ^xm ` and ^xk ` from reconstructed versions of Eqs. (31) and (32), by using a non-linear least square solver with lower and upper bounds to avoid divergence problem. Step 4: After obtaining updated parameters in Step 3, update FE system matrices and go to Step 2. Use previous solution as initial guess for the non-linear least squares solver. Continue until eigenvalues and eigenvectors converge to their experimental counterparts. 3. CASE STUDY In this case study, the applicability and the accuracy of the method are demonstrated by using simulated experimental modal data derived from a FE model of SM-AG19 scaled aircraft structure of GARTEUR (Group for Aeronautical Research and Technology in Europe). The FE model shown in Figure 2, has 44 nodes, 40 elements and 264 dofs and its overall dimensions are: 2000 mm wing span and 1500 mm fuselage. Material properties of aluminum used in the FE model are:
E = 70 GPa,
U = 2800 kg/m3 Q = 0.3. The simulated experimental modal data are obtained as follows: 1. It is assumed that an experiment similar to the one used in the work of Kozak et al. [10] is carried out, and the first 10 elastic modes are obtained. By using the FE model of the system with 264 dofs, the first 10 elastic modes are numerically calculated. 2. In order to simulate the experimental extraction of the modal vectors, only the 66 elements are retained in each eigenvector by keeping the dofs corresponding to the measurements points in [10]. 3. Finally, the 10 incomplete eigenvectors are polluted with random multipliers changing between 0.95 and 1.05, thus giving ± 5% error to the exact values.
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Figure 2. FE Model of GARTEUR SM-AG19 test bed
The method presented in this paper is applied according to the following scenario: Only the locations of measurement points and the first 10 elastic experimental mass normalized mode shapes and corresponding natural frequencies are available for the structure. The objective is to derive a FE model of that structure which is dynamically equivalent to the actual FE model. Thus it is expected to have a good correlation between the pseudo-experimental modal data and the modal data obtained from the eigen solution of the FE model determined. The following procedure is applied in determining the FE model: 1. Measurement points are connected by beam elements to construct mesh only FE model of the structure without any geometric or material properties. Accordingly, a mesh only FE model with 44 nodes, 40 elements and 264 dofs is obtained. Only available geometric properties are element lengths. 2. Arbitrary geometric and material properties are assigned to the beam elements to derive a stiffness matrix. This stiffness matrix is used to calculate the transformation matrix between primary (measurement) and slave dofs in order to apply Guyan’s Expansion. Then, incomplete pseudo-experimental normal modes are expanded into the size of the FE model. 3. 110 structural identification equations are obtained by using the expanded normal modes. Half of these equations are obtained from mass orthogonality and the other half are obtained from stiffness orthogonality. To reduce the number of unknowns (which is 240 in this example) below the number of equations, similar elements are collected into 4 groups with the assumption that elements within the same group have the same structural properties. Definition of each group is given in Table 1. Thus the total number of unknowns is reduced to 24. 4. By least square solution of 110 equations, unknowns are calculated. These are the initial estimates of the structural parameters. By assigning these initial estimates of structural parameters to relevant beam elements stiffness and mass matrices (of sizes 264x264) of the whole system are obtained. Comparisons of the eigen solutions of the initial FE model with corresponding pseudo-experimental modal data are given in Table 2 and Figure 3. 5. In the last step, the iterative solution procedure presented in section 2.5 is applied and a converged FE model whose eigen solutions are in very good correlation with pseudo-experimental modal data is obtained. Comparisons of eigen solutions of the converged FE model with corresponding pseudo-experimental modal data are given in Table 3 and Figure 4.
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Table 1. Definition of element groups Group No
Group Definition
1
Wing
2
Fuselage
3
Vertical Stabilizer
4
Horizontal Stabilizer
Table 2. Correlation between pseudo-experimental modes and the modes of the initial FE model PseudoExperimental Modes 1 2 3 4 5 6 7 8 9 10
PseudoExperimental Natural Frequencies (Hz) 5.62 16.73 37.37 37.93 38.02 43.63 46.27 54.83 67.57 73.13
Corresponding Modes of the Initial FE Model 1 2 3 4 5 6 7 8 10 9
Natural Frequencies of the Initial FE Model (Hz) 4.96 15.33 35.09 35.46 35.47 38.28 62.01 68.20 74.85 71.74
Difference of Natural Frequencies (%) -11.8 -8.4 -6.1 -6.5 -6.7 -12.3 34.0 24.4 10.8 -1.9
Figure 3. MAC matrix between pseudo-experimental modes and the modes of the initial FE model
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Table 3. Correlation between pseudo-experimental modes and the modes of the converged FE model PseudoExperimental Modes 1 2 3 4 5 6 7 8 9 10
PseudoExperimental Natural Frequencies (Hz) 5.62 16.73 37.37 37.93 38.02 43.63 46.27 54.83 67.57 73.13
Corresponding Modes of the Converged FE Model 1 2 3 4 5 6 7 8 9 10
Natural Frequencies of the Converged FE Model (Hz) 5.65 17.02 37.30 37.94 38.03 43.63 46.28 54.63 67.57 73.16
Difference of Natural Frequencies (%) 0.5 1.7 -0.2 0.0 0.0 0.0 0.0 -0.4 0.0 0.0
Figure 4. MAC matrix between pseudo-experimental modes and the modes of the converged FE model
The excellent agreement observed in Table 3 and Figure 4 shows the accuracy of the method developed. 4. CONCLUSIONS In this paper, a noble method is presented to determine spatial FE models of structures by using experimentally measured modal data and connectivity information. The method aims to eliminate considerable time and effort required to obtain a complex and accurate initial FE model which is a must in every state-of-the art model updating scheme. The theory of the method is presented for structures that can be dynamically modeled by beam elements, although the general concept can be applied to structures that can be modeled by using other types of elements as well. In this method, first of all, a mesh only stick FE model is constructed by connecting measurement points with beam elements. Secondly, structural identification equations are derived from mass and
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stiffness orthogonality of experimentally obtained incomplete normal modes. Then, from the solutions of structural identification equations initial estimates for geometric and material properties of beam elements are obtained, and thus spatial stiffness and mass matrices of the whole structure are constructed. Finally, starting from this initial FE model, an iterative procedure is applied to obtain an ultimate FE model whose eigensolutions correlating well with experimental modal data. The applicability of the method is demonstrated on a GARTEUR scaled aircraft structure. Based on the performance of the identified spatial FE model in predicting modal characteristics of the structure, it is concluded that the method presented deserves to be tested on real aircraft structures, which will be the next step in the ongoing research. 5. REFERENCES [1] M. Baruch, Y. Bar Itzhack, Optimal Weighted Orthogonalization of Measured Modes, AIAA Journal, 16(4), 346-351, 1978 [2] A. Berman, E.J. Nagy, Improvement of a Large Analytical Model Using Test Data, AIAA Journal, 21(8), 11681173, 1983 [3] J. Sidhu, D.J. Ewins, Correlation of Finite Element and Modal Test Studies of a Practical Structure, Proc of 2nd IMAC, 756-762, Orlando, Florida, 1984 [4] B. Caesar, Update and Identification of Dynamic Mathematical Models, Proc. of 4th IMAC, 394-401, Los Angeles, California 1986. [5] H. P. Gypsin, Critical Application of the Error Matrix Method for Localization of Finite Element Modeling Inaccuracies, Proceedings of 4th IMAC, 1339-1351, K.U. Leuven, 1986. [6] J. Carvalho, B. N. Datta, A. Gupta and M. Lagadapati, A Direct Method for Model Updating with Incomplete Measured Data and without Spurious Modes, Mechanical Systems and Signal Processing, 21, 2715-2731, 2007 [7] D. Göge, 2003 Automatic Updating of Large Aircraft Models Using Experimental Data from Ground Vibration Testing, Aerospace Science and Technology, 7, 33-45, 2003 [8] J. M. W. Lee, G.R. Parker, Application of Design Sensitivity Analysis to Improve Correlation Between Analytical and Test Mode, MSC 1989 Wold Users’ Conference Proceedings, Los Angeles, California, USA, 1989 [9] M. T. Kozak, M.D. Cömert, H. N. Özgüven, A Model Updating Routine Based on the Minimization of a New Frequency Response Based Index for Error Localization, 25th IMAC, Orlando, Florida, 2007 [10] M. T. Kozak, M. Öztürk, H. N. Özgüven, A Method In Model Updating Using Miscorrelation Index Sensitivity, Mechanical Systems and Signal Processing, 23(6), 1747-1758, 2008 [11] R. M. Lin, D. J. Ewins, Model Updating Using FRF Data, 15 International Seminar on Modal Analysis, 141-162, Leuven-Belgium, 1990 [12] P.O. Larsson, P. Sas, Model Updating Based on Forced Vibration Testing Using Numerically Stable Formulations, 10th IMAC, San Diego, USA, 966-974, 1992 [13] R. J. Guyan, Reduction of Stiffness and Mass Matrices, AIAA Journal, 3(2), 380, 1965 [14] J. He, Z. Fu, Modal Analysis, Butterworth Heinemann, 2001
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Optimization of substructure dynamic interface forces by an energetic approach T. Weisser(1), L.-O. Gonidou(2), E. Foltête(1), N. Bouhaddi(1) (1)
FEMTO-ST Institute, Applied Mechanics Department, 24 rue de l’Epitaphe, 25000 Besançon, France [email protected], [email protected], [email protected] (2)
Centre National d’Etudes Spatiales, Direction des Lanceurs, Rond point de l’Espace - Courcouronnes, 91023 Evry Cedex, France [email protected] ABSTRACT A method based on an energetic approach has been developed to optimize vibration isolation at low frequencies. The aim is to minimize the dynamic response of a main structure (car body, rocket core…) subjected to interface forces generated by connected substructures (engine, booster…). At first, using a direct approach, an eigenvalue problem is formulated by minimizing the average dissipated power flow resulting from external load work. In order to keep a strong mechanical meaning, a modal approach has been developed to obtain an analytical expression of these eigensolutions. Then, the energy introduced in the structure at a given frequency is characterized by projecting the junction forces on the previously calculated basis. Both approaches have been numerically studied on an academic discrete system. The method has then been applied to a simple multimode coupled structure. 1. Introduction Industrial structures are often referred to as complex structures. They are composed by an assembly of several substructures, whose mechanical properties generally differ, joined at their interfaces by different junction types. However, their local dynamic behavior may not always be compatible, resulting in difficulties determining the global response of the structure. To reduce vibration propagation, engineering techniques usually attempt to decouple the substructures by using passive isolators, which tends to be ineffective or even impossible as the complexity of the interface increases. It is therefore necessary to have design methods to perform vibration isolation, i.e. to minimize the vibrational power transmission and to control the power flow through the interface. The vibration transmission problem between the boosters and the rocket core of the European space launcher Ariane 5 perfectly illustrates this stake. During atmospheric flight, combustion of the solid propellant generates acoustic modes of the boosters resulting in harmonic oscillations of their structures. One of these modes induces an important dynamic response at the junction with the rocket core, which could have an impact on the dynamic environment. For such reasons, it is necessary for industrials to have access to design methods to determine optimal power flow ways and thus to ensure vibration isolation [1]. Power flow analysis (PFA) has become a widely recognized technique. It is based on developing expressions for the vibrational power flow through coupled substructures by means of mobility or impedance functions. The expression of the time-averaged power is then derived at the input and output of each junction of the substructure. The basic approach has been discussed by Goyder and White [2,3] and Pinnington and White [4] with applications on beams and plates representing simple machinery foundations. It has been further extended to complex structures as simple periodic structures, allowing model size reduction [5], or frameworks, by using a direct dynamic-stiffness method [6]. The equality between the time-averaged power of a system and the energy dissipated by its damping is a well known property. In order to investigate the related dissipation characteristics, Miller et al. [7] first proposed to perform an eigenanalysis of the system’s power matrix to determine frequency ranges and mode combinations which cause the junction to dissipate power. Su et al. [8] hence developed a power flow expression using the eigenanalysis of the real part of the mobility matrix and derived upper and lower bounds T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_70, © The Society for Experimental Mechanics, Inc. 2011
801
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802 of the power flow by considering the obtained eigenmodes and matrix theory, which was reformulated and investigated by Ji et al. [9,10] in a mobility-based power flow mode method. To overcome the strong dependency of the mobility matrix on the physical parameters of the system, the frequency and the external forces, Xiong et al. [11] developed a complementary damping-based power flow mode theory and formulated power flow design theorems to control required energy flow dissipation levels and patterns. 2. Problem statement and formulation The general theoretical problem can be stated as follows. Let substructure
S1 be excited by an external force
f ext . Interface forces f j are then exerted on substructure S 2 through junctions. This will generate displacements at the interface and a dynamic response to optimize the structural interface forces
γ
of the whole structure. The purpose of this study is
f j by minimizing/maximizing the dissipated power flow at the
junction.
͡
͢
ɇ
Fig. 1: Representation of the interface between two substructures
The proposed method is based on the eigenanalysis of the interface dynamic flexibility matrix whose eigenvalues and eigenvectors respectively represent power flow intensity factor and associated force distribution. Particular attention is paid to this set of basis vectors: it spans the power flow space and allows a complete characterization of the interface forces by projection. It is therefore possible to consider a method to design power flow optimized junctions by avoiding the most dissipative configuration at the interface. 2.1.
Dissipated power flow of a dynamic system
The complex power
Pc transmitted through a multiple point, n degree-of-freedom (dof), interface subjected to
interface forces is given by
Pc = where
1 T
T
³
* f rms (t ) vrms (t ) dt =
0
1 2T
T
³f
*
(t ) v(t ) dt
(1)
0
f rms (t ) and v rms (t ) are vectors of size n respectively denoting the Fourier transform of the interface
force and velocity normalized to their root-mean-square-level (the asterisk denotes the transpose complex conjugate). After integrating over a single period, this complex power can be expressed as
Pc =
1 * f v 2
(2)
The averaged dissipated power, also known as active power in electrical engineering, is then extracted from the real part of the complex power
Pactive = Pdiss = Re( Pc )
(3)
Considering that both force and velocity are harmonic, and noticing that v = j ω x , where of vibration and x the displacement vector, the averaged dissipated power becomes
ω § ω · Pdiss = Re¨ j f * x ¸ = − Im f * x 2 © 2 ¹
( )
In order to express
ω
is the frequency
(4)
Pdiss as a function of the system’s parameters, the dynamic flexibility matrix Γ(ω ) is n degrees-of-freedom interface system
derived from the dynamic matrix equation of the associated
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(− ω
2
)
M + jωB + K x = f
(5)
x = Γ(ω ) f
(6)
By substituting equation (6) into equation (4) and normalizing the quotient, the averaged dissipated power can finally be re-expressed on a discretized form by N Pdiss =−
2.2.
§ f * Γf Im¨¨ * 2 © f f
ω
· ¸¸ ¹
(7)
Stationarity of the averaged dissipated power
The averaged dissipated power formulation given by equation (7) can be interpreted as the ratio of two N Pdiss at a given frequency ω = ω 0 comes to differentiating (7) with respect to the real injected interface force f = f r
quadratic forms and considered as a Rayleigh quotient. Thus, optimizing
N ∂ Pdiss ∂s =0⇔ =0 ∂f r ∂f r
(8)
According to the stationarity property of the quotient, this leads to solve, at
ω = ω 0 , the following equivalent
eigenproblem
(Γi − sν where
Id ) fν = 0
(9)
Γi is the real matrix of size n , denoting the imaginary part of the complex matrix Γ . It is assumed
that S is a non-positive diagonal matrix of the power flow eigenvalues sν arranged in ascending order, i.e.
s1 ≤ s 2 ≤ ≤ s n < 0
and that
(10)
F = [ f 1 , f 2 , , f n ] represents the orthogonal matrix of the normalized power flow eigenvectors,
both satisfying the following orthogonal relations
F T F = In
(11)
F T Γi F = S
(12)
where the subscript T denotes the transpose and I n is the identity matrix of size
n.
The derived frequency dependent eigenmodes precisely characterize the power flow at the interface: the eigenvalues and eigenvectors respectively give quantitative and qualitative information about the vibration transmission mechanisms at a given frequency. 3. Frequency analysis of the power flow eigenproblem 3.1.
Power flow eigenmodes properties
As stated above, the imaginary part of the dynamic flexibility matrix
Γi admits n eigenvalues at a given
frequency. However, it depends on the mass, elastic and damping system’s parameters. Thus, a parametric analysis of the power flow eigenproblem is performed. For the sake of simplicity, a two degree-of-freedom academic discrete system will be considered in what follows (see Fig. 2 and Table 1).
k1 c1
m1
kj cj
m2
k2 c2
Fig. 2: Two dof damped spring-mass system
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Substructure 1 2 1000
Mass (kg) -1 Stiffness (N.m ) -1 Junction stiffness (N. m ) Damping
Substructure 2 2 1000 2000
Proportional:
a = 0.001, b =0.5
Table 1: Initial system’s parameters
Eigenvalues
At first, the eigenvalues’ behavior is studied with regard to the frequency (see Fig. 3). It confirms that both eigenvalues are strictly negative and shows that a two dof system admits two power flow resonances. An analytical study has shown that these are very close but not equal to the dynamic vibration eigenfrequencies of the system. One should also notice that the curves cross each other: the eigenvalues have been sorted so as to follow their associated eigenvector. In this particular case, these are invariant, non frequency dependent, and similar to the dynamic vibration modes. This property will be discussed in section 3.2.
-10
-5
0
5 10 15 Frequency (Hz) Fig. 3: Power flow eigenvalues of the two dof system vs. frequency
Then, the influence of the system’s mass distribution is investigated. Figure 4 presents the variation of the eigenvalues versus frequency for three different values of mass m 2 . The eigenvalues’ behavior is mainly affected by a shift of the resonance peaks. The damping rate and its model are also investigated. Initially, the system is proportionally damped, i.e. C = aK + bM . Thus, it is possible to reduce it to modal damping rates. The effect of increasing these rates is a smooth decrease of the power flow resonances’ amplitudes, as encountered with dynamic vibration mode damping. However, by changing the damping model by a matrix that does not verify Caughey’s condition, i.e. KM
−1
C = CM −1 K , huge changes are introduced (see Fig. 5). -10
-6
-4
-10
-2
-10
0
5 10 15 Frequency (Hz) Fig. 4: Power flow eigenvalues for three different values of mass m2 : full line +5%; dashed line +100%; dotted line -50%
-10 Eigenvalues
Eigenvalues
-10
-10
-10
-8
-6
-4
-2
0
5 10 15 Frequency (Hz) Fig. 5: Power flow eigenvalues for three different values of dampers c2 and c3 : full line +5%; dashed line +100%; dotted line -50%
In both cases, if one focuses on the anti-resonance, a singular phenomenon can be seen. Many articles in the literature (e.g. [12]) show that, in structures with crossing, double-order eigenvalues, small perturbations of the system’s symmetry can compromise this property. These perturbations can generate interactions, sometimes
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805 in a very localized way, between the eigenvalues, whose loci will come closer until they reach a minimal distance and then diverge. This phenomenon, referred to as eigenvalue loci veering, also implies a hybridization of the associated eigenvectors, whose deformed shapes will be exchanged in a rapid but continuous way. To visualize this, a simple eigenvector veering criterion (EVC) is proposed. It is based on a frequency dependent modal assurance criterion (MAC) and calculated as follows, T
EVC ij(1, 2) =
(ω j )
f1(ωi ) f 2 2
f1(ωi )
2
(13)
(ω j ) 2
f2
Fig. 6: Eigenvector veering criterion for mass m2 +5%
Fig. 7: Eigenvector veering criterion for mass m2 +100%
Figures 6 and 7 represent the normalized scalar product between the first and second power flow eigenvectors at each frequency (from 0Hz to 150Hz): orthogonality or colinearity respectively result in a deep blue or red color. On the one hand, the blue colored diagonal of the visualization matrix confirms the fact that power flow eigenvectors constitute a basis for the power flow space at each frequency, even at the power flow resonance (around 6Hz). On the other hand, the red areas illustrate the exchanged deformed shapes between the eigenvectors. Intermediate colors give information on the quickness and the level of the veering phenomenon. It is thus possible to decompose the interface forces in the power flow space, above characterized by the force basis F , into the form n
f j = Fα = ¦ α ν f ν
(14)
υ =1
αυ
where the
are complex coefficients representing the weight of each eigenvector in f j . These are easily
determined by means of the basis’ orthogonal property, which yields to
α σ = f σT f j = α σR + jα σI
(15)
One should notice that, due to the real nature of the injected force f ext and assuming that studied structures are generally lightly damped,
α υI << α υR
and then becomes negligible. Moreover, by comparing the
α υR , a
truncated power flow subspace can be constituted to only retain the most significant eigenvectors, i.e. f1 , f m ⊂ F , with m << n . Finally, by substituting these power flow mode parameters into equation (4),
[
]
an approximate expression of the averaged dissipated power can be derived. It is related to the optimal interface force configuration expressed by equation (14) truncated at the m first power flow eigenmodes and, at a given frequency
Pdiss (ω 0 ) ≈ −
ω0 , is given by ω0 2
[(αυ fυ ) Γ (αυ fυ )] = − ¦ υ m
*
i
=1
ω0 2
m
αυ ¦ υ =1
2
f υT Γi f υ = −
ω0 2
m
αυ ¦ υ =1
2
sυ > 0
(16)
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806 3.2.
Power flow modes determination by a modal approach
This direct approach does not allow determining an explicit relationship between the system’s parameters and its power flow modes. However, as stated above, their behavior with regard to frequency has shown similarities with the dynamic behavior of the structure. The aim of this section is thus to develop a simple expression of the power flow modes depending on dynamic modal parameters. At first, the conservative natural vibration modes of the system are determined by solving
(K − ω M )y 2
ν
It is assumed that that Y
=0
ν
(17)
Λ is the diagonal spectral matrix containing the n eigenvalues ωυ of the structure and 2
= [ y1 , y 2 , , y n ] represents the orthogonal matrix of the associated normalized eigenvectors, both
satisfying the following orthogonal relations
Y T MY = I n
Y T CY = β
Y T KY = Λ
Assuming that the damping matrix C verifies Caughey’s condition (see section 3.1),
β
becomes a diagonal
βυ ,υ . The dynamic flexibility matrix is then easily computed and
matrix containing the modal damping rates takes the form
(18)
(
Γ = Y − ω 2 I n + jωβ + Λ
)
Y T = YH (ω ) −1Y T
−1
(19)
Hence, the equivalent power flow eigenproblem given at equation (12) can be re-expressed by
Γi = Y H i (ω ) −1 Y T
(20)
and finally allows re-formulating the power flow eigenvalues expression as follows −1
−1
FSF T = Y H i Y T S = ZH i Z T assuming that Z
(21)
= F TY .
Then, supposing the mass matrix is proportional to the identity matrix, i.e. M
ZZ T =
1 In m
= mI n , one finds that (22)
Substituting equation (22) into equation (21) leads to
SZ = where both
Hi
−1
1 −1 ZH i m
(23)
and S are diagonal matrices. Consequently, an expression of the power flow eigenvalues is
given by the general term of the latter matrix
sυ (ω ) = −
βν ,υ ω 1 <0 m ων2 − ω 2 2 + (βν ,υ ω )2
(
)
(24)
Furthermore, one obtains that SZ = ZS , which means that Z is also a diagonal matrix and therefore, power flow and natural vibration eigenvectors can be related as follows
Z = F TY =
1 m
In
(25)
Equations (24) and (25) clearly state the relationship between both power flow and natural vibration eigenmodes. However further investigations are needed to generalize this interesting property, formulated in the particular assumption of a diagonal mass matrix.
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807 4. Application 4.1.
Model description SS1 SS2
y ș
x
Fig. 8: Two junction coupled beams
In this section, the power flow mode method is applied to a multimode coupled structure. It is constituted of two clamped-free beams having identical geometric properties but different material parameters. They are also discretized in the same way (same 2D-beam element formulation, same number of nodes) and a proportional damping assumption is made. These two substructures are coupled by means of two complete junctions (see Fig. 8). The second one is excited at its free end by a constant harmonic force of 10N, with a 45° positioning. A dynamic study of the structure i s carried out to determine its natural vibration modes and to compute the dynamic response at the junction points (see Fig 9): six resonances can be seen on this frequency band. -2
10
28 -4
36 42
Amplitude (m)
10
-6
10
-8
10
-10
10
0
50
100 150 Frequency (Hz)
200
250
Fig. 9: Structure harmonic response at nodes 28, 36 and 42 in the y direction
4.2.
Power flow modes
After having computed the dynamic flexibility matrix of the whole structure and extracted its imaginary part, the power flow eigenproblem is solved at the interface nodes between the two substructures. Figure 10 presents the variation of the eigenvalues versus frequency. These are all negative and six power flow resonances can be seen at about the same frequencies as natural vibration ones. One should also notice the importance of the two first eigenvalues, containing the principal power flow part. Both loci veering and crossing can be observed. -20
Eigenvalue
-10
-10
-10
0
50
100 150 Frequency (Hz)
200
Fig. 10: Power flow eigenvalues of the system vs. frequency
250
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808 To evaluate if the external force applied to the second substructure implies an important vibration transmission at the junctions with the first one, interface forces are determined by a substructuring approach: 1 SS 1 1 SS 1 f j = Z SS + Z SS ji X i jj X j
(26)
i and j respectively denote internal and junction degree-of-freedom, Z SS1 is the dynamic SS 1 stiffness matrix of the first substructure and X its dynamic response (the same computation can be where subscripts
performed with regard to the second substructure).
1 3 Eigenvectors
0.5 2
0 -0.5
1 0
100 150 200 Frequency (Hz) Fig. 11: Interface forces projection coefficient on the subspace constituted by the three first power flow eigenvectors
Thus, Figure 11 represents the
50
α υR
coefficients determined by the projection of
f j on the subspace
constituted by the three first power flow modes. The dominant influence of the two first modes is again observed whereas the third one seems to have a negligible influence over 50Hz. 5. Conclusion A power flow mode method has been developed to characterize the interface forces at the junction between two substructures. It is based on the minimization/maximization of the averaged dissipated power flow at the interface and allows determining quantitative and qualitative parameters of the vibration transmission, respectively by means of the eigenvalues and eigenvectors of the dynamic flexibility matrix. These parameters have been studied with regard to both frequency and systems’ parameters, enlightening complex properties. A complementary approach has been developed to relate these power flow modes to the natural vibration modes of a structure. Furthermore, the method has been applied to a multimodal couple system to visualize the interface forces decomposition on a principal power flow subspace. On going works focus on modal truncation and density of the power flow space, and a method generalization to multi-interfaces structures. References [1] [2]
[3] [4] [5] [6] [7]
Gonidou L.-O., Dynamic characterization of structural interfaces. In Proceeding of The Spacecraft and Launch Vehicle Dynamic Environments Workshop, 2007. Goyder H.G.D. and White R.G., Vibrational power flow from machines into built-up structures, Part I: introduction and approximate analyses of beam and plate-like foundations. Journal of Sound and Vibration 68, 59-75, 1980. Goyder H.G.D. and White R.G., Vibrational power flow from machines into built-up structures, Part III: power flow through isolation systems. Journal of Sound and Vibration 68, 97-117, 1980. Pinnington R.J. and White R.G., Power flow through machine isolators to resonant and non-resonant beams. Journal of Sound and Vibration 75, 179-197, 1981. Cuschieri J.M., Vibration transmission through periodic structures using a mobility power flow approach. Journal of Sound and Vibration 143, 65-74, 1990. Langley R.S., Analysis of power flow in beams and frameworks using the direct-dynamic stiffness method. Journal of Sound and Vibration 136, 439-452, 1990. Miller D.W., Hall S.R. and Von Flotow A.H., Optimal control of power flow at structural junctions. Journal of Sound and Vibration 140, 475-497, 1990.
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[12]
Su J., Moorhouse A.T. and Gibbs B.M., Towards a practical characterization for structure-borne sound sources based on mobility techniques. Journal of Sound and Vibration 185, 737-741, 1995. Ji L., Mace B.R. and Pinnington R.J., A power mode approach to estimating vibrational power transmitted by multiple sources. Journal of Sound and Vibration 265, 387-399, 2003. Ji L., Mace B.R. and Pinnington R.J., Estimation of power transmission to a flexible receiver from a stiff source using a power mode approach. Journal of Sound and Vibration 268, 525-542, 2003. Xiong Y.P., Xing J.T. and Price W.G., A power flow mode theory based on a system’s damping distribution and power flow design approaches. In Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 461, 3381-3411, 2005. Pierre C., Mode localization and eigenvalue loci veering phenomena in disordered structures. Journal of Sound and Vibration 126, 485-502, 1988.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Converting CT Scans of a Stradivari Violin to a FEM
Michael Pyrkosz1, Charles Van Karsen1, George Bissinger2 1
Dynamic Systems Laboratory, Mechanical Engineering-Engineering Mechanics Michigan Technological University 1400 Townsend Drive Houghton, MI 49931, USA
2
Physics Department East Carolina University Greenville, NC 27858, USA
Abstract This study is part of an ongoing project to reverse engineer the structural and acoustic behavior of the Titian Stradivari violin. Several violins, ranging from some low quality factory fiddles, to those from Stradivari’s “golden era” have been measured using a medical Computed Tomography (CT) scanner. This paper discusses the process used for converting the CT scan data into a Finite Element (FE) model. With comprehensive density data from the CT scans, accurate frequencies and modeshapes from a true 3D modal analysis and a complete acoustic scan over a sphere, the Titian FE model has unique potential to allow extraction of the elastic moduli of this violin without the disassembly needed to test the various substructures. There will also be a discussion of determining material properties to be used in the model that will be correlated with experimental data and updated at a future date. Nomenclature Orthotropic axis directed longitudinally along the grain of the wood L R Orthotropic axis directed radially across the grain in the wood Orthotropic axis directed tangentially across the grain in the wood T HU Hounsfield Unit, on a scale from -1024 to 3072 Density of element or material i ȡi Linear coefficients A, B Average value of the voxels that are contained in element i, in HU xi Elastic modulus along axis i Eii Shear modulus in ij plane Gij İii Strain along axis i Ȗij Shear strain in ij plane ıii Stress along axis i IJij Shear stress in ij plane 1. Introduction There are many theories about how Stradivari made his instruments, and why they are superior to other violins. Most of these theories point to the material properties of the wood. Properties that are critical to the structural vibrations of a material are density, elasticity, and internal damping. The most important acoustic properties of the material are the acoustic impedance or admittance, including the radiation efficiency. High fidelity solid finite element models of several different violins, including a Stradivari violin, are constructed using geometry information gathered from a medical CT (Computed Tomography) scan. These models will be updated with material properties to more closely match the experimental results. By developing models of the different instruments, cross comparisons can be made easily.
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The data from a medical CT scanner provides accurate geometry and density information. This information is then interpreted in order to build the finite element (FE) model. This paper focuses on the process of going from the CT scan data to a usable FE model. Dr. George Bissinger, of East Carolina University, has done extensive research on violins, especially those of Antonio Stradivari. This project is an extension of his work to reverse engineer a violin (labeled VIOlin Computer Aided Design Engineering Analysis System, or VIOCADEAS) [1], [2]. VIOCADEAS is an approach to studying the violin that involves combining simulation and experimental modal analysis to determine the material properties that govern its dynamic behavior [1], [2]. One of the important aspects of Bissinger’s approach is developing Finite Element (FE) models with violin-specific geometry and density information through the use of medical CT X-ray scans. Bissinger had CT scans performed on two Stradivari instruments, (Titian 1715, and Willemotte 1734), as well as a Guarneri del Gesu, a contemporary of Stradivari, (Plowden, 1735), [1]-[3]. This data will be utilized to develop FE models for these specific violins. Keeping the density and geometry parameters of the model constant simplifies the task of determining elastic moduli for the different types of wood, through correlation and updating with experimental data [2]. The experimental modal analysis of the violins being studied is another important aspect to Bissinger’s approach. A significant portion of the data in VIOCADEAS is a database of normal modes of several violins ranging in quality from “bad”, “good”, and “excellent” [2], as well as several experimental violins known as the Hutchins-Schelleng violin octet, [4]. The total population of the current study includes a larger number of violins for comparisons. In addition to the old Italian instruments, CT scans have been performed on three factory violins, as well as several intermediate quality violins. The overall goal is to eventually have correlated models for each violin to compare. However, since this paper is primarily concerned with the process used for converting the CT data to a FE model, the discussion will mostly be limited to one violin in particular, the Titian Stradivari. The Titian is unique in that not only is it a considered to be one of Stradivari’s finest violins, it is the only one of his instruments to have measurements of 3D mobility, radiativity around a sphere, and a high quality CT scan [1]. This means that updating the solid model can be extended beyond just surface normal motion. Also, although there have been a few vacuum FE models of violins constructed over the years, none have included accurate anisotropic density data, vibroacoustic analysis, or correlation with experimental data. 2. Method 2.1. Computed Tomography (CT) Scan One of the difficulties in modeling a violin is that it is not an easily definable engineering structure. Its shape is a complex arrangement of curves that are handcarved from various species of woods, whose material properties are distinctly heterogeneous. For these reasons the CT scanner presents itself as a useful tool for reverseengineering a violin. Computed Tomography (CT) is a medical imaging method similar to an X-ray, where individual cross-section slices of density information of a patient are taken. By taking a multitude of these single slice images along the length of the body, three-dimensional density information is collected and digitized. This data can then be used to reconstruct various aspects of the patient (organs, skeletal structure, etc.) to be used in diagnosis and treatment planning. First, accurate geometry information can be determined through edge detection. Secondly, accurate density data is available at any location within the structure. A CT scanner digitizes density information on a scale of 4096 values (i.e. 12 bits), which are commonly referred to as Hounsfield Units (HU). The scale used by the CT scanner for the violin data ranges from -1024 to 3071, where -1024 represents the density of air and 0 represents that of water. The digitized data is sampled in three dimensions. The resolution in the XY directions is defined by the pixels per unit length of the individual slices. The resolution in the Z direction is defined by the number of slices per unit length. The old Italian instruments were scanned with an XY resolution of 1.462 pixels/mm (or 0.684 mm/pixel). The Z resolution of all the violin scans was 1 slice/mm. Where the smallest unit of 2D image data (one slice) is commonly referred to as a pixel, the smallest unit of 3D volume data is generally referred to as a voxel (Volumetric pixel). The voxel size of the 3 3 violin data therefore ranges from 0.343 mm to 0.604 mm .
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It has been shown that the typical behavior of a CT scanner is that the calibration curve consists of two linear relationships, one for values less dense than water, and another for values more dense than water [5]. Fortunately, most wood is less dense than water and therefore only one relationship will be required for the violin data. Error! Reference source not found. shows a picture of two violins being CT scanned. Note that the violins are supported by foam which can be easily separated out later due to its lack of density.
Figure 2-1: Violins being CT scanned
2.2. Interpretation of the CT Scan Data The first step in processing the CT-scan data is to import it into a CT interpretation program. This type of program allows the user to segment the images and render them as three-dimensional objects. The program used here doing this was Mimics by Materialise. Segmentation is a process of separating specific structures of interest based on density information. This is accomplished by creating a mask using a threshold of HU. For processing the violin data, thresholding is primarily used to separate the violins from the foam supports. An upper threshold can also be used for removing various metal components. It may be tempting to use thresholding to separate out different species of wood, since spruce is a softwood, and maple is a hardwood, and ebony is considerably more dense than both. However, there is enough overlap in density between summer and winter growth that this proves impractical. Once the thresholding step is complete, the violins can be separated into different masks. Since the threshold effectively removed the foam supports from the mask each violin should be floating freely, and separating them is trivial. Unfortunately, the scans contain some artifacts that interfere with this process. Line-artifacts can be caused by abrupt changes in density from one material to another that exceeds the dynamic range of the processing
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electronics. The result is “scatter” noise, which appears as bright and dark lines that radiate away the offending area in the scan. Metallic objects are particularly troublesome since iron and steel can completely extinguish or reflect the X-ray. Mask editing tools within the CT interpretation software can be used to remove the effects of scatter on the calculated mask. Figure 2-2 demonstrates the process of correcting the mask to remove scatter caused by a fine tuner. It should be noted that this method only effectively changes the shape of the calculated object; the HU values within the mask, and therefore the calculated density, will still be affected by the noise. One possible correction may be to edit the images directly, but the values that are affected will have to be interpolated based on the nearby slices.
Figure 2-2: Example of mask editing to remove artifacts
It is also necessary to separate out the different parts of each violin. The reason for this is to simplify material property assignment of each wood part based on its species and grain orientation. Again, the mask editing tools in the CT interpretation software can be used for separating out the different parts. Once the masks have all been cleaned of artifacts and the parts separated, each mask can be used to create 3D objects. In calculating the 3D object, the surfaces of the object are triangulated based on the edges of the mask. This effectively creates a basic triangular surface mesh of the object. Different interpolation methods are available to account for voxels that are only partially filled by the volume of the object. By separating all the parts into separate masks, the best set of parameters can be applied when calculating each 3D object. For example, the thicknesses of the top and bottom plates of the violin are critical since they will act as the primary resonators of the instrument. These were compared to measurements of the Titian top and bottom plates to ensure accuracy. However, obtaining accurate rib thickness is not as straight forward. Due to the ribs being so thin (approximately 1.0 mm), the majority of the voxels that contain information on the ribs also contain a fair amount of air. This leads to the average density within a voxel to be estimated considerably lower, and using the same threshold on the ribs as the backplate, (since they are both maple) creates holes in the ribs. Therefore the rib dimensions are estimated from contours around the rib voxels rather than on the specific density data.
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Figure 2-3: Screen capture from Mimics depicting separation of parts and 3D object calculation
2.3. Meshing At this point the 3D objects exist as triangular surface mesh over the boundary between the air and solid material. The next step is to remesh this surface to improve the mesh quality. This optimizes the mesh for FEA. This includes reducing the amount of detail, reducing the number of triangles, and improving the quality of the triangles. For structural vibration simulation fine detail in the mesh is not generally necessary and can lead to slow computation time and numerical instability. Reducing the amount of detail increases efficiency and stability, without sacrificing significant accuracy. Reducing the number of triangles also improves computation time, but can lead to lower quality triangles. Therefore it is necessary to remesh using a quality threshold based on various shape parameters. Finally, it is necessary to ensure that none of the triangles intersect. Once the 3D object has been remeshed, it is exported to the FEA preprocessor which is used to generate a volumetric mesh within the surface mesh. The preprocessor used on the violins is LMS Virtual.Lab since it is has tools for performing both structural and acoustic studies. The volume enclosed by the surface mesh is filled in using tetrahedrons. It should be apparent that the quality of the created solid mesh is largely dependent on that of the surface mesh. One important consideration here is the type of tetrahedron elements to use. Linear elements contain only four nodes that are connected by straight edges. These are commonly called TETRA4 elements. Parabolic elements contain ten nodes, four on the vertices of the tetrahedron and six intermediate nodes on the edges. These intermediate nodes define curved edges between the vertices, and increase the overall number of degrees of freedom without having to decrease the elements size. These are called TETRA10 elements. TETRA4 elements are easier to implement, but TETRA10 elements are preferred as they improve accuracy. Figure 2-4 depicts the result of computing a solid tetrahedron mesh for the topplate of the Titian Stradivari.
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Figure 2-4: Screen capture of tetrahedron mesh in Virtual.Lab
2.4. Material Property Definition Before finite element analysis can be performed several material properties need to be defined. Among these are density, elasticity, and structural damping. Most of these properties will eventually need to be updated during the correlation process so that the final model will reflect the structural characteristics of each particular violin. It is also noted here that the different material properties can be defined for each of the separate parts to account for each species of wood and grain orientation. The CT scan data provides highly detailed density information. This data can be extracted and placed into the model. After the volumetric mesh has been created in the FEA preprocessor, it is imported back into the CT interpretation software for extracting the density information and assigning it to the material properties. The range of gray values or Hounsfield Units in the CT scan data is divided up into a number of groups that are assigned a specific density value. To do this, the relationship between HU and density must be defined within the software. This relationship has been shown to be linear [5]; from -1024 HU = ȡair to 0 HU = ȡwater. (see Equation 1).
Ui
A xi B
(1)
gm , xi is the average value of the voxels, in HU, that define the (mm)3 gm gm 3 element, A | 975.4 , and B | 1.00 u 10 . 3 (mm) HU (mm)3 Where ȡi is the density of the element in
This is a fairly straight forward calculation and is easily compared to the specific weight of some known wood species (specific weight is density normalized with respect to water). Note that the range of gray values (i.e.
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Hounsfield units) contained within the mask is divided into a smaller number of approximate values. decreases the total number of material groups generated simplifying data management.
This
Mechanically, wood is typically treated as an orthotropic material; that is, mechanical properties are unique along orthogonal axes [6]. These axes consist of the longitudinal (L) axis along the grain, the radial (R) axis across the grain normal to the growth rings, and the tangential (T) axis across the grain but tangent to the growth rings. Due to the structure of the wood, the properties such as elastic and shear moduli, and poisson’s ratios are different on each of these axes. Most of the studies that have been done on measuring the properties of wood are done by bending beams; and the results are only given in one direction, usually the longitudinal. There is data on a few species of wood for properties along the other axes, but this is limited. This issue is complicated further by elasticity of the wood being effected by moisture content. Other properties of interest are the speed of sound traveling within the wood and the internal damping. This is primarily a function of the elasticity and density, and as a result also dependent on the grain direction and moisture content. In general, since the cross-gain elastic moduli are significantly smaller than the longitudinal elastic modulus, the speed of sound a cross the grain is usually smaller than it is along the grain. Also, the speed of sound decreases proportionally with an increase in temperature or in moisture content, depending on species [6]. The internal friction or damping of wood depends greatly temperature and moisture content [7]. Internal damping is also particularly troublesome because there are no hard numbers available for most species of wood. Bissinger has measured the total damping of the Titian Stardivari violins as a sum of radiation damping and internal damping [1]. By computing radiation efficiency and radiation damping a semi-empirical internal damping value can be assigned to each violin. However, this means that the internal damping property may not be fully updated until an acoustic simulation is performed to ensure that the total damping is realistic. A few studies have been done to show a linear regression between the longitudinal modulus of elasticity and density for certain samples of softwoods, (primarily Norway spruce) [7]. These are simply average properties of the samples used, however, in this spirit, it may be possible to determine a linear relationship for the density and elasticity changes within a sample. During material assignment within the CT interpretation software such a relationship between density and elasticity may be included, such that the mesh groups created for different density ranges would also have a proportional elastic modulus assigned. However, this elasticity assignment does not account for the orthotropic nature of the wood; that is, elasticity is different for each density group created, but equal along each orthogonal axis. To account for the orthotropic properties, elasticity and Poisson’s ratio must be adjusted for the materials used in the model. It can be shown that an orthotropic material is a specific case of an anisotropic material. This is shown Equation 2, which gives the compliance tensor of an orthotropic material.
H LL ½ °H ° ° RR ° °°H TT °° ® ¾ °J LR ° °J RT ° ° ° ¯° J TL ¿°
ª 1 «E « LL « Q LR «E « LL « Q LT « « ELL « « « « « « « « ¬
Q RL ERR
Q TL ETT
1 ERR
Q TR ETT
Q RT ERR
1 ETT
zeros
1 GLR zeros
1 GRT
º » » » » V ½ » ° LL ° » °V RR ° »° » °®V TT °°¾ » ° W LR ° »° » °W RT °° » ¯° W TL ¿° » » 1 » » GTR ¼
(2)
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The compliance matrix can be inverted to determine the stiffness matrix to be used in the material definition. Matlab is used to solve transverse terms based on average ratios from similar species listed in “The Encyclopedia of Wood” [6]. The result is then exported into a format compatible with a Nastran MAT9 material definition for anisotropic materials. 3. Results One question that arises is how complex does the model need to be in order to accurately simulate each violin in question. It is a fairly straightforward task to create an anisotropic material in the model, and using CT interpretation software makes it possible to account for the non-uniform density distribution in the wood. However, including both of these details in the model is a little more complicated, since each density group needs to be assigned a different anisotropic elasticity matrix, each containing 9 independent values. This adds up to a lot of properties to keep track of, and will be more than a little cumbersome to correlate later on. In order to gain a better understanding of what effect different modeling complexities have on the result, an initial study was done using the same mesh of the topplate from the Titian Stradivari changing the features one at a time. Five case studies were performed varying between isotropic and anisotropic elasticity, uniform and distributed density, and utilizing TETRA4 and TETRA10 elements within the mesh. The results of the first five modes and modeshapes for each of these cases are shown in Table 3-1 and Figure 3-1. Blue indicates the nodal lines, and red indicates areas of maximum deflection. The modes vary tremendously, but this is not a surprise since the different elastic moduli and densities vary considerably. What is interesting is the effect on the different modeshapes. There does not seem to be much change in the first modeshape regardless of how the model is set up. Nor does the density distribution based on the CT scan data seem to make a difference when compared to the uniform density model. The reason for this is unknown. It is possible that the density groups created require more resolution to see a noticeable difference. Remember groups were created by dividing the total range of HU values contained within the mask, which includes points effected by the scatter from fine tuner.
Table 3-1: Topplate study results Element Type Density Elasticity Mode 1 Mode 2 Mode 3 Mode 4 Mode 3
Case 1
Case 2
Case 3
Case 4
Case 5
TETRA4 Uniform Isometric 332.3 494.9 821.4 922.1 1131.0
TETRA4 Distributed Isometric 273.3 394.5 669.1 753.4 901.0
TETRA4 Uniform Anisometric 219.2 330.6 535.4 556.7 622.5
TETRA10 Uniform Isometric 178.7 321.5 487.2 522.7 654.5
TETRA10 Uniform Anisometric 131.1 196.1 328.9 365.8 408.9
On the other hand, the change from isotropic to anisotropic elasticity has a dramatic effect. With the softer crossgrain moduli the structure bends more easily about the longitudinal axis than it does against the longitudinal axis, which can be seen in how the modeshapes change. What is peculiar that the effect is more dramatic with TETRA4 elements than it is with TETRA10 elements. In fact, the TETRA4 elements with anisotropic elastic moduli model (Case 3), is the only case where the modeshapes change order. The fifth modeshape actually drops below both the 3rd and 4th modeshape.
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C as e2
C as e3
C as e4
C as e5
Mode1
Mode2
Mode3
Mode4
Mode5
Figure 3-1: Modeshape results from topplate study
The next question becomes which one of these models more closely represents the actual toplate of the Titian Stradivari. This question is not so easy to answer, given the price-tag associated with the instrument. Based on experiments that have been done on violin topplates case 5 does appear to yield the most reasonable results both in mode frequency and shape. In order to better understand how to best model the wood, and to get a better idea of what the material properties ought to be, a simpler structure must be modeled that can be readily tested, and a correlation analysis performed. 4. Conclusion This project aims to determine the structural and acoustic properties of priceless, old, Italian violins, and compare them to that of cheap factory fiddles, as well as some middle class instruments. This is done by developing highfidelity FE models specific to each violin based on a CT scan of the instrument. It has been shown that this is possible with the aid of modern engineering tools like Mimics and Virtual.Lab. What remains is correlating the models to experimental data and updating the material properties. Several iterations of this correlation and updating process may have to be performed to get values representative of the actual instruments. Furthermore, in order to start out with as good of values as possible some testing, modeling, and correlation work will be done on some samples of the wood species of interest. This will make the process a lot easier, and reduce the number of iterations needed to match the model with the data.
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5. References [1] Bissinger, George, “Structural acoustics of good and bad violins,” J. Acoust. Soc. Am. Vol. 124 (3), pp. 1764-1773 (September 2008) [2]
Bissinger, George, “A Unified Materials-Normal Mode Approach To Violin Acoustics,” ACTA Acustica united with Acustica Vol. 91, pp. 214-228 (2005)
[3]
Bissinger, G., Oliver, D., “3-D Laser Vibrometry Focuses On Legendary Old Italian Violins,” Souns and Vibration magazine Vol. 41/Num. 7, pp. 10-15 (July 2007)
[4]
Fletcher, Neville H., Rossing, Thomas D., “The Physics of Musical Instruments” 2nd edition, Springer Science+Business Media, LLC, New York, NY (1998)
[5]
Saw, C. B., Loper, A., Komanduri, K., Combine, T., Huq, S., Scicutella, C., “Determination of CT-toDensity Conversion Relationship for Image-Based Treatment Planning Systems” Medical Dosimetry, Vol. 30, No. 3, pp. 145-148 (2005)
[6]
U. S. Department of Agriculture, “The Encyclopedia of Wood,” Skyhorse Pub., Inc., New York, NY (2007). Originally published by U.S.D.A. Washington, D.C. (1999)
[7]
Bucur, Voichita, “Acoustics of Wood,” CRC Press, Inc., Boca Raton, FL (1995)
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
LAN-XI – The Next Generation of Data Acquisition Systems Niels-Jørgen Jacobsen Brüel & Kjær Sound & Vibration Measurement A/S Skodsborgvej 307, DK-2850 Nærum, Denmark ABSTRACT Designing a general purpose high-quality sound and vibration data acquisition system is not a trivial matter. A number of divergent requirements must be fulfilled and experience and innovation play an important role in achieving this challenging objective. With the introduction of the LAN-XI concept, a hitherto unseen combination of requirements can be fulfilled in the same data acquisition system thereby significantly increasing the versatility of the system and at the same time reducing the amount of measurement equipment and accessories needed. This paper will highlight some of the technologies used in LAN-XI including Precision Time Protocol (PTP), Power over Ethernet (PoE), Dyn-X, REq-X and TEDS. 1
Introduction
Chapter 2 describes some of the numerous requirements that have to be taken into consideration when designing a data acquisition system for sound and vibration measurements. This includes the ability to make very accurate measurements in a fast, safe and easy manner regardless of whether the system is to be used in the field, in a lab, or in a centralized or distributed setup. In addition, modularity, scalability and plug & play are functionality expected from a modern data acquisition system. Most data acquisition systems, however, only fulfil a very limited number of these requirements and consequently make sound & vibration measurements complicated, timeconsuming, costly, less accurate and in worst cases even impossible. In Chapter 3 the philosophy and technologies behind the LAN-XI concept are explained. It is shown how the combined use of several different technologies ranging from powering the data acquisition system, over sample synchronisation techniques to correcting the measured data in real-time makes it possible to design a data acquisition system meeting the most demanding requirements for sound & vibration measurements. The conclusion is presented in Chapter 4. 2
Requirements to a Sound & Vibration Data Acquisition System
Planning, setting up and performing sound & vibration measurements can be very different from one test to another. A number of factors have to be taken into account including the nature of the test object, the test environment and the operating conditions. In addition, high quality data must be obtained in the shortest possible time with the lowest possible cost. 2.1
Nature of Test Object
Sound & vibration measurements are performed on the smallest components like hearing aids to the largest structures like aircraft, ships and bridges. Measurements range from a few channels to hundreds of channels and a versatile data acquisition system must therefore be scalable, modular and easily reconfigurable. The same data acquisition system capable of doing a large multi-channel measurement using up to several rack systems one day, should be easily dividable into multiple systems the next day for doing small-scale measurements at different locations without compromising performance, ease-of-use or cost.
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In general, it is advantageous to place the data acquisition system as close as possible to the test object to shorten the length of the transducer cables. Apart from significant cost savings on expensive high-quality transducer cables and the lower risk of setup and measurement mistakes due to reduced cable “infrastructure”, short cables minimise the risk of adding measurement noise. For large test objects, this requires that small front-ends – or individual modules - can be freely distributed around – or inside - the test object and that the measurements can be made 100% sample synchronously between all measurement channels. Many traditional systems are, however, neither distributable nor capable of ensuring sample synchronisation between front-ends. Newer systems have offered various cable-based synchronisation techniques between the individual front-ends, but all with the disadvantage of requiring extra cabling. 2.2
Test Environment
Sound & vibration measurements can roughly be divided into in-situ field, lab and test cell measurements. Field measurements are often performed in harsh environments placing high demands on the robustness of the data acquisition system. In addition, the data acquisition system must be light weight, small and truly portable, have low power consumption and have the option of battery operation and the possibility of distributing the front-ends. For fixed lab or test cell setups, a centralised data acquisition system with one or more rack systems has traditionally been used. Long transducer cables have been required with the disadvantages previously mentioned. In both test environments the use of distributed front-ends could significantly reduced the required cabling. Figure 1 shows a traditional test cell setup using long transducer cables between connector panels in the various test cells and the operator room.
Figure 1. Traditional test cell setup requiring extensive cabling.
2.3
Operating Conditions and Data Quality
In many sound & vibration measurements the signal levels are often unknown and/or uncontrollable. Traditionally this has been tried compensated for by performing numerous trial runs and auto-ranging procedures to set the input attenuators in the most optimal way. However, this is both a very time-consuming and error prone procedure. Consequently, to get the measurements right the first time, a high dynamic range throughout the complete measurement is required to avoid erroneous results from overload and under-range situations. Another characteristic of many vibration measurements is the high content of low frequency components due to the large size of many structures (bridges, buildings, aircraft, ships etc.). The data acquisition system including transducers must be able to measure down to DC.
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2.4
Time and Cost Constraints
Sound & vibration measurements are often subject to extreme time and cost constraints and can often not be redone. Planning, setting up and performing the measurements in a fast, safe and easy manner is therefore of outmost importance. Ease-of-use can be significantly improved using true Plug & Play functionality throughout the complete measurement chain. Ideally it should be possible just to attach the transducers to the data acquisition system, connect the data acquisition system to the PC, open the data acquisition software and then just press the “Start” button and get the results right the first time. As a result of the extreme time constraints there is, however, an impending risk of setup and measurement mistakes. Intelligent user feedback from the data acquisition system is consequently mandatory to quickly and easily detect and correct these potential errors. The information should preferably come from each front-end, module and channel in terms of identification, calibration, conditioning and measurement status. 2.5
Versatility
A key requirement to most data acquisition systems is versatility. A versatile data acquisition system must support different types of sound & vibration applications like structural dynamics, rotating machinery diagnostics, acoustics, electro-acoustics and vibro-acoustics. This gives a range of new requirements to the data acquisition system including powerful generator functionalities, tacho probe support, silent operation mode and support of low-frequency process channels. These requirements will, however, not be discussed further in this paper. 3
The Philosophy and Technologies behind LAN-XI
The LAN-XI concept has been developed from the philosophy “Less is More” to ensure very accurate sound & vibration measurements in a fast, safe and easy manner at the lowest possible cost. By using a very modular and highly reconfigurable architecture, a minimum of data acquisition hardware, accessories and cables are required and the same data acquisition system can quickly be configured for field, lab or test cell use based on a centralised or distributed setup. Core technologies behind the LAN-XI concept are Precision Time Protocol (PTP), Power over Ethernet (PoE), Dyn-X, REq-X and Transducer Electronic Data Sheets (TEDS). 3.1
Precision Time Protocol (PTP)
With LAN-XI a new technique is introduced to ensure sample-synchronous measurements over the same LAN connection used for transferring measurement data. This simplifies the data acquisition system’s cabling and makes it possible to perform sample-synchronous measurements over long distances, eliminating the effect of delays over the cables and switches. The technique is based on the Precision Time Protocol (PTP) described in the IEEE 1588-2002 standard [1] of which an updated version was approved in March 2008, see IEEE 1588-2008 [2]. PTP enables precise synchronisation of PTP devices on a network, e.g. Ethernet. Synchronisation with sub microsecond accuracy can be achieved using hardware generated time stamps. PTP synchronisation thus provides a whole new set of possibilities for combining data acquisition systems located in different places, closer to the actual measurement points and with long distances between. Only a LAN connection is required. By measuring the delays between the individual PTP devices, all clocks on the network can be set to exactly the same time. This is done using a hierarchy of a Master clock and one or more Slave clocks. The Master clock is the most accurate clock present on the network and if a PTP device with a more accurate clock is added to the network, the Master clock will pass the role as Master to the better clock and become a Slave. The basic principle of PTP synchronisation is shown in Figure 2.
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Figure 2. Basic principle of PTP synchronisation. The Master clock continuously sends out time stamped synchronisation messages to the Slaves. When the Slave clocks receive the synchronisation messages they are time stamped as well. The difference between the time stamp of the Master and the Slave is equal to the network propagation delay plus the offset between the Master and the Slave clock. The Slave now adjusts its clock, thereby reducing the difference between the two clocks to the network propagation delay. All time stamping is implemented in the hardware of the PTP devices to avoid the variable delays that would be caused by software time stamping. The PTP synchronisation technique assumes that the propagation is the same from the Master to a Slave as from a Slave to the Master. Using this assumption, the Slave sends a time stamped Delay Request to the Master that promptly returns a Delay Response stamped with the time at which it received the request. The difference between these two time stamps is the network propagation delay and the Slaves can thus adjust their clocks to match the Master clock. This simplified description does not take oscillator errors into account. A simple servo implementation handles that. On top of the PTP synchronisation, the LAN-XI concept also corrects the unavoidable phase drifts of the Slave clocks by continuously measuring and counter-adjusting the Slave clocks. The PTP is independent of the network topology and it is self-adjusting to actual system setup in terms of selecting the most accurate clock and adapting to the actual delay in the network. This makes it very easy to set up a data acquisition system using PTP to synchronise multiple front-ends. As all synchronisation is done by the PTP devices on the network, a data acquisition system with PTP complying front-ends can operate on a standard network. Switches used in standard LAN do not include any special features to support PTP, but special PTP switches are available from manufacturers of network backbone devices. However, the typical phase deviation measured at 25.6 kHz in a network using a standard 1 Gigabit LAN switch is less than 1 degree, see Figure 3.
Figure 3. Typical phase deviation using a standard 1Gigabit LAN switch. Phase error versus time.
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The “channel x bandwidth” performance is practically unlimited using LAN-XI. With 1 Gigabit LAN data backbone in each LAN-XI front-end measurements from 2 to more than 1000 channels can easily be measured with the same system. 3.2
Power over Ethernet (PoE)
Power over Ethernet (PoE) is a technology that allows power to be transmitted along with measurement data and sample synchronisation (PTP) on a LAN cable in an Ethernet network. The technology is specified in the IEEE 802.3af-2003 standard [3] and updated in the IEEE 802.3-2005 standard [4]. Devices called Power Sourcing Equipment (PSE) supply the required power. A PSE can be an in-line power injector or a LAN switch/hub. A device powered by a PSE is called a Powered Device (PD). Standard LAN cables can be used, but for long distance operation CAT-6 LAN cables are required. An example of communication between a PSE and a PD is shown in Figure 4.
Figure 4. Example of PoE communication between a power source and a powered device. When the PSE is turned on it starts a detection cycle in order to find out if any PD needs power. A PD that needs power will indicate this by placing a “signature resistance” (19 – 26.5 k:) and “signature capacitance” (150 nF) between the powering pairs. This is recognized by the PSE and power is turned on. The PSE injects between 36 V and 57 V (usually 48 V) into the cable and must be able to provide up to 15.4 W with max. 400 mA. In order to prevent overloading as well as powering open lines, the PSE will turn off the power if the resistance becomes less than 15 k: or greater than 33 k: or the capacitance becomes greater than 10 PF. The IEEE 802.3 standard specifies a set of different power classes used for PoE, see Table 1. Power Class 0 1 2 3 4
Usage Default Optional Optional Optional Reserved for future use
Input Power Level [W] 0.44 – 12.95 0.44 – 3.84 3.84 – 6.49 6.49 – 12.95 -
Table 1. Power classes for PoE.
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The PD can indicate the required power to the PSE. The communication is done in a manner similar to that described for turning on the power. If the PD requires more power than the PSE is able to supply, the power to the line is disabled and the detection cycle restarted. With a maximum cable length of 100 m, the available power for the PD will drop from 15.4 W to 12.95 W when accounting for cable loss. 3.3
PTP and PoE – One-cable Operation
The combined used of PTP and PoE in the LAN-XI concept for data acquisition systems has made it possible to drastically reduce the amount of cabling in distributed front-end setups by significantly reducing the length of the transducer cables and using the same LAN cable for data transfer, sample synchronisation and power supply, see Figure 5 (compare with Figure 1) and Figure 6. The benefits are obvious and include reduced noise and cable cost, faster setup and easier maintenance, less risk of setup and measurement errors, inexpensive LAN switches that can replace expensive Patch Panels and no need for additional power outlets. To support the concepts of distributed front-ends and field use, the LAN-XI modules are cast in magnesium for maximum stability and light weight and battery operation is supported.
Figure 5. Using LAN-XI, cabling between test cells and operator room is drastically reduced.
Figure 6. Using LAN-XI, cabling can be drastically reduced for measurements on large structures.
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3.4
Dyn-X Technology
With the introduction of the Dyn-X (Dynamics eXtreme) technology, data acquisition systems for the first time fully match or outperform the dynamic performance of high-quality transducers as explained in Jacobsen et al. [5]. A Dyn-X input channel using a single input range has a useful analysis range of 160 dB narrowband and more than 125 dB broadband, see Figure 7.
Figure 7. 160 dB analysis in one range. FFT analysis of a 1 kHz signal 80 dB below full scale (7 Vrms). Noise and all spurious components measure 160 dB below full scale input. The Dyn-X technology utilises a special analog input design to provide a very high dynamic range of the analog circuit, pre-conditioning the transducer signal before feeding it to the ADC. A Dyn-X input channel has no input attenuators, but uses one of two input ranges. A general purpose 10Vpeak range or an extended 31.6Vpeak range. The digitising is performed synchronously in two specially selected, high-quality, 24-bit delta-sigma ADCs, and both data streams are fed to the DSP environment where dedicated algorithms in real-time merge the signals while obtaining an extremely high accuracy match in gain, offset and phase, see Figure 8.
Figure 8. A simplified block diagram of the Dyn-X technology. With no setting of input ranges, and with no need to be concerned about overloads, under-range measurements and the accuracy of the measurements done, the ease and safety of measuring are drastically increased using Dyn-X technology. And with no need for trial runs to ensure correct input ranges for the various input channels, the certainty of getting the measurements right the first time is also significantly increased.
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3.5
REq-X Technology
Transducer Response Equalisation eXtreme (REq-X) is a recent technique that flattens and stretches the frequency response of accelerometers, microphones and couplers in real-time, see Gade et al. [6]. This extends the useful frequency range of the transducers, improves the accuracy of the measurements and expands the use of existing transducers. As an example, for a correctly mounted accelerometer REq-X will increase the usable frequency range from 1/3 of the accelerometer’s resonance frequency to 1/2 - an increase of 50%. Consequently, an accelerometer optimised for low frequency measurements can now be used for more general purpose tasks. The REq-X technique corrects the time signal of a transducer by using the impulse response of the inverse of its calibrated frequency response as shown in Figure 9. Both amplitude and phase are corrected in real-time.
Figure 9. Basic principle of REq-X. Upper red curve shows transducer response before equalisation.
3.6
Transducer Electronic Data Sheet (TEDS)
Transducers with Transducer Electronic Data Sheet (TEDS) contain information about their sensitivity, type number, serial number, manufacturer, calibration date etc. When a transducer with TEDS is connected to an input module supporting TEDS, it is automatically detected and its data read into the hardware and analyzer setups. TEDS is specified in the IEEE 1451.4-2004 standard [7]. 3.7
Plug & Play Functionality
To ease setting the LAN-XI data acquisition system up, each module has its own built-in network interface that can be configured to use dynamic or static IP addressing. Using dynamic IP addressing (default), the modules automatically receive their IP addresses from a DCHP server on the network. If modules are connected directly to a PC, the modules will use “link-local” (“auto-IP”) and addresses in the 169.254.xxx.xxx range are selected. A Windows® XP/Windows Vista®/Windows® 7 PC will do the same and the PC and modules can now communicate. The dynamic IP addressing together with the Dyn-X, REq-X and TEDS technologies ensure true Plug & Play functionality. You basically just attach the TEDS transducers to the LAN-XI modules, connect the modules/frontends to the PC, open the data acquisition software and then press the “Start” button and get the results right the first time. If, however, errors occur due to setup or measurement mistakes, the LAN-XI data acquisition system includes intelligent user feedback from each front-end, module and channel in terms of identification, calibration, conditioning and measurement status to quickly and easily detect and correct these potential errors.
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4
Conclusion
Designing a modern data acquisition system for sound & vibration measurements is not a trivial matter as numerous often divergent requirements have to be taken into consideration. Thus many data acquisition systems are either optimised for dedicated use or are the result of a compromise. It has been shown with the description of the LAN-XI concept for data acquisition systems how a hitherto unseen combination of requirements can be fulfilled in the same system by combining a number of different technologies like PTP, PoE, Dyn-X, REq-X and TEDS. The same data acquisition system can now be configured from 2 to more than 1000 channels and be used in a centralised or distributed setup either in the field or in a lab environment without compromising data quality, performance or ease-of-use. The LAN-XI concept significantly reduces the amount of measurement equipment, cabling and other accessories needed thereby requiring less investment in hardware, less setup and maintenance time and lower risk of potential setup and measurement errors. References [1]
IEEE Standards Association Standard for a Precision Clock Synchronization Protocol for Networked Measurement and Control Systems IEEE 1588-2002
[2]
IEEE Standards Association Standard for a Precision Clock Synchronization Protocol for Networked Measurement and Control Systems IEEE 1588-2008
[3]
IEEE Standards Association LAN/MAN - Specific requirements Part 3: Carrier Sense Multiple Access with Collision Detection (CSMA/CD) Access Method IEEE 802.3af-2003
[4]
IEEE Standards Association LAN/MAN - Specific requirements Part 3: Carrier Sense Multiple Access with Collision Detection (CSMA/CD) Access Method IEEE 802.3-2005
[5]
Jacobsen, N-J; Thorhauge, O New Technology increases the Dynamic Ranges of Data Acquisition Systems IMAC XXIV Conference, 2006
[6]
Gade, S; Schack, T; Herlufsen, H; Thorhauge, O Transducer Response Equalisation IMAC XXVI Conference, 2008
[7]
IEEE Standards Association Standard for a Smart Transducer Interface for Sensors and Actuators - Mixed-Mode Communication Protocols and Transducer Electronic Data Sheet (TEDS) Formats IEEE 1451.4-2004
BookID 214574_ChapID FM_Proof# 1 - 23/04/2011
BookID 214574_ChapID 73_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Ultrasonic Guided Wave Modal Analysis Technique(UMAT) for Defect Detection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Figure 2 Von Mises stress distributions across the thickness of a 1mm thick aluminum plate for the A0 mode at 500 kHz and the S0 mode at 1.35 MHz (captured in static frames of some FEM wave propagation animation studies). Finite Element Simulation Results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
Figure 3
Geometry of the aluminum plate with an annular array loading.
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Figure 4 Steady state modal vibration fields for out-of-plane displacement (a) defect free plate and (b) plate with a 1 cm2 surface defect when the A0 mode was chosen. Significant differences are shown in the figures. As predicted, excellent defect detection results can be obtained.
Surface defect
D
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Figure 5 Steady state modal vibration fields for out-of-plane displacement (a) defect free plate and (b) plate with a 1 cm2 surface defect when the S0 mode was chosen. Only minor differences are shown in the figures. As predicted, defect detection possibility is poor. Experimental Results .H\ H[SHULPHQWV ZHUH FRQGXFWHG WR GHPRQVWUDWH WKH IHDVLELOLW\ RI DSSO\LQJ WKH QRYHO 80$7 PHWKRG WR GHWHFWGHIHFWVLQERWKPHWDOOLFDQGDQL VRWURSLFFRPSRVLWHSODWHV7KHILUVWH[SHULPHQWZDVFDUULHGRXWRQ DQDOXPLQXPSODWHXVLQJSLH]RHOHFWULFZDIHUDFWLYHVHQVRUV 3:$6V >@7KHDOXPLQXPSODWHZLWKWKH 3:$6V DWWDFKHG RQW KH VXUIDFHLV VKRZQ LQ)LJXUH $V SRLQWHG RXWLQ WKH SLFWXUH RQHRIWKH 3:$6V
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Figure 6 Aluminum plate with 5 PWASs attached to the surface for transient guided wave tests and UMAT tests. A 2 cm (0.8”) long notch defect was introduced to the aluminum plate. The orientation of the defect was parallel to the direct wave path between the transmitter and the receiver PWASs 2 and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Figure 7 Transient guided wave signals taken from the defect free plate and the plate with the notch defect. 5 cycle 352 kHz tone-burst inputs were used. Only small changes were shown in the signals after the defect was introduced, especially for the signals received by the receiver PWASs 2 and 4.
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Figure 8 Vibration signals under continuous guided wave inputs for the defect free plate (shown in blue) and the plate with the notch defect (shown in red). The frequency was 352 kHz. Significant changes introduced by the defect were observed.
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Figure 9 (Left) Composite plate with the annular array attached to the surface. The 25 measurement points and the point where an impact was applied to generate a defect are also marked on the plate. (right) Experimental setup for the laser measurements. Good measurement consistency was achieved with the setup shown in the picture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Figure 10
UMAT amplitude changes from the defect free case to the case with the impact damage. Excellent results were obtained at many positions.
Concluding Remarks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cknowledgement 7KLVZRUNLVVXSSRUWHGE\$)265XQGHU6775*UDQW)$ &WKURXJKSURJUDPPDQDJHUV'U 9LFWRU*LXUJLXWLXDQG'U'DYLG6WDUJHO References >@ (ZLQV'- Modal Testing: Theory, Practice and ApplicationVHFRQGHG5HVHDUFK 6WXGLHV3UHVV/7'%DOGRFN+HUWIRUGVKLUH(QJODQG >@ &KLPHQWL'( ³*XLGHG:DYHVLQ3ODWHVDQG7KHLU8VHLQ0DWHULDOV&KDUDFWHUL]DWLRQ´ Appl. Mech. Rev.vol. 50(5) >@ 5RVH-/ ³8OWUDVRQLF*XLGHG:DYHVLQ6WUXFWXUDO+HDOWK0RQLWRULQJ´ Key Engineering Materials 9RO 270-273 >@ 5RVH-/ Ultrasonic Waves in Solid Media&DPEULGJH8QLYHUVLW\3UHVV1HZ@ *LXUJLXWLX9 ³7XQHG/DPE:DYH([FLWDWLRQDQG'HWHFWLRQZLWK3LH]RHOHFWULF:DIHU $FWLYH6HQVRUVIRU6WUXFWXUDO+HDOWK0RQLWRULQJ´J. Intel. Mat. Sys. Struct16
BookID 214574_ChapID 74_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Damage Diagnosis of Beam-like Structures Based on Sensitivities of Principal Component Analysis Results NGUYEN Viet Ha, GOLINVAL Jean-Claude University of Liege, Aerospace & Mechanical Engineering Department, Structural Dynamics Research Group, Chemin des chevreuils, 1 B52/3, B-4000 Liège 1, Belgium NOMENCLATURE K, M, C X p U, V,Ȉ
stiffness, mass and damping matrix observation (snapshot) matrix vector of parameters matrices of left and right singular vectors and of singular values
D kji , E jik
projection coefficients
H
FRF matrix angular frequency sensitivity variation first and second derivatives of the vector '
Z
' d I , d II
ABSTRACT This paper addresses the problem of damage detection and localization in linear-form structures. Principal Component Analysis (PCA) is a popular technique for dynamic system investigation. The aim of the paper is to present a damage diagnosis method based on sensitivities of PCA results in the frequency domain. Starting from Frequency Response Functions (FRFs) measured at different locations on the beam, PCA is performed to determine the main features of the signals. Sensitivities of principal directions obtained from PCA to beam parameters are then computed and inspected according to the location of sensors; their variation from the healthy state to the damaged state indicates damage locations. It is worth noting that damage localization is performed without the need of modal identification. Once the damage has been localized, its evaluation may be quantified if a structural model is available. This evaluation is based on a model updating procedure using previously estimated sensitivities. The efficiency and limitations of the proposed method are illustrated using numerical and experimental examples. 1. Introduction A dynamic transformation, resulting from a variety of causes, e.g. structural damage or nonlinearity onset, may disturb or threaten the normal working conditions of a system. Hence, questions of the detection, localization and severity estimation of those events have attracted the attention of countless engineering researchers in recent times. The detection, localization and assessment of the damage allow to reduce maintenance costs and to ensure safety. In the last decade, the problem of damage localization and assessment has been approached from many directions. Often based on monitoring modal features, these processes can be achieved by using an analytical model and/or promptly by measurement. Damage can cause change in structural parameters, involving the mass, damping and stiffness matrices of the structure. Thus many methods deal directly with these system matrices. The Finite Element Method (FEM) is an efficient tool in this process [1]. The problem of detection may be resolved by this method through model updating or sensitivity analysis. For damage localization and evaluation, model updating is utilized to reconstruct the stiffness perturbation matrix [2]. This may be combined with a genetic
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_74, © The Society for Experimental Mechanics, Inc. 2011
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algorithm [3] or based on modal parameter sensitivity [4]. In these cases, a well fitted numerical model is essential to compare with the actual system. Methods using measurement are also widely used because of their availability in practice. Yan and Golinval [5] achieved damage localization by analyzing flexibility and stiffness without system matrices, using time data measurements. Koo et al. [6] detected and localized low-level damage in beam-like structures using deflections obtained by modal flexibility matrices. Following localization, Kim and Stubbs [7] estimated damage severity based on the mode shape of a beam structure. Rucka and Wilde [8] decomposed measured FRFs by continuous wavelet transform in order to achieve damage localization. Cao and Qiao [9] recently used a novel Laplacian scheme for damage localization. Other authors have located damage by comparing identified mode shapes [10] or their second-order derivatives [11] in varying levels of damage. Sampaio et al. [12] extended the method proposed in [11] through the use of measured FRFs. Natural frequency sensitivity has also been used extensively for the purposes of damage localization. Ray and Tian [10] discussed the sensitivity of natural frequencies with respect to the location of local damage. In that study, damage localization involved the consideration of mode shape change. Other authors [13-15] have located damage by measuring natural frequency changes both before and after the occurrence of damage. However, such methods, based on frequency sensitivity with respect to damage variables require an accurate analytical model. Jiang and Wang [16] extended the frequency sensitivity approach by eliminating that requirement. However, an optimization scheme is still needed to estimate the unknown system matrices through an identified model using input-output measurement data. It is not only the issue of localization that has become the subject of recent study; the assessment of damage is also increasingly attracting the interest of researchers [2-4, 7, 14]. Yang et al. [17] estimated damage severity by computing the current stiffness of each element. They used Hilbert-Huang spectral analysis based only on acceleration measurements using a known mass matrix assumption. Taking an alternative approach, other authors have used methods involving the updating of a finite element model of the examined structure and have used sensitivity analysis to discover the effective parameter. For example, Messina et al. [13] estimated the size of defects in a structure based on the sensitivity of frequencies with respect to damage locations where all the structural elements were considered as potentially damaged sites. Teughels and De Roeck [18] identified damage in the highway bridge. They updated both Young’s modulus and the shear modulus using an iterative sensitivity based FE model updating method. This study focuses on the use of sensitivity analysis for resolving the problems of damage localization and evaluation. Natural frequencies are known to be successful in characterizing dynamical systems. Mode shapes, meanwhile, have been considered effective in model updating, since these shapes condense most of the deformation database of the structure. Here, we use not only sensitivity of frequency, but also of mode shape, a subject which appears less developed in the literature. A modal identification is not necessary for the objective of localization. In monitoring the distortion of a sensitivity vector, the localization may be carried out in the first step. An analytical model is then needed for model updating, and this enables the assessment of the damage. 2. Sensitivity analysis for Principal Component Analysis The behavior of a dynamical system depends on many parameters related to material, geometry and dimensions. The sensitivity of a quantity to a parameter is described by the first and higher orders of its partial derivatives with respect to the parameter. Sensitivity analysis of modal parameters may be a useful tool for uncovering and locating damaged or changed components of a structure. On one hand, we know that the dynamic behavior of a system is fully characterized by its modal parameters which result from the resolution of an eigenvalue problem based on the system matrices (when a model is available). On the other hand, principal component analysis (PCA) of the response matrix of the system is also a way to extract modal features (i.e. principal directions) which span the same subspace as the eigenmodes of the system [19]). The second approach based on PCA is used in this study to examine modal parameter sensitivities. Let us consider the observation matrix X muN which contains the dynamic responses (snapshots) of the system where m is the number of measured co-ordinates and N is the number of time instants. We will assume that it
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depends on a vector of parameters p. The observation matrix X can be decomposed using Singular Value Decomposition (SVD): X = X(p ) = UȈVT (1) where U and V are two orthogonal matrices, whose columns represent respectively left and right singular vectors; Ȉ contains singular values of descending importance: V1 ! V 2 ! ... ! V m . A sensitivity analysis is performed here by taking the derivative of the observation matrix with respect to p: wX wU T wȈ wVT ȈV U VT UȈ wp wp wp wp
(2)
Through this equation, the sensitivity of the system dynamic response shows its dependence on the sensitivity of wU wȈ wV each SVD term. So, the determination of , and is necessary. Junkins and Kim [20] developed a wp wp wp method to compute the partial derivatives of SVD factors. The singular value sensitivity and the left and right singular vector sensitivity are simply given by the following equations: wV i wpk
UTi
wUi wX Vi ; wpk wpk
m
¦ j 1
D kji U j ;
wVi wpk
m
¦E
k ji V j
(3)
j 1
The partial derivatives of the singular vectors are computed through multiplying them by projection coefficients. These coefficients are given by equation (4) for the off-diagonal case and by equation (5) for the diagonal elements. T T ª § ª § § T wX · º · § T wX · º 1 wX · 1 k T wX «V i ¨ UTj » « D kji V V U V ; E V U V V U V j z i (4) i ¸ j ¨ i j ¸ ji j ¨ j i ¸ i ¨ i j ¸ », wpk wpk V i2 V 2j «¬ © wpk ¹ V i2 V 2j «¬ © wpk ¹ © ¹ »¼ © ¹ »¼
D iik Eiik
wV 1 § T wX Vi i ¨ Ui V i © wpk wpk
· k k ¸ ; D ii E ii ¹
wV 1 § T wXT Ui i ¨¨ Vi Vi © wpk wpk
· ¸¸ , ¹
j
i
(5)
The sensitivity analysis for PCA may also be developed in the frequency domain, e.g. by considering frequency response functions (FRFs) [19]. As the dynamical system matrices depend on a vector of parameters p, the FRF matrix takes the form: H Z , p
1
ª Z 2 M(p) i Z C(p ) K(p )º (6) ¬ ¼ where Z represents the circular frequency. With regard to sensitivity analysis, the partial derivative of equation (6) with respect to one parameter pk may be written [19], [24]:
w Z 2M i ZC K
H Z , p
§ wM wC wK H Z , p ¨ Z 2 iZ wpk wpk wpk wpk © Equation (7) provides a way of determining the derivative of the FRF matrix needed means of the partial derivative of the system matrices. wH wpk
H Z , p
· (7) ¸ H Z , p ¹ for the sensitivity analysis by
Let us consider the FRFs for a single input at location s, and build a subset of the FRF matrix (6): ª h1 Z1 h1 Z2 ... h1 ZN º « » « h2 Z1 h2 Z2 ... h2 ZN » s H (Z ) « » ... ... ... ... « » «¬ hm Z1 hm Z2 ... hm ZN »¼
(8)
where m is the number of measured co-ordinates and N is the number of frequency lines. This matrix is the frequency domain analog of the observation matrix X. The rows in (8) represent the responses at the measured degrees of freedom (DOFs), while the columns are “snapshots” of the FRFs at different frequencies. We consider that this matrix depends on a given set of parameters. We can assess its principal
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components through SVD by (1) where the left singular vectors give spatial information, the diagonal matrix of singular values shows scaling parameters and the right singular vectors represent modulation functions depending on frequency. In other words, this SVD separates information depending on space and on frequency. The sensitivity of the i th principal components can be computed by (3)-(5). First, we compute the SVD of the FRF matrix in (8) for the set of responses and the chosen input location. Then, the partial derivatives of (8) are determined using equation (7). For a particular input, only a subset of the derivatives in (7) is needed. 3. Damage localization based on sensitivity analysis of the FRF matrix In the following, sensitivity analysis is used to resolve the problem of damage localization and evaluation. We present now some simplifications that may be carried out in experimental practice. Giving the FRF matrix Hs for a single input at location s of the system and its SVD, the sensitivity computation of the principal components (PCs) requires the partial derivatives wHs / wpk which are a subset of wH / wpk . This quantity may be assessed by (7) requiring the partial derivative of the system matrices with respect to system parameters. If the parameter concerned is a coefficient ke of the stiffness matrix K, the partial derivatives of the system matrices are such that wM / wpk and wC / wpk equal zero and wK / wpk wK / wke . Although only a subset of wH / wpk is needed for a particular input s, i.e. wHs / wpk , which corresponds to the s th column of wH / wpk , the calculation of (7) demands the whole matrix H , which turns out to be costly. However, we can compute wHs / wpk by measuring only some columns of H , as explained below. We recall that our parameter of interest is a coefficient ke of the stiffness matrix K. Equation (6) shows that FRF matrices are symmetric if system matrices are symmetric. In experiment, the number of degrees of freedom (DOF) equals the number of response sensors. So, the FRF matrix has the same size as the number of sensors. Let us consider for instance a structure instrumented with 4 sensors. The FRF matrix takes the symmetrical form: ªa b c d º «b e f g » » (9) H Z « «c f h i » « » ¬d g i k ¼ Assuming that ke accords to the second DOF only, we have: wM / wpk
0 ; wC / wpk
ª0 0 0 0 º « » wK wK «0 1 0 0 » wpk wke «0 0 0 0 » « » ¬0 0 0 0 ¼ Equation (7) allows us to deduce the partial derivative of the FRF matrix: ª> b e f « wH wK «> b e f H Z , p H Z , p « wpk wpk «> b e f «> b e f ¬
0 and:
(10)
g@b º » g @e » » g @f » g @ g »¼
(11)
To compute the sensitivity of Hs , only the s th column of wH / wpk is needed, which is written in (12) in setting Hke
>b
e f
g @ . This relies entirely upon the column corresponding to ke in the FRF matrix in equation (9). T
wHs wpk
Hke Hke , s
(12)
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Hke , s is the s th element of the vector Hke . Thus, the sensitivity of Hs with respect to ke does not involve the
entire matrix H ; only the column relating to ke is needed. 4. Localization indicators When wHs / wpk has been computed, the sensitivity of principal components can be determined next using equations (3)-(5). The sensitivities of the left singular vectors are good candidates for resolving localization problems of linear-form structures, e.g. chain-like or beam-like structures. In each working condition of the system, we can compute the sensitivity wUi / wpk . The reference state is denoted by wUi R / wpk , and the deviation of the current condition may be assessed as follows: ' ' wUi / wpk wUi / wpk wUi R / wpk (13) Other indicators may be utilized to better locate dynamic change, such as: 1 1 § d Ij ' j ' j 1 and d IIj 2' j ' ¨' r r 2 © j 1
· ¸
j 1 ¹
(14)
where r is the average distance between measurement points. The indicators d I and d II , effectively comparable with the first and second derivatives of vector ' , may allow the maximization of useful information for damage localization. The formula d II is widely identified in the literature of damage localization, e.g. in [11, 12]. However, the previous methods compared mode shape vectors or FRF data. In this study, the sensitivity of singular vectors is the subject under examination. 5. Damage evaluation Once the damage has been localized, it may then be assessed by a technique of model parameter updating. We first assemble the modal features of the system coming from the SVD of the FRF matrix (8) into a vector v called the model vector. Principal components (PCs) in U or their energies in Ȉ may be considered to construct the model vector. In the literature, PC vectors have been often considered as more convenient for damage detection so they will be used here. The model vector v of a system is usually a nonlinear function of the parameters p [ p1 ... pnp ]T where np is the number of parameters. The Taylor series expansion (limited to the first two terms) of this vector in terms of the parameters is given by: np
va
v(p)
wv
¦ wp k 1
where va
v
p pa
pk
va S 'p
(15)
k
represents the model vector evaluated at the linearization point p pa . S is the sensitivity
matrix of which the columns are the sensitivity vectors. The changes in parameter are represented by 'p p pa . The residual vector measures the difference between analytical and measured structural behaviors [21]: rw Wvr Wv (v m v(p)) (16) where v m represents measured quantities. The weighting matrix Wv (according to a weighted least squares approach) takes care of the relative importance of each term in the residual vector r. Substituting (15) into (16) and setting ra v m va leads to the linearized residual vector: rw Wvr Wv (v m v a S 'p) Wv (ra S 'p) (17) Let us define the penalty (objective) function to be minimized as the weighted squared sum of the residual vector: J
min rwTrw
min r T Wr ,
W
WvT Wv
Equation (17) may be solved from the objective function derivative wJ / w'p equation: Wv S 'p Wvra , for which the solution is: 'p
(ST W S) 1 ST W ra
(18) 0 , which produces the linear
(19)
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In our approach, we will assume that the number of measurements is larger than the number of updating parameters, which yields an overdetermined system of equations. Note that the conditioning of the sensitivity matrix S plays an important role in the accuracy and the uniqueness of the solution. The solution may be implemented using QR decomposition or SVD, which allows to check the conditioning of S [21]. The updating process is shown in Figure 1 where the i th PC is considered as the model vector. The two separated branches correspond to: 1) the analytical model which will be updated; 2) experimental damage responses, input position and correction parameters. The choice of correction parameters may be based on the damage localization results.
Analytical model
K, M, C
H wH / wpk
Z M iZC K H Z wM / wp i Z wC / wp 1
2
2
k
k
ls H input s, parameters p Experiment:
wK / wpk H
l , Ȉ, V l] [U
Hs , wHs / wpk
[U, Ȉ, V ]
svd(Hs )
ra 'p
wUi / wpk o Si
Kn
, W
Kn 1 'p
li U U i
(ST W S)1 ST W ra
pn
s
l ) svd(H
pn 1 'p
pn | pn 1
(min J)
+
p
Figure 1: Updating diagram 5.1.
Choice of weighting matrix
The positive definite weighting matrix is usually a diagonal matrix whose elements are given by the reciprocals of the variance of the corresponding measurements [22]. This matrix may be based on estimated standard deviations to take into account the relative uncertainty in the parameters and measurements: 2 W diag(w1,w 2 ,...,w i ,...,w m ) ; W Var 1 with Var diag(V 12 ,V 22 ,...,V i2 ,...,V m ) (20) where m is the number of measurements and V i is the standard deviation of the i th measurement. The relationship between W and the variance matrix Var is assumed to be reciprocal because a correct data has a small variance but presents a significant weight in the estimate. Similarly, in our problem of damage evaluation, the weight may be intensified according to damaged locations in order to improve the efficiency of the technique, i.e. to accelerate the convergence and raise the accuracy.
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6. Application to damage localization 6.1. Numerical example of a cantilever beam Let us examine a steel cantilever beam with a length of 700 mm, and a square section of dimension 14 mm. The beam is modeled by twenty finite elements as illustrated in Figure 2. The input location is chosen at node 7 and the snapshot matrix is assembled from FRFs corresponding to the vertical displacements at nodes 1 to 20.
Figure 2: Cantilever beam We model the damage by a stiffness reduction of a beam element. Four states are examined: the reference (healthy) state, and 3 levels of damage (L1, L2, L3) induced by a reduction of stiffness of respectively 10%, 20% and 40%. The damage is assumed to occur in element 12. Note that the maximum deviations on the first three frequencies from the reference state are 0.70%, 1.41% and 2.81% for the 3 levels respectively. For illustration, sensitivity analysis results are shown in Figure 3 according to the parameter pk = k15. Note that similar results were obtained for other stiffness parameter in various positions. The FRFs were considered in the frequency range from 0 Hz to 165Hz at intervals of 1 Hz. Figure 3 shows the sensitivity difference ' wU1 / wk15 of the first left singular vector with respect to the coefficient associated to the 15th DOF in the stiffness matrix and its derivatives d I , d II .
Figure 3: ' wU1 / wk15 , d I , d II for 3 levels with pk
k15
It is observed that the ' wU1 / wk15 curves are discontinuous at DOFs 11 and 12; index d I shows a discontinuity with large variations around element 12 and finally, index d II allows us to discover explicitly the position of the damaged element. The method reveals itself robust when damage develops in several elements. Localization results are shown in Figure 4 in two different cases of damage. We note that index d II does not indicate the same level of damage in the damaged elements. However, the damage locations are accurately indicated. The difference in magnitude is due to unequal sensitivities for various damage locations, as discussed in [10].
Figure 4: d II for 3 levels, damage at elements 12, 13, 14 (left, pk
k8 ); 7 and 16 (right, pk
k18 )
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Damage evaluation In this section, the beam is examined with the model of 10 elements. We consider firstly the damage occurring in a single element. For example, the stiffness of element 5 is subjected to reduction. FRF matrices are considered for an excitation at node 9. Frequency lines are selected from 0 to 200 rad/sec. Because the damage localization is readily identified, the first PC and its derivative with respect to the damaged element are considered. For the sake of efficiency and accuracy, only the stiffness of the damaged element is taken into account in the parameter vector p. The diagonal weighting matrix defined by (20), as discussed above, is intensified in entries according to damaged elements in order to increase the calculation efficiency. For this purpose, the standard deviation of the principal component (PC) vector elements is preliminarily assumed to be 10% of the corresponding element in the considered PC. And in particular, to take advantage of sensitivity vector, the entries matched to the damaged elements are assigned a lower variance so as to give a larger weighting in the algorithm. A fitting variance for measurement in the damaged elements can improve significantly the convergence speed. In the tests below, the standard deviation values according to the damaged elements were multiplied by a factor between 1% and 3%. The evaluation of some of the damage is shown in Figure 5. Several levels are considered so that the stiffness of element 5 is reduced of 5%, 10%, 20%, 30% and 50%. The data has also been perturbed with 5% of noise, but it still shows satisfactory results.
Figure 5: Evaluation of damage, noise free ( ____ ) and noise at 5% ( __ __ __) Now let us examine the problem of some elements being damaged; those elements may be side by side or distant from each other. The results of various occurrences of damage are given in Figure 6 for the same or for different levels, respectively. 5% of noise was taken into account.
Figure 6: Evaluation of damage when stiffness reduction occurred in some elements
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It is worth noting that the accuracy of the result depends on several factors when many elements are damaged, namely, the relative position of the damaged elements and the difference between their levels of damage. Input position also plays an important role on the number of iterations of the updating process. Generally, the estimate is more effective if the degrees of damage of different elements are not too different. Data recorded from an excitation close to the damaged location often accelerates the convergence speed. 6.2.
Experiments involving a mass-spring system
The next example involves the system of eight DOFs shown in Figure 7 and for which data are available in [23]. The system comprises 8 translating masses connected by springs. In the undamaged configuration, all the springs have the same constant: 56.7 kN/m. Each mass weighs 419.5 grams; the weight is 559.3 grams for the mass located at the end which is attached to the shaker.
Figure 7: Eight degrees of freedom system The acceleration responses and also the FRFs of all the masses are measured with the excitation force applied to mass 1 – the first mass at the right-hand end (Figure 7). The FRFs are assembled so as to localize the damage by the proposed method. Frequency lines are selected from 0 to 55 Hz at intervals of 0.1562 Hz. Two types of excitation are produced: hammer impact and random excitation using a shaker. First, several experiments were implemented with the system in the healthy state (denoted “H”) and in the damage state (denoted “D”). Then damage was simulated by a 14% stiffness reduction in spring 5 (between masses 5 and 6). As the excitation was applied only on mass 1, the partial derivative was taken with respect to the first DOF (equations (9)-(12)). The damage index d I is shown in Figure 8, where the healthy states are denoted “H” and the damaged states “D”. The damage index marks a clear distinction between the two groups – healthy and damaged. Healthy states show regular indexes in all positions, so they do not display any abnormality. By contrast, all the “damaged” curves reveal a high peak in point 6 or 5 where the slope is the most noticeable. Damage evaluation In order to evaluate the damage, it is necessary to build an analytical model. The first PC and its sensitivity are assessed in all conditions for the damage evaluation. So, for the sake of consistent data, a frequency line is selected from 6 to 29 Hz, which covers only the first physical mode and removes noisy low frequencies. The variations in stiffness of spring 5 are shown in Figure 9 for impact and random excitations respectively. In both types of excitation, the results are very satisfactory. All false-positives indicate small stiffness variations in the element concerned, and they show any detection of damage. By contrast, all damaged states give an evaluation very close to the exact damage – a stiffness reduction of 14%.
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Figure 8: d I by impact (left) and random excitation (right)
Figure 9: Evaluation of damage in cases of impact (left) and random (right) excitation 7. Conclusions The sensitivity computation of principal components by analytical methods has been verified in [19] in both the time domain and the frequency domain. The contribution of the present study is its application of sensitivity analysis in the frequency domain to the problem of damage localization and evaluation. Damage localization is achieved in the first step as a result of the difference in principal component sensitivity between the reference and the damaged states. The method has proved efficient in damage localization in circumstances where either only one or where several elements are involved. As sensitivity computation from FRFs is easy, the technique should be suitable for online monitoring. The analytical model is not required in damage localization but it is necessary in carrying out damage evaluation. 8. References [1]. Huynh D., He J. and Tran D., “Damage location vector: A non-destructive structural damage detection technique”, Computers and Structures 83, pp. 2353-2367, 2005. [2]. Koh B. H. and Ray L. R., “Localisation of damage in smart structures through sensitivity enhancing feedback control”, Mechanical Systems and Signal Processing, 17(4), pp. 837 – 855, 2003. [3]. Gomes H. M. and Silva N. R. S., “Some comparisons for damage detection on structures using genetic algorithms and modal sensitivity method”, Applied Mathematical Modelling, pp. 2216-2232, 2008.
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[4]. Bakir P. G. and Reynders E., Roeck G. D., “Sensitivity-based finite element model updating using constrained optimization with a trust region algorithm”, Journal of Sound and Vibration 305, pp. 211-225, 2007. [5]. Yan A. and Golinval J.-C., “Structural damage localization by combining flexibility and stiffness methods”, Engineering Structures 27, pp. 1752-1761, 2005. [6]. Koo K. Y, Lee J. J, Yun C. B and Brownjohn J. M. W., “Damage detection in beam-like structures using deflections obtained by modal flexibility matrices”. Proceedings of the IMAC-XXVII, USA, 2009. [7]. Kim J. T. and Stubbs N., “Improved damage identification method based on modal information”, Journal of Sound and Vibration, 252(2), pp. 223-238, 2002. [8]. Rucka M. and Wilde K., “Application of continuous wavelet transform in vibration based damage detection method for beam and plates”, Journal of Sound and Vibration, pp. 536-550, 2006. [9]. Cao M. and Qiao P., “Novel Laplacian scheme and multiresolution modal curvatures for structural damage identification”. Mechanical System and Signal Processing 23, pp. 1223-1242, 2009. [10]. Ray L. R. and Tian L., “Damage detection in smart structures through sensitivity enhancing feedback control”, Journal of Sound and Vibration, 227(5), pp. 987-1002, 1999. [11]. Pandey A. K., Biswas M. and Samman M. M., “Damage detection from changes in curvature mode shape”, Journal of Sound and Vibration, 142, pp. 321-332, 1991. [12]. Sampaio R. P. C, Maia N. M. M. and Silva J. M. M., “Damage detection using the frequency – response – function curvature method”, Journal of Sound and Vibration, 226(5), pp. 1029-1042, 1999. [13]. Messina A., Williams E. J. and Contursi T., “Structural damage detection by a sensitivity and statisticalbased method”, Journal of Sound and Vibration, 216 (5), pp. 791-808, 1998. [14]. Jiang L. J., “An optimal sensitivity-enhancing feedback control approach via eigenstructure assignment for structural damage identification”, Journal of Vibration and Acoustics, 129(6), pp. 771-783, 2007. [15]. Koh B. H. and Ray L. R., “Feedback controller design for sensitivity-based damage localization”, Journal of Sound and Vibration 273, pp. 317-335, 2004. [16]. Jiang L. J. and Wang K. W., “An experiment-based frequency sensitivity enhancing control approach for structural damage detection”, Smart Materials and Structures 18 (2009), Online at stacks.iop.org/SMS/18/065005. [17]. Yang J. N., Lin S. and Pan S., “Damage identification of structures using Hilbert-Huang spectral analysis”. Proc. 15th ASCE Engineering Mechanics Conference, New York 2002. [18]. Teughels A. and De Roeck G., “Structural damage identification of the highway bridge Z24 by FE model updating”. Journal of Sound and Vibration 278, pp. 589-610, 2004. [19]. Todd Griffith D., “Analytical sensitivities for Principal Components Analysis of Dynamical systems”, Proceedings of the IMAC-XXVII, Orlando, Florida, USA, February 9-12, 2009. [20]. Junkins J. L. and Kim Y., “Introduction to Dynamics and Control of Flexible Structures”, AIAA Education Series, Reston, VA, 1993. [21]. Link M., “Updating of analytical models-basic procedures and extensions”, in: J. M. M. Silva, N. M. M. Maia (Eds.), Modal Analysis and Testing, Kluwer Academic Publishers, pp. 281-304, London 1999. [22]. Friswell M. I. and Mottershead J. E., “Finite Element Model Updating in Structural Dynamics”, Kluwer Academic Publishers, 1995. http://institute.lanl.gov/ei/software-and-data/data [23]. [24]. De Lima A.M.G, Faria A.W., Rade D.A., “Sensitivity analysis of frequency response functions of composite sandwich plates containing viscoelastic layers”, Composites Structures, 2009.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Coriolis Flowmeter Verification via Embedded Modal Analysis Matthew Rensing, Research Engineer, Sheet Dynamics, Ltd., 1775 Mentor Ave. Mailbox Suite #227, Cincinnati, OH 45213 [email protected] Timothy J. Cunningham, Principal Engineer, Micro Motion, Inc., 7070 Winchester Circle, Boulder, Colorado 80301 [email protected] NOMENCLATURE Z: frequency radians/second f: frequency Hz GI: phase difference or delay radians Gt: time delay microseconds FCF: flow calibration factor (gm/sec )/ microsec : mass flow rate lbm/min m C: geometric constant, dimensionless E: Young’s modulus psi I: beam moment of inertia in4 L: length of flowtube in H: frequency response function (FRF )
x : velocity response in/sec f: input force lbf v pickoff : pickoff voltage volts
idriver : measured driver current amps icommand : commanded driver current amps R: residue O: pole M, C, K: physical mass, damping, and stiffness
ABSTRACT Verification of industrial flowmeters can be a costly endeavor, requiring significant factory downtime, but is often required to ensure measurement accuracy. A new feature, called Smart Meter Verification, now available on Coriolis meters uses embedded experimental modal analysis to confirm flowmeter accuracy. Smart Meter Verification has been successfully employed by numerous customers and has generated significant revenue. This paper outlines the evolution of this product from an offline, lab-based, concept to an embedded product. The embedded modal analysis fits a single degree of freedom model to the primary drive mode, based on frequency response measurements made at select tones near the drive mode. The estimated stiffness from each verification’s modal analysis is compared to the stiffness from a factory baseline to ensure that the meter is still within calibration. Simultaneous multi-tonal excitation of the meter is used to maximize the usable signal generated from a limited excitation power budget, to cope with a time-varying system, and to minimize the complications of nonlinearity. MATLAB-based rapid software prototyping was used to prototype these embedded algorithms on a real-time DSP platform. Excitation of the flowmeter and measurement of response is performed using the transducers already on the device, and the model fit and control logic are performed using the existing embedded electronics. This has allowed the verification technique to be applied without changing the design of the flowmeter hardware. INTRODUCTION Experimental modal analysis typically is used as a tool for testing structures in the laboratory environment. These one-off lab tests can be used to confirm the dynamics of the structure or to correlate a structural model. Structural health monitoring goes a step further by performing ongoing testing and tracking the dynamics over time. Changes in the dynamics are used to indicate structural changes. In both of these approaches the structure is instrumented with transducers to measure the response and perhaps also to provide the excitation. These transducers, data acquisition systems, and analysis software are not necessary for the day to day operation of the structure. The tests usually require significant involvement of test personnel. Micro Motion has developed a product called Smart Meter Verification that uses experimental modal and structural health monitoring techniques to ensure accurate measurement by its Coriolis mass flowmeters. The embedded digital signal processing (DSP), electronics, and transducers that are used to measure flow are multi-
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_75, © The Society for Experimental Mechanics, Inc. 2011
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tasked to perform ongoing modal analyses of the flowmeter structure. The modal results are tracked to provide assurances of the health of the meter. This new diagnostic has been a success in the marketplace. The story of the development of Smart Meter Verification is an interesting case study. This paper will highlight some of the things that we learned in going from lab-based modal analysis to a reliable real-time product. We will first discuss the theory of operation of Coriolis flowmeters to get a background for the rest of the paper. CORIOLIS FLOWMETER OPERATION A Coriolis mass flowmeter directly measures the mass flow rate of a fluid by vibrating (driving) a fluid-conveying tube at resonance. Figure 1 shows a simplified “U” shaped tube geometry. Flow enters one leg of the tube and exits the other leg. The cross product of the moving fluid with the tube vibration develops Coriolis forces, as shown in Figure 1a., with FC being the Coriolis force on the fluid and Ft being the equal and opposite force on the tube. The tube is commonly vibrated in a fundamental bending mode, Figure 1b. A more realistic dual “U” tube flowmeter is shown in Figure 2. The flow enters from the pipeline and is split at an inlet manifold between the 2 U-shaped flowtubes. The flow is then rejoined at an exit manifold and continues down the pipeline. The tube vibrates in a balanced, out-of-phase fundamental bending mode, like a tuning fork. The Coriolis forces act on the tubes to perturb the vibrational motion, giving rise to a spatially varying phase angle along the tube as shown in the top view of Figure 2. The difference between phase angles at two or more two locations, which is called phase delay, is used to calculate mass flow rate.
Figure 1. Coriolis Flowmeter Operation
Figure 2. Coriolis Mass Flowmeter
The mass flow rate calculation more commonly uses time delay (Gt) rather than phase delay (GI). For a tube vibrating at resonance frequency Z, the two are related by Equation (1).
G t GI / Z
(1)
The mass flow rate through a Coriolis sensor is directly proportional to this Gt with the proportionality constant , is given by being the flow calibration factor (FCF) [1]. The mass flow rate through a sensor, m
m
FCF G t
(2)
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FCF therefore is defined in units of mass flow rate/time delay. A typical set of units for FCF is (gm/sec)/Psecond. The transducers used to measure flow (called “pickoffs”) are voice coil velocity transducers, with the magnet mounted on one tube and the coil on the other tube, as shown in Figure 2. These transducers produce a voltage proportional to the tube velocity. Since the tube is vibrating at resonance the voltage is sinusoidal. The flowmeter’s DSP based electronics process the pickoff voltages to measure flow. Not shown in the figure is the driver transducer, another voice coil. Sinusoidal current is applied to the coil, which, in conjunction with the magnet, produces equal and opposite forces on the flow tubes. The electronics also use the pickoff signals in a closed-loop control scheme to maintain the resonant frequency at a set amplitude. CORIOLIS FLOWMETER VERIFICATION Coriolis flowmeters contain no moving parts and typically last for 10 or more years. They are also unique among flow measurement devices in that the mass flow measurement is unaffected by the process fluids; that is, the flow calibration factor (FCF) is unchanged over the life of the flowmeter. However in many applications, such as custody transfer where the flowmeter is used essentially as a cash register or the pharmaceutical industry where processes need to be tracked rigorously, a method of verifying the accuracy of the flowmeter is highly desirable. Micro Motion's original electronics were of course analog, being developed in the 1980s. Their diagnostic capabilities were extremely limited. Micro Motion introduced digital signal processing into their attached electronics in 2002. DSP allowed us to expand our diagnostic capabilities. When we were developing the next generation DSP-based electronics, an embedded self diagnostic was a key requirement. With our understanding of the structural dynamics of the flowmeter we felt we could develop this diagnostic. MODAL ANALYSIS AND CORIOLIS FLOWMETERS BACKGROUND Micro Motion first started using experimental modal analysis in the late ‘80s to improve the product development process. At about the same time we adopted finite element analysis. Initially experimental modal analysis was used to verify the frequencies of its flowmeters. Modal analysis was later used to correlate finite element models to achieve better modeling fidelity. These modal analyses were done in the typical manner in that a test specimen was instrumented in the lab and, in our case, impact tested. Modal parameters and mode shapes were extracted from the experimental data. The data was correlated to finite element models which we used to improve our modeling techniques. Experimental modal testing done in this way had its downsides. First, the experimental setup, acquisition, and analysis were time-consuming. Gaining access to test points on the flowmeter required modifications of the standard production units, specifically cutting holes in the case, which needed special tooling and changed the dynamics of the sensor. Adding the accelerometers mass-loaded the structure significantly, which also changed the dynamics. To improve the modal data acquisition, Micro Motion took advantage of the driver and pickoffs, normally used to drive the tubes and measure flow, for the modal analysis actuation and response. These transducers eliminated the need to cut access holes in our meters and eliminated any additional mass loading or damping caused by attaching accelerometers. Modal analyses using burst random excitation into the driver and standard data acquisition with a signal analyzer gave us very good results. With this significant improvement in modal results, and improved FE models, we developed new insight relating the modal results to the flow measurement. MODAL THEORY LEADS TO BREAKTHROUGH DIAGNOSTIC A key breakthrough in Micro Motion's Smart Meter Verification product started with recognizing that the flow calibration factor is directly related to the tube stiffness. Equation (2) can be derived from first principles, for example starting with the Housner differential equation describing a fluid-conveying beam [2, 3]. These derivations result in a term corresponding to the FCF as shown in Equation (3). m
C
EI Gt L3
(3)
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where C is a dimensionless geometric constant related to the boundary conditions and beam properties. The EI 3 term, corresponding to the FCF, has units of force/length, the units of stiffness. L
Going through these derivations in detail to show this relationship between stiffness and flow calibration factor is beyond the scope of this paper. However a much simpler dimensional analysis also shows that the FCF is related to stiffness by showing that they have the same units. Rearranging Equation (2) FCF
m Gt
(4)
shows that the units of the FCF are mass flow rate/time delay. The FCF is shown dimensionally as § Mass · ¨ Time ¸ © ¹ Time
(5)
For example, FCF is commonly expressed in units of (gm/sec)/Psec. In a consistent system of units, mass can be represented by force/(acceleration of gravity), taking advantage of Newton’s Second Law. Plugging this into equation (5) shows that the flow calibration factor has units of stiffness, (Force/Length). § Mass · ¨ Time ¸ © ¹ Time
§ Force * Time 2 / Length · ¨ ¸ Time © ¹ Time
Force Length
(6)
With this understanding the stage was set to use the modal analysis derived stiffness as a calibration diagnostic. Developing the capability to do these modal analyses in real time was necessary to implement this diagnostic. RAPID PROTOTYPING WITH MATLAB/SIMULINK/REATIMEWORKSHOP/DSPACE In the mid-90s Micro Motion started to investigate the use of modal filters to improve its digital signal processing. These modal filters required extremely accurate modal parameters, with the mode shape coefficients extracted from the driver and pickoff transducers. These modal analyses taxed our standard signal analyzers in terms of data storage and programming flexibility. To improve our data acquisition and modal analysis capabilities we acquired a dSPACE rapid prototyping system. The dSPACE system leverages Matlab core numerical routines and Simulink’s graphical signal processing capabilities. Signal flow diagrams implemented in Simulink can be converted into real-time executable code for the dSPACE processor. dSPACE has A/D and D/A I/O blocks for Simulink to handle the transducer I/O. We implemented our techniques for performing modal analysis using the onboard transducers via the rapid prototyping system. This modal approach required files larger than 100 MB taking approximately 20 minutes of data collection to obtain the required frequency resolution to extract the modal parameters of interest. Extracting the modal parameters was a manual process requiring engineering judgment and a fair amount of time for each iteration. However, the modal parameters were precise enough that we could correlate them to the FCF. We decided that if we could embed this modal analysis we would have a viable diagnostic product. DEVELOPMENT OF EMBEDDED METER VERIFICATION Micro Motion had shown that it could extract good stiffness measurements using the onboard transducers in a time consuming, off-line measurement process. The challenge was being able to implement that measurement process into an extremely robust one operating in an embedded environment. Our initial development was done using a dSPACE rapid prototyping system connected to the meter’s transducers, but techniques were developed with an eye toward implementation in the embedded hardware that is an inherent part of the flowmeter. For customers to benefit from this verification process it had to be simple to use, with minimal user interaction. This meant no additional instrumentation would be required, no additional processing hardware would be connected, and minimal restrictions would be placed on the user’s process. The embedded implementation of Meter Verification would have to run on the existing processing platform and use the existing transducers already on the meter. These restrictions provided significant challenges beyond the initial laboratory proof-of-concept – the sections below lay out how we addressed some of these challenges.
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TRANSDUCERS AND ELECTRONICS The meters on which the verification algorithm was developed employ a single driver for input and two pickoffs for output (with phase measured between the two pickoffs – see Figure 2). Using these as inputs and outputs for verification means that we used a single-input multiple-output (SIMO) system model (the model is explored in more detail below). Using the same transducers and electronics to measure flow and to perform meter verification enhances the correlation between the measured meter stiffness and the flow calibration factor. Both the driver and pickoffs are voice coil devices, converting current to force as a driver, and velocity to voltage as a pickoff. Since the pickoff measurements are in terms of velocity, our model was formulated in terms of velocity. The transducers do suffer from some limitations as compared to laboratory-grade instruments (e.g. accelerometers). The primary limitation of the transducers is a mild nonlinearity; the calibration of the transducers is a function of the instantaneous engagement (i.e. the position of the permanent magnet relative to the wire coil, which is simply the distance between the flow tubes), and is also affected by the rotational component of tube vibration (the alignment of the coil and the magnetic field changes when the flow tubes are not perfectly parallel). Another limitation of the transducers is their temperature dependence; the calibration of the transducers changes slightly with temperature, but this effect becomes significant given the large range in allowed operating conditions for the meters and the sensitivity of the stiffness measurement. The electronics used in the embedded processing platform also provided challenging limitations. Only two highspeed analog to digital converters were available on the board, limiting the number of system parameters that could be simultaneously measured. The embedded processor itself was a fixed-point device, with limited compute cycles and system memory available. Finally, the current amplifier used to drive the excitation had limited output current available. We addressed these limitations imposed by the electronics and transducers in a number of ways, as described below. EXCITATION With copious off-line computing resources available, burst random excitation was a viable approach. However, this technique was inappropriate for an embedded approach. In order to maintain a flow measurement during verification, we had to maintain the closed-loop excitation of the fundamental bending mode with minimal interference from other excitations. The limited excitation power available also weighed against a burst random excitation. The first excitation technique we considered as an embeddable replacement for the burst random approach was a sinusoidal excitation at a second frequency near the resonance frequency. The method would generate a frequency response function by measuring the excitation and response at the non-drive frequency, stepping the excitation sequentially among four frequencies near the drive. The FRF coefficients measured at these frequencies (plus the FRF coefficient for the drive frequency) would be used as inputs to the model fit. Ultimately, this approach proved unsatisfactory on several counts. The measurement was quite time consuming; the digital filtering then in use to resolve the narrow frequency bands required a few minutes to settle when conditions changed, pushing the total measurement time near 10 minutes. This long measurement time, while undesirable, was not in and of itself a dealbreaker. However, the fact that the system was not perfectly stationary over this time frame was a problem. Small changes in temperature and/or fluid density meant that the FRF coefficients at different frequencies were taken from slightly different systems. These differences required normalizing the data for the curve fit, increasing the computational difficulty and increasing the variability of the results. We had much better success by simultaneously applying and measuring responses to the four “test” frequencies near the resonance (in addition to the ongoing resonance excitation). This had some drawbacks (increased computational load to simultaneously measure FRF coefficients at 5 frequencies, having to divide the excitation power budget amongst all the tones) but the benefits of making simultaneous measurements of the same system greatly outweighed these problems. The additional tones were generated open-loop at fixed frequencies and added to the closed-loop excitation of the resonance. Determining the exact frequencies for the additional tones involved balancing several concerns. Our desire to maximize the response amplitude (for signal-to-noise purposes) results in tones closely clustered near the lightly-damped resonance, but tightly spaced tones require additional time and tighter filtering to resolve. Much consideration went into determining the optimal spacing of these additional test tones, the results of which we consider proprietary. Figure 3 shows a typical frequency response function curve with four test tones and the drive tone indicated.
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A few practical issues associated with the excitation arose that we had to deal with. The first problem was that the additional test tones influenced the closedloop excitation of the resonance, which sought to keep the drive amplitude constant. This affected the stability of the closed-loop control and influenced the ongoing flow measurement. We handled this by simply digitally filtering out the known test frequencies from the pickoff signals that are fed back to the closed-loop controller (after the FRF coefficients are measured), using tightly designed and phaseadjusted notch filters. Other problems included working within the available power budget for excitation and minimizing nonlinear effects from the transducers. Both of these were addressed by reducing the amplitude of the closed-loop resonance drive. During normal operation a large amplitude excitation was used to produce a good phase Figure 3. FRF Estimation measurement to determine flow; during meter verification, though, we used a lower amplitude at the resonance to free up drive current to be distributed amongst the additional test tones (which required more input force to achieve reasonable response levels). The lower overall motion of the meter also dramatically reduced the nonlinear behavior of the transducers. A final issue we encountered with the excitation is that the sudden application of the four test tones could act as an impulsive shock to the system. This impulse upset the closed-loop drive and took some time to die out in our lightly damped system. For this reason the test tones had to be started out at low amplitude and slowly ramped up to full power. This ramping extended the total measurement time somewhat, but ultimately proved faster than waiting for impulses to the meter to decay. AMPLIFIER CALIBRATION At its core the meter verification algorithm fitted a model to FRF coefficients measured at select frequencies. These FRF coefficients were defined as:
H{
x v pickoff | f idriver
(7)
With two pickoffs and a single driver, we had to measure three quantities to compute the FRF coefficients. However, as outlined above, only two high-speed A/D channels were available on the embedded hardware. To address this we split the measurement process into two steps. First, the current amplifier was calibrated by computing an FRF of the commanded current to the measured current. Then the pickoff responses were measured with respect to the commanded current. We’d then combine the two responses to produce the desired system FRF defined in Equation (7):
H|
v pickoff idriver
v pickoff icommand icommand idriver
(8)
We used a hardware switch to toggle between feeding the measured current or a pickoff response into one of the A/D converters. The fact that pickoff and commanded current measurements were now taken at two points in time may seem to be an issue, but this was not problematic in the way that measuring at sequential frequencies had been. This was because the current amplifier is much more time-invariant than the meter itself. The amplifier calibration was performed immediately before the system FRFs were computed, and was measured driving at the same amplitude and frequencies as were used for the system FRFs. The only expected short-term causes of change in the amplifier’s behavior were temperature changes or saturation of the amplifier (i.e. requiring more current or voltage than the hardware can supply). Saturation is unlikely to be an issue, since calibration was done under the same conditions as the measurement. Since the electronics are not thermally coupled to the flow
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tubes, temperature changes were not likely to affect the amplifier either. In any case, temperature effects were less pronounced on the current amplifier than on the flowmeter itself. While this amplifier calibration process did add some time to the total measurement and briefly interrupt the measurement of flow, it did effectively cope with the limited capabilities of the embedded hardware. FRF MEASUREMENT FRF coefficients were measured for the two pickoffs at the five excited frequencies (the resonant frequency and the four test tones). Since the excitations were tonal we computed the FRF coefficients by separately demodulating and filtering each tone, rather than any sort of Fourier-based approach. In this way we avoided leakage effects and reduced the amount of processing required. Still, the filtering required was somewhat taxing. To minimize the required processing and speed the transient response time we used a multi-stage filtering approach. The filters for each stage were carefully designed to minimize the filter order. The final filter stage had to provide a lowpass cutoff sharp enough to ensure that none of the other tones interfered (effectively forming a frequency band around each tone). Once we separated the input and output signals by frequency using this demodulation and filtering we applied H1 FRF average:
H
x f H ffH
(9)
An H1 FRF technique attempts to minimize noise on the output measurement and assumes the input measurement is comparatively clean [4]. 25 averages were used in computing each FRF coefficient; with the decimation already applied to the input and output signals, this resulted in a set of FRF coefficients once every 5 seconds. Along with the FRF coefficients we also calculated the coherence. This was generally quite excellent (at least 0.99999), and can be attributed to the excellent SNR of the tonal excitations and the tightly filtered signal processing. At this point we also addressed the temperature dependence of the transducers. A flowtube temperature measurement was already in place for compensating the flow and density measurements. This temperature measurement was combined with the known temperature dependence of the transducers to rescale the FRF coefficients back to the measurements that would have been made at a nominal temperature. MODEL FITTING With the FRF coefficients measured, several different forms of a second-order dynamics model could be fit to estimate the modal parameters. We ultimately selected a single DOF pole-residue model for this algorithm after testing was done with several other candidate forms. The starting point for the pole-residue model (which is based on a velocity rather than position output variable) was of the form
H Z
R jZ R jZ jZ O jZ O
(10)
The pole O and the residue R are both complex-valued. The second term of Figure 4. Pole Residue Model Compared to 2nd order System Equation (10), corresponding to the negative frequency half of the response, was then dropped. As demonstrated in Figure 4, the elimination of the negative frequency contribution has a negligible effect near the positive-frequency resonance. We preferred this form over a full second-order system model to better fit the limited resources available on the embedded hardware.
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We then extended the basic pole-residue model to accommodate the two-output system by fitting two residues and a single pole to the FRFs, since the pole of the system is independent of the measurement location. This system of equations is shown in Equation (11). Note that the equations are dependent on the frequency. The system of equations is expanded for each of the 5 frequencies measured, producing an overdetermined system with 10 complex equations and 3 complex unknowns.
H L Z º » ª RL º jZ » « » RR H R Z » « » » «¬ O »¼ jZ ¼
ª «1 0 « « «0 1 ¬
ª H L Z º « » ¬ H R Z ¼
(11)
Solving this overdetermined system of equations (in the least-squares sense) is simple in, say, MATLAB, but requires some thought in an embedded system. We chose to first simplify the complex pseudo-inverse into a real inverse, then solve with standard matrix techniques and software emulation of floating-point math (the processing load in doing so was tolerable, since the model was fit only once per 5 seconds). The simplification, starting from a generic system of equations Ax b , is performed as
Ax b AH Ax x
AH b
A A H
1
(12)
AH b
H
Note that A A is a strictly real 3x3 matrix whose inverse can be easily found. EXTRACTING SYSTEM PARAMETERS As discussed earlier, our primary goal with meter verification was to measure the meter stiffness, monitoring for changes that may indicate a flow calibration factor change. Obtaining stiffness and other physically-meaningful system parameters from the pole-residue model was straightforward, and can be derived by equating the system of Equation (10) to an MCK model,
H
jZ M Z jCZ K 2
(13)
Since the stiffness is compared to a baseline stiffness, though, we found it necessary to compensate for any expected changes to the stiffness. Micro Motion has long compensated for the change in tube stiffness as a function of temperature, using experimentally-derived coefficients for the change in Young’s modulus as a function of temperature. These correction coefficients were applied in conjunction with the measured tube temperature to the stiffness derived from the pole-residue model, correcting it back to the equivalent stiffness at a nominal temperature. The calculations used to extract stiffness and other quantities from the pole-residue model are shown in Figure 5. This graphical representation was taken from the rapid prototyping environment used to initially develop the algorithms, and comprised only one of the many subsystems associated with implementing the algorithm. Throughout the development process we made extensive use of this rapid prototyping framework. The graphical signal-flow approach made it easy to rapidly change techniques, debug problems, and try new ideas. Only after the final form of the algorithm was determined did we go about hand-writing code for the embedded platform, using the rapid prototyping diagrams as the reference for the implementation. PRESENTATION OF STIFFNESS RESULTS TO THE CUSTOMER Presenting diagnostic results to customers is a very involved topic. This paper will just discuss briefly how we present the stiffness as a Meter Verification diagnostic using the deviation from a factory baseline.
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1 Residues
|u| 1
Im (u)
u Invert
2
1 M
(Left and Right )
2 K
(Left and Right )
Gain
2 Pole
|u|
2
K K corr
3 Temp Re(u)
-1 Gain 1
|u|
Temp
Stiffness Corrections 3 Zeta
4 wn
Common
Common
Figure 5 – Variable extraction as implemented in Simulink/dSPACE rapid prototyping environment
During the standard calibration of the flowmeter, a factory baseline stiffness is calculated and stored in the transmitter. Customer initiated verifications are normalized as a percent deviation from the factory baseline, which we define as the stiffness uncertainty. The stiffness uncertainty normalization is shown below.
stiffnessuncertainty
stiffness ( stiffness
measured
)
1 %
(14)
factory baseline
The standard deviation of the stiffness uncertainty is much better than the baseline accuracy of the flowmeter, as shown in Figure 6. Under lab conditions this deviation is on the order of 0.01%. While many field effects are compensated for, there can be residual uncompensated error in the stiffness uncertainty. The spec limits for the stiffness uncertainty has been set to account for these potential residual errors. Over the entire range of process conditions, temperature, pressure, flow rate, density, etc. there is less than a 0.3% (3V chance of a false alarm (i.e. indicating that the sensor has changed when it hasn’t). The meter verification Figure 6. Presentation of Meter Verification Results diagnostic results are discussed more thoroughly in the references [5, 6, 7] CONCLUSION Micro Motion Smart Meter Verification embeds experimental modal analysis in an industrial device. Data acquisition, control, and the modal parameter estimation process have been automated to produce the desired physical estimates of the model parameters. The stiffness estimate that comes out of the modal analysis provides an independent estimate of the flowmeter calibration.
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Embedding a modal analysis required a significant amount of development work to overcome the constraints of the embedded hardware. The embedding efforts were greatly aided by the use of a DSP rapid prototyping system. Smart Meter Verification is extremely robust and provides customers with a diagnostic that ensures that their Coriolis flowmeter is operating at the specified factory accuracy. Micro Motion has been able to meet a significant customer need by embedding modal analysis into a successful product. REFERENCES [1] Micro Motion TUTOR, http://www.emersonprocess.com/micromotion/tutor/index.html. [2] Lange U., Levien A., Pankratz T., Raszillier H., Effect of detector masses on calibration of Coriolis flowmeters, Flow Measurement Instrumentation, Volume 5 Number 4, 1994. [3] Stack C., Garnett R., Pawlas G., A Finite Element for the Vibration Analysis of a Fluid-conveying Timoshenko Beam, 34th SDM conference proceedings, 1993, AIAA [4] Brown, D., Allemang, R., Phillips, A., Structural Measurement Lecture Notes, UC-SDRL Experimental Techniques Seminar Series, Chapter 6. 2004. [5] Cunningham T., Stack C., Connor C., Using Structural Integrity Meter Verification, Micro Motion White Paper WP-00948, www.micromotion.com, 2007. [6] Cunningham T., Using Structural Integrity Meter Verification to Track Corrosion in Coriolis Flowmeters, Micro Motion White Paper WP-01196, www.micromotion.com, 2009. [7] Cunningham, T., An IN-SITU Verification Technology for Coriolis Flowmeters, Proceedings of the 7th International Symposium on Fluid Flow Measurement, 2009.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
BAYESIAN FINITE ELEMENT MODEL UPDATING FOR CRACK GROWTH Juan M. Caicedo, Assistant Professor, Department of Civil and Environmental Engineering, University of South Carolina, 300 Main Street, Columbia, SC 29208 Boris A. Zárate, Post-doctoral Fellow, Department of Civil and Environmental Engineering, University of South Carolina, 300 Main Street, Columbia, SC 29208 Victor Giurgiutiu, Professor, Department of Mechanical Engineering, University of South Carolina, 300 Main Street, Columbia, SC 29208 Lingyu Yu, Post-doctoral Fellow, Department of Mechanical Engineering, University of South Carolina, 300 Main Street, Columbia, SC 29208 Paul Ziehl, Associate Professor, Department of Civil and Environmental Engineering, University of South Carolina, 300 Main Street, Columbia, SC 29208 Nomenclature ȣ D M P() ݂ ሺȣǢ ܦሻ ȣ୪ ȣ୪ ݑௗ ݑ
Parameter to be updated Observation Model Probability Posterior PDF Lower bound for parameter being updated Upper bound for parameter being updated Experimental displacement Displacement calculated from finite element model
Abstract This paper describes a probabilistic structural health monitoring framework to determine crack growth on structural members using model updating. The framework uses Bayesian inference to estimate crack lengths. On the proposed framework data from embedded piezoelectric wafer sensors (PWAS) and acoustic emission sensors is used for model updating. This paper presents preliminary results obtained using simulated data of a steel specimen. As a first step, the crack length is estimated using calculated displacements at the tip of the specimen. Results show that Bayesian inference can be used to estimate crack lengths on structural members. Introduction The focus on the interstate highway system shifted in the 80’s and 90’s from expansion to preservation and operation [1]. During the 60’s, 70’s and 80’s the focus of the transportation agencies was in the development and expansion of the interstate highway system, creating a $1 trillion transportation network to operate and maintain in the coming decades. This shift from expansion to operation puts a heavy load for state DOTs who are required to maintain and operate the developed network with limited resources. The South Carolina DOT alone had a $5.3 billion maintenance backlog in 2005 [2]. In addition, the nation’s infrastructure is aging and major upgrades are becoming a necessity. Many of the nation’s 590,750 bridges are being used well beyond their design life and a routine and detailed inspection should be performed to ensure the safety of these everyday structures. Bridge inspection plays an important role within bridge maintenance systems, providing vital information for scheduling maintenance and upgrades to the transportation network, which directly impacts the cost to maintain transportation networks. Visual inspection, defined as determining the health of the structure using the five senses with the help of very basic tools is the most commonly used method of nondestructive evaluation. Unfortunately, visual inspection has
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proved to be highly inconsistent between inspectors as shown on a report from the Federal Highway Administration Nondestructive Evaluation Validation Center [3]. Part of this report includes the results from an experimental program where a group of 49 bridge inspectors from different DOTs performed 10 inspections in 7 bridges with different characteristics. The inspections included six routine inspections; two inspections following their DOTs procedures; and two in-depth inspections. Routine inspections used the standard condition rating system which rates the condition of members of the structure from 0 to 9 with zero indicating failed condition and 9 indicating an excellent condition. The Inspector field notes were collected for in-depth inspections and used to evaluate the accuracy of each inspection. In addition to the inspection results, information about the working environment, such as light intensity and noise level, and about the inspector, such as visual acuity, was collected to evaluate possible human factors affecting the evaluation. The results showed a large variation in the visual acuity of the inspectors. Additionally, 5 of the 49 inspectors showed signs of color vision deficiencies corresponding to the mean color deficiency of the general population. The results of the study showed a large variation in the assigned condition ratings during routine inspections. In general, four or five different condition ratings were assigned by inspectors on the same primary bridge element. Visual acuity, color vision, bridge inspection training and fear of traffic were determined to influence the condition rating of primary structural elements. During in-depth inspection of the B544 Bridge, all inspectors identified paint system failure, however only 5% of the inspectors identified missing rivet heads. The in-depth inspection was performed using lifts, thus it was determined that fear of heights affects inspector’s performance. Damage on structural members can be produced by several factors such as corrosion, damage from cars colliding with girders or columns, and fatigue due to repetitive loads. In this paper we focus on damage created by cracks on structural steel members. A framework to identify crack length is discussed and preliminary results showing the capabilities of the framework are presented. Model Updating Framework The framework described on this paper is designed to infer the probability of the length of the crack given some experimental data. Bayesian inference is used to update a probability density function of the length of the crack based on experimental data. Let, Ȃ, represent a chosen model (e.g. finite element model from a real structure), which is a function of the crack length 4, and some experimental information obtained from the structure D (i.e. strain measurements, displacement measurements, etc). The Bayes’ theorem can be written as
P(4 D, M ) v P( D 4, M ) P(4 M )
(1)
where, P(4 D, M ) corresponds to the probability density function (PDF) of 4 for the chosen model M after being updated with the observation D, or posterior PDF. P(4 M ) is the PDF of the parameters 4 for the chosen model M before updating, or prior PDF, and P(D 4, M ) is the likelihood of occurrence of the measurement D given the vector of parameters 4 and the model M. Model updating requires the comparison of experimental and theoretical data. Therefore, features from the data records that are easily calculated with a numerical model are commonly used for updating. For example, in structural dynamics the natural frequencies and mode shapes of the structure are commonly used for model updating [4]. Modal parameters can be obtained from field measurements and can be easily calculated from numerical models. On the proposed framework Acoustic Emission (AE) sensors and Piezoelectric Wafer Active Sensing (PWAS) are used. PWAS allow for the imaging and quantification of the crack on their active form and provide important information for the development of a prior PDF. AE sensors are passive sensors that capture acoustic signals that are emitted by the crack. AE sensors are used to estimate the amount of energy released by the crack. The amount of energy released by the crack can also be computed from finite element models for comparison. PWAS are active sensors that allow for the imaging and quantification of cracks in steel and other metallic structures in the absence of crack growth, and perform active interrogation at will. This complements passive (e.g. AE) sensing that relies on damage progression for quantification. In both Lamb wave generation and sensing
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(a)
(b)
( )
Figure 1. PWAS pulse-echo crack detection experiment. (a) plate with built-in rivets and a simulated 12.7 mm crack; (b) baseline signal containing reflections from the plate edges and the rivets at 100 mm; (c) reading containing reflections from the crack, plate edges, and rivets; (d) subtracted signal containing reflection from crack only PWAS couple their in-plane motion with the particle motion of Lamb waves on the material surface, which is excited by the applied oscillatory voltage through the d31 piezoelectric coupling. The interaction between the PWAS and the structure is achieved through an adhesive layer, in which the mechanical effects are transmitted through shear effects. When generated by PWAS, optimal Lamb wave excitation and detection can be obtained when the PWAS length is an odd multiple of the half wavelength of particle wave modes. An example of using PWAS active sensing to detect crack on aluminum plate is given in Figure 1[5]. The PWAS embedded sensing and monitoring technique has achieved significant success on damage detection in metallic plate or pipe structures. The sensing and monitoring feasibility is demonstrated through various methods either using propagating guided waves in pitch-catch, pulse-echo or phased array patterns, or using high frequency impedance spectrum method. Additionally, the PWAS transducers have also shown their potential for passive sensing in AE and impact detection. The small size of the PWAS is shown in Figure 2. The Acoustic Emission method is a non-destructive technique that can be used for Civil Engineering applications. When cracks grow, energy is released at the location of the crack tip in form of waves. Acoustic Emission (AE) sensors (Figure 3) can be used to measure these waves. Several sensors can be used to locate the approximate location of the crack. The intensity of the acoustic emission can also be used to estimate the severity of the crack. AE data can be used to estimate the amount of energy released by the crack. This information can be
(a)
(b)
Figure 2. PWAS: (a) 7-mm round PWAS compared to a dime; and (b) linear phased array of ten 5mm square PWAS with a total dimension of 50 mm.
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Figure 3. Acoustic Emission Sensor used on the model updating framework. Energy released from the crack can be estimated from the finite element model and related to the experimental data. Numerical simulations The preliminary results presented on this paper focus on numerical simulations of a steel specimen. The goal is to demonstrate that the probability of the crack length can be inferred based on a clip gage on another location on the specimen. The specimen modeled is shown in Figure 3 and it is made of steel A572 grade 50. A displacement gage was simulated at the end of the slot to measure the deformation of the slot along the test. The numerical model was tested under cyclic loading. The cyclic tension load was applied with a minimum of 0.5 KN and a maximum of 50 KN. First, a numerical model is cracked such that the clip gauge measures 0.5mm (Figure 4). This value is assumed as experimental data. Then, a second numerical model with the same characteristics is updated to find the probability of the length of the crack. The finite element model was built in Abaqus/standard using plane stress assumptions and a direct cycle fatigue analysis. The material is modeled as elastic-plastic with strain hardening. The entire strain stress curve up to the rupture point was introduced to the model. The boundary and load conditions of the model were set by constraining the displacements of the nodes at the inferior and superior holes, equal to the displacements of a node at the center of each circle. The displacements in the X (horizontal) direction and Y (vertical) direction of the node located at the center of the inferior hole were restraint for setting the CompactTensionSpecimen
d = 2.50 +0.0 02in Ͳ 0.0 0in
3. 25in
6.00+/Ͳ 0.05in
2. 75in h
h=0. 25in
12.0+/Ͳ 0.10in
5.5 0+/Ͳ 0.10 in
5 .50in 3. 25in
6.00+/Ͳ 0.05in
1.25 in 1.2 5in
0.75in or0.50in
10.00+/Ͳ 0.05in
2.50in
12.50+/Ͳ 0.10in
Figure 4. Specimen specifications and instrumentation
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boundary conditions. The model was loaded by applying a cyclic load at the node located at the center of the superior hole. The crack is assumed to develop at the middle of the specimen. For the crack to develop two different surfaces are created. The node at the tip of the slot is assumed as the initial crack tip, from which the crack will develop. The Paris law is used to relate the crack growth to the relative fracture energy release rate [6]. The specimen was divided in several parts to allow the creation of a structure mesh. Shell elements of 4 nodes with reduce integration (CPS4R) were used for the construction of the mesh. This type of elements has plane stress assumptions and 2 degrees of freedom per node [7]. The element size increases as the elements get away from the crack tip. The smallest element along the crack surface has a length of 0.25 mm and the largest a length of 25 mm for a total of 2302 elements. A constant thickness of 0.5 in was specified for the specimen. Assuming that no prior information about the crack length is known, the prior PDF would correspond to a uniform distribution. The likelihood PDF for a function which is proportional to a Gaussian distribution of the difference between the numerical model and the measured data, the posterior probability can be calculated as id fe § 1 n u j u j (4 ) f p (4; D ) exp¨¨ ¦ V uj ¨ 2 j1 © 0
2
· ¸ 4 4 4 l u ¸¸ ¹ otherwise
(2)
where, n is the number of displacement measurements obtained (in this case one), u idj is the j-th measured displacement, u jfe (4) is the j-th displacement at the finite element model being updated, V ju is the standard deviation of the error in the j-th displacement, and 4 l and 4 u are the lower and upper bounds of the parameter 4 . This equation is proportional to the probability of the structural parameters 4 given a set of experimental modal parameters D and could have one or several local maxima. Equation (2) measures the probability of the numerical model as the parameters 4 vary. Furthermore, an optmization of the equation (2) will result in the values of the parameters 4 that best represent the model. For this particular example the identified the standard deviation of the error was assumed to be 0.005 mm. Figure 4 shows the posterior PDF describing the probability of the crack given a measurement of 5mm on the clip gauge. This result is confirmed by simulating the finite element model with a crack of 8 to 10 mm, and obtaining gauge values close to 0.5 mm.
Figure 5. Finite element model
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Probability Density Function (1/mm)
0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0
5
10
15
20
25
30
Crack length (mm)
Figure 6. Posterior PDF Conclusions and future work A Bayesian model updating framework to detect crack length on steel specimens was presented in this paper. On the proposed framework data from acoustic emission and PWAS sensors are used to update the posterior PDF of crack length. A numerical example is presented as preliminary to show the capabilities of the framework for crack length identification. A a clip gauge readings is used to determine the posterior probability density function of the crack length. Future developments include the inclusion of acoustic emission and PWAS data onto the updating process. Acknowledgments
This work is performed under the support of the U.S. Department of Commerce, National Institute of Standards and Technology, Technology Innovation Program, Cooperative Agreement Number 70NANB9H9007. References [1] FHWA, (1999), “Asset Management Premier”. U.S. Department of Transportation, Federal Highway Administration, Office of Asset Management. [2] ASCE, (2005), “2005 Report Card for America’s Instrastructure”, available online at http://www.asce.org/files/pdf/reportcard/2005reportcardpdf.pdf. [3] Moore, M., Phares, B., Graybeal, B., Rolander, D., (2001), Raliability of Visual Inspections for Highway Bridges, Volume I: Final Report, Federal Highway Administration report FHWA-RD-01-020, June. [4] Zarate, Boris Adolfo, and J. M. Caicedo. 2008. Finite element model updating: Multiple alternatives. Engineering Structures 30 (12):3724-3730. [5] Giurgiutiu, V., Structural health monitoring with piezoelectric wafer active sensors. Academic Press (an Imprint of Elsevier), 2008. [6] Hibbitt, K. and I. Sorensen, ABAQUS theory manual. User Manual and Example Manual, Version, 2002. 6(3) [7] Abaqus, I., ABAQUS Analysis: User's Manual. 2007: Dassault systèmes.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
A non-destructive, contactless technique for the health monitoring of ancient frescoes L. Collini Dipartimento di Ingegneria Industriale, Università di Parma Viale G.P. Usberti 181/A, Parma, 43100 Italy. [email protected]
ABSTRACT. In this paper an innovative non-destructive, contactless technique applied to the health monitoring of ancient frescoes is presented. The problem of the health monitoring of artistic frescoes without a direct interaction with structures and paintings is of great concern in the field of art restoration and preservation. In artistic frescoes, the partial detachment of plaster portions is a typical and serious problem. Both layer-to-layer detachments and delaminations and surface cracks are usually present in ancient wall paintings. At present, the standard procedure of diagnosis consists of manual inspection, but produces only approximate information. This paper describes an acoustic, non-invasive, experimental technique of diagnosis, based on the acoustic-structural interaction which occurs when a fresco wall is excited by a loudspeaker. The analysis of the acoustic pressure field and of its alterations allows the assessment of detachments, since the acoustic modal parameters are affected by the acoustic system boundary conditions, i.e. the portion of analyzed fresco. The reconstruction of the modal behavior of the analyzed portion of the fresco is made by a scanning laser Doppler which measure the velocity field of the observed surface. It is a non-contact measure technique that provides a great accuracy. Experiments carried out on fresco artificial specimens show the potential of the technique. Keywords: NDT, frequency-based identification method, fresco health monitoring. INTRODUCTION 1.1 Typical fresco structure and causes of deterioration. The fresco is the most widespread traditional technique of wall decoration. Its application has known few variations in the centuries and has remained unchanged since today. The fresco painting technique is born in remote times before the Buon Fresco of the Italian Renaissance. It needs the use of pigments to base of colored sands or metallic oxides, able to withstand the discoloring effect of the mortar. The shades are simply dissolved in water, or in water of mortar or diluted grassello, and smoothed against plates of marble to eliminate all the lumps or coarse parts, [1]. The preparation of the wall for the fresco typically implies the realization of a plaster made of many layers: the first layer is called rinzaffo, it is composed of mortar, sand and brick crocks in ratio 1:2:1, and it has the function to stick to the supporting wall and carry the following layers. The intermediate layer is called arricciato; it is realized with mortar and river sand (ratio 1:2) and has a leveling function. The finishing layer has mainly an aesthetic function, and is realized in mortar and dust of marble thin (1:2). The finishing layer must be applied only in the amount that can be painted during a day of work, since the color is given in mixed together with the mortar layer when still fresh (from here the name fresco) and under curing. The pigments are incorporated in the carbonate crystals that are formed after the chemical hardening reaction, and give a bright and transparent color to the treated support, [1]. Antique frescoes are affected by particular environmental factors (variations of temperature and moisture), pollutants (superficial deposits of dirt, biodeteriogens and chemical attacks) and physical (building structure subsidence, seismic vibrations). These factors can bring surfaces to different degrade pathologies: delamination, salty efflorescences, sub-efflorescences, separations, stains, pulverization, [2]. Particularly, the pathologies from separation as detachments, stains, sub-efflorescences bring to a lack of adherence located within the support layers behind the pictorial film. The presence of air, moisture and salts between the layers can bring to the separation and fall of plaster and fresco layers. For example, figure 1 shows a detail of a sixteenth-century fresco in the Cappella del Comune at the Dome of Parma (picture taken by the authors in year 2008), where can be
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noticed, inside the outlined red zone, a wide separation of the arriccio, that has locally caused the complete loss of the fresco, here highlighted by the green line. 1.2 Diagnostic methods for frescoes. Although analytical methodologies and techniques for detecting the chemicalphysical characteristics of works of art are already known and used since a long time, very often it is made reference to the manual skill of the restorer. After a visual and manual inspection, he is generally able to judge the health of the work. The base idea of the techniques for non-destructive (ND) non-invasive (NI) diagnosis is to replace the human senses with measuring tools, in order to automate, standardize and speed up the procedures. Some fundamental characteristics are expected from these methodologies of investigation: no remarkable measurement intrusion; remote measurements possibly without physical contact; ample frequency response; high sensibility; digital recording of data; instruments portability. A monitoring techniques family is based on the vibration of the fresco: a classical example is the impact test, in which the surface of the fresco is struck with a small instrumented hammer Figure 1. Ancient fresco of the Cappella able to measure the intensity of the impulse, and the state of del Comune at the Parma Dome. maintenance of the fresco is deduced on the basis of surface vibration velocity, by analyzing the mobility differences due to local damages. For smaller works of art, it is possible to use standardized means of investigation, for example Xrays. This introduces nevertheless other limitations, due to the necessity to move the work from its usual location and hence not generally applicable. Another very common test is thermography, that consists in the measure of the surface temperature distribution of a material after a thermal impulse. Anomalies in such distribution are clue of a possible defectology. Other less diffused methods are based on ultrasounds, interferometry or radar prospection. The Laser Doppler Vibrometry (LDV) is an innovative technique that allows the vibration velocity measurement of the elements under test without contact, and that also allows to reach a qualitative and quantitative characterization of superficial layers and building structures. To produce the structure excitation impulse, piezoelectric or mechanics actuators are commonly used. Such technique has already been applied in ND diagnostic monitoring of antique frescoes and, in general, of works of art, as for example wall structures, ancient manufactured items, ceramics and mosaics, [3-8]. Experimental application of LDV on a test wall of 1 m2 supporting two layers of plaster containing defects artificially realized, has highlighted a different surface vibratory response near the damaged points, [9]. Other results of vibration-based diagnosis both on test panels and on real frescoes and icons can be found in [10]. In this works, the impulse is generated with piezoelectric actuators to transfer the necessary quantity of energy to the structure. LDV often operates in conjunction with acoustic stimulation. An hybrid acoustic-LDV diagnostic technique is compared in [11] with the traditional hammer excitation, and an acoustic stimulation is applied in [12] to a wide number of artificially aged structures with the purpose to locate and to characterize the defects. An interesting scanning instrumentations designed to automatically detect the fresco damages by acoustic excitation and microphone acquisition is reported in [13]. Finally, the preliminary operations to the creation of an acoustic device that associates the signal acquired through a microphone to the signal acquired through LDV is described in [14]. With such method, that shows effectiveness only for not too small fresco separations, a damage is identified by a sensitive indicator to the perturbation of the phase of the FRF around an anti-resonance point. In this work a ND&NI technique based on acoustic excitation and LDV identification of mechanical vibrations is presented. The measuring methodology is validated on a test panel realized in laboratory, firstly excited by the instrumented hammer. Different delamination thicknesses are taken in consideration, and the adoption of an acoustic absorption coefficient to individuate possible sub-superficial discontinuities is finally discussed. 2. EXPERIMENTAL SET-UP AND TESTS 2.1 Realization of the laboratory panel. A test panel has been realized with the purpose to investigate on the possibility to use LDV for the diagnosis of the health state of antique frescoes, particularly for the survey of detachments, cracks and delaminations. The test panel has been built with the same traditional technique of an antique fresco to emulate its layered structure. The panel, as shown in figure 2, is made of a concrete square tile
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0,5x0,5 m wide. To emulate the gap that arises after a separation among the plaster layers or between the plaster and the wall when a detachment in the fresco occurs, a series of interposed materials is laid on the panel during its preparation. In particular, 9 sample disks of different materials have been inlaid with known geometry and location. In order to not bring out of alignment the disks during the panel preparation, a drop of glue is laid between disks and cement. After that, a layer of plaster 8 mm thick has been applied above the cement and the disks. For simplicity, only one layer has been applied. As shown in figure 2, all the disks have the same shape: they are circular, with diameter of 80 mm and variable thickness between 0,1 mm and 2,5 mm. The center of each disk is placed on a grid, 160 mm one from another and 90 mm from the edge of the panel, to form a regular array. To each row has been assigned a letter (A-B-C), and to each column a number (1-2-3). Such arrangement allows to recognize each disk typology during the data survey, to relate the dynamic answer of the delaminated areas to the respective material and its thickness. The choice of the materials is for something particularly yielding and that doesn't absorb excessive water to not alter the water concentration in the plaster. The materials inlaid in the various positions of the grid, and the relative partial and total thicknesses are listed in table 1. After the glue dried up, the concrete has been dampen up to saturation. This absorbs a lot of water, that quickly enough spreads inside the material. To prepare the plaster on the dry Figure 2. The test panel. cement means to alter the concentration of water in the inside layers of the same plaster causing a great brittleness and crumbliness of it. The following step has been the choice of the lime mortar. An industrial premixed plaster has been preferred rather than to realize it by mixing sand and lime, both for saving time and for demands of precision of execution and inexpensiveness: the cost of the premixed is more or less equal to the sum of the costs of lime and sand. The chosen mortar doesn't contain cement, this type of mortar has considerably shorter drying times. The plaster has been realized with 5 kg of lime premix and around 1 liter of water, mixed slowly to have a homogeneous mixture. During the application, the superficial state has been worked the most homogeneously possible with trowel and float. A surface too wrinkled and rich of imperfections would be able to alter the reading of the laser spot. To make easier the data reading, the points of measure have been gently smoothed down with sandpaper. No pigments have been applied to the surface because they are not influent on the measurements. The finished panel ready for the measurements is illustrated in figure 3.
Figure 3. Experimental set-up.
2.2 Base principles and experimental details. To proceed to the experimental tests it has been necessary to build a frame system to carry the structure in a floating way. First, the panel has been installed on a steel frame; a rubber layer has been added between the panel and the framework to damp undesired vibrations and flutters. Such "elastic bed" has subsequently proved non to influence the diagnosis of the detachments, since the vibratory behavior associated to it (below 50 Hz) clearly appears far away the frequencies of interest for the measures. The frame has been therefore fixed to a support structure by means of ropes and connecting rods, as shown in figure 3. The ropes have been made to pass through the 8 vertexes; turnbuckles have been added to the 4 lower ones, to adjust their tightness. The influence of such realized fixing system, i.e. with the panel elastically suspended, on its vibratory behavior is practically negligible. The impulse on the panel has been transmitted with two different techniques: (i) impact test, with an instrumented hammer Brüel & Kjær 8202 with a rubber tip and a built-in force transducer; (ii) acoustic test, with the two-way loudspeaker Turbosound TXD 121, that consists of a 12” reflex-loaded low frequency driver and a 1” high frequency compression driver with 600 watts program (300 watts r.m.s.), and has a frequency response 60Hz – 20kHz ± 4dB.
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The vibration measurement on the panel test points has been realized in both cases with the laser vibrometer Polytec OFV 505 that gives a resolution of 0,05 mms-1/Hz in the range of 10mms-1/V for a distance range from 0,1 to 100 meters. Measurements have been done aiming the laser ray to the center of each disk, and in 4 points between adjacent disks, equally spaced from the disk centers. These 4 points are indicated in figure 2 with red crosses and in the following identified with Roman numbers from I to IV. These points have been assumed as "healthy": they have been considered sufficiently distant from every disk to disregard their effect. The aim of each test is to acquire the dynamic response in terms of auto power spectrum Gxx(f) the velocity signal measured by the Laser Doppler Vibrometer. The auto power spectrum gives information on how the signal power is distributed in the frequencies domain. Gxx(f) is gotten multiplying the frequency spectrum S(f) of the temporal signal for his own conjugated complex:
Gxx f
S * f S f
(1)
and may therefore be calculated as square of the module of S in the frequencies domain (f). Comparing the spectral diagrams of “healthy” points with the “defected” ones allows to see which peaks are linked to the structure and which are linked to the defects realized in a known way. The first phase of characterization through impact hammer has the double purpose of validate the investigation technique and to detect the range of frequencies in which to analyze by the acoustic impulse. The resultant diagrams from the acquisition of vibration signals give a FRF (Frequency Response Function) in the frequencies domain Ȧ, that is the relationship between the superficial velocity v of the panel and the impact force F of the hammer. This quantity, defined mobility M, can be considered as the inverse of the mechanical impedance Z:
M=
v Ȧ 1 = Z Ȧ F Ȧ
(2).
With the purpose to reduce the noise and the influence of the low repeatability of the impact strength, in every point has been calculated the average of 4 consecutive measures. Subsequently, the diagram of the coherence of the signal has been plotted. The tests with acoustic impulse generally offer the possibility to overcome the necessity of physical access to the point of measure (remote investigation). In fact, if a sufficient quantity of energy is transferred by the air to the surface panel and also to the deepest layers, the acoustic wave can induce vibrations in every point on the panel surface. The acoustic waves can be produced in an ample range of frequencies. In particular, the impulse signal that has been used is a burst-random type on a range of frequencies between 700 and 8000 Hz. The exact measure of the force in this case is particularly complex: the calculation of the FRF doesn’t result practical. In order to get a FRF, it would be in fact advisable to measure the impulse signal through a microphone positioned in proximity of the panel surface, near the laser measurement points. In this case, however, the auto power spectrum of the measured velocity signal can be good enough to observe the presence of defects. A first series of measurements has been made to identify the best configuration; two parameters have been considered in this stage of execution: 1. the loudspeaker height referred to the ground; 2. the loudspeaker distance and orientation relatively to the panel. Different combinations of these parameters have been tried. A second series of acoustic tests has been developed, in order to create an alternative acoustic method for the acquisition of the signal. In this kind of test the signal acquired through LDV, in the same way of the “traditional” test, is associated to the signal acquired through a microphone. In particular, the authors tried to study the possible correlation between the dynamic response of the panel surface in correspondence of a defect and a defined characteristic parameter identifying the surface acoustic absorption. A similar study is reported in [15]. The base principle of this theory is now recalled. The theory considers the power transmitted by an incidental acoustic wave Wi on a material as split in three parts: a fraction Wr is reflected, a fraction Wa is absorbed, and a fraction Wt is transmitted beyond the material itself. Coefficients of reflection, absorption and transmission, are then respectively defined as: Rw =
W W Wr ; Aw = a ; Tw = t Wi Wi Wi
(3).
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The relationship among the coefficients Rw, Aw and Tw can simply be determined by the Law of conservation of energy: Wi = Wr + Wa + Wt. From this, it is deduced that: Rw + Aw + Tw = 1. A fundamental parameter for the evaluation of the phonoabsorbent performances of a material is the “coefficient of apparent acoustic absorption” D. This parameter is defined as energy fraction not reflected by the material: D = 1 – Rw. The equipment used for this second series of tests consists of a sound card Edirol FA 101 and a microphone Behringer ECM8000 for audio analyzers linear measurement in real time, with a frequency response 15 Hz – 20 kHz. The microphone has been positioned at 4,5 cm from the point under examination while the acoustic box used for the excitation has been set 35 cm behind the microphone. The acoustic box is a home-made speaker system consisting of two coaxial plastic tubes of 120 and 95 mm in diameter and 330 mm in length. This second experimental set-up is showed in figure 4. The laser vibrometer is positioned behind the system in line with the speaker and the microphone. The commercial software Adobe Audition v1.5 has been used for the contemporary stereo recording of the signals. It is a professional software for the multi-track audio recording on hard disk. The analysis of the data and the creation of the signals reproduced by the loudspeaker have been performed with the Aurora plug-in. The frequency response is here calculated from two signals: the input is the pressure transmitted to the system through the loudspeaker and measured by the microphone; the output is the vibration velocity as measured by the LDV. As excitation impulse signal a logarithmic-sweep on a range of frequencies between 200 Hz and 15 kHz has been chosen. Since the signal is distorted and contaminated by noise, a de-convolution, that is an Figure 4. Experimental set-up of the algorithm-based process used to reverse the effects of LDVµphone test. convolution on recorded data, is therefore necessary to obtain the system response. Usually many impulse responses are obtained, but only the last one is the response to the linear impulse, the previous ones being the products of harmonic distortion. The responses to the linear impulse of the laser and the microphone respectively correspond to the velocity variation of the panel and the pressure variation near the point under examination. Holding eq. (3), the coefficient of absorption can be obtained. 3. RESULTS AND DISCUSSION 3.1 Impact test. Some results of the dynamic tests on the test panel are summarized in the graphs of figure 5, that shows the FRF of points from A1 to C3 in comparison with not-defected points. It can be noticed that in a range between 2500 and 3500 Hz the defected points show very high peaks not found over the healthy areas. Outside of this range, many peaks in the FRF appear; these have not been considered because they are found both in the FRF of the defected and not-defected points, and certainly correspond to resonance frequencies of the structure. At low frequencies, under 30 Hz, the measurement points show a different behavior. It has been noticed that the peaks below 30 Hz strongly depend on the panel hanging conditions, because aiming the laser in the same point and tightening or loosening the connecting turnbuckles that bind the frame, extremely different dynamic response around this frequency are obtained. Because of this, the zone under 50 Hz has not been investigated. Besides, in the range between 2500 and 3500 Hz the coherence of the defected points is far higher than for the healthy ones. 3.2 Acoustic test. With regard to the results of the acoustic tests, in all tested configurations the peaks of the auto power spectrum of the delaminated areas can be clearly distinguished in the range between 2500 and 3500 Hz. A good match with those already noticed in the hammer tests is found, as figure 6 illustrates. Another similar range shows up, around 5500 – 5600 Hz. During the acoustic tests it has been noticed that more than the distance or the height of the loudspeaker, the factor that mostly influences the quality of the signal is the relative orientation of the loudspeaker with respect to the panel. The test in which the speaker is turned to 270° has given the signal of inferior quality, while when the acoustic wave is turned toward the wall (as in the cases 90° and 180°) the signal is somehow reflected and the graphs result cleaner, being greater the ratio signal/noise. When the speaker is turned 270° it is in fact turned toward the open space of the laboratory and the signal doesn’t get reflected, scatters and gets dirty. With regard to the signal coherence, it is next to 1 in the whole characteristic range of the defect when the orientation is 0° (situation in which it prevails the direct wave), and progressively goes dirtying with the increase of the loudspeaker-panel relative angle. In conclusion, the selected configuration is the one with the speakers directly facing to the panel (orientation 0°), lying 120 cm from it and raised 55 cm on the ground.
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In comparison with the FRF obtained with the impact test, a frequency range is still found, in which the auto power spectrum value increases considerably in presence of delaminated layers, of one or two orders of magnitude if compared with not defected zones. Such range corresponds to the characteristic one of the damaged points previously detected by means of impact test. In correspondence of the healthy points the mobility is lower than in the defected ones, being present only some peaks associated at the resonance of the structure as previously underlined. This can clearly be seen also looking at the power auto-spectrum produced by acoustic excitation, that assumes extremely low values between 2500 and 3500Hz. None of the 4 healthy points produce peaks of meaningful value. If a possible correlation among mobility in the various lacked points and thickness of the correspondent disk is investigated, it can be noticed that there is not a direct correlation among thickness of the material used for reproducing the void and mobility in the dynamic response. In other words, the thickness of a subsuperficial void in a fresco does not seem to influence the vibrational behavior of the superficial layer laying over it. The trend of the coefficient of apparent D is depicted in figure 7. It presents some characteristics peaks at frequencies in the range where the auto power spectrum is higher: between 2500 and 4000 Hz and between 8000 and 12000 Hz the defected points show higher D values. This behavior is absent in not-defected points. These tests show values of the coherence close to 1 in correspondence of frequencies where D = 1: it indicates a totally absorbent surface. 4. CONCLUSIVE REMARKS The laboratory activity presented in this paper focuses on the applicability of a completely not-invasive notdestructive technique for the diagnosis and the health Figure 5. FRF functions of impact test: (a) monitoring of antique frescoes. Measurements have delaminated points (d.p.) A1 (green line), A2 (yellow been conducted on a test panel built following the line), A3 (red line) and healthy point (h.p.) II (gray traditional technique of a fresco, and in which some line); (b) d.p. B1 (green line), B2 (yellow line), B3 defects have been artificially introduced. The (red line) and h.p. I (gray line); (c) d.p. C1 (green measurement technique is based on a traditional modal line), C2 (yellow line), C3 (red line) and h.p. IV (gray analysis system with impact hammer and on an acoustic line). system for structural excitation. An optic laser system is used for the measure of the superficial vibrations of the fresco. The analyses, even though at a preliminary state of the experimentation, have been able to conduct to some meaningful observations. The tests with the impact hammer have shown that there are substantial differences in the surface vibrations between healthy points and points corresponding to detachments or subsuperficial separations: the surface mobility demonstrate to be a valid index of the presence of a separation between support and plaster. The shape of the FRF of defected points strongly depends on the material used to create the void. Factors that influence the shape of the FRF could be the mass, the density, the elastic constant and the damping factor of the material. The maximum value of the mobility however is not directly proportional to the thickness of the defect.
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The results of the acoustic tests confirm the presence of some peaks characteristic of the defects, also found with the impact hammer. In both cases, in a range between 2500 and 3500 Hz, the function of reference (FRF or auto power spectrum) assumes extremely high values if a defect is present under the external surface of the panel while it shows low values and doesn't introduce significant peaks for the healthy points. On the other hand, using a system of acoustic excitation it is not possible to determine, if not in an approximate way, the value of the input force. Because of this, the realization and the reading of the FRF graphs result particularly difficult. Nevertheless, these tests show that the analysis of the auto power spectrum of the Figure 6. Frequency auto-spectrum range of delaminated points signal in velocity is sufficient to detect the obtained by acoustic test (black segments) and maximum range in which the characteristic peaks of mobility frequency of FRF obtained by impact tests (green the defects are found. By comparison with triangles). results from ref. [9-13], it is noticed that when the test is realized on a true wall the FRF function grows of one or two orders of magnitude on a range of frequencies between 2 and 4kHz only if the defect is present. The coherence diagrams are similar to those obtained in this work; also in this case the form of the functions strongly depends on the defect morphology. The present study showed that substantial differences among defected and not defected points in the fresco can be noticed also from the analysis of the coefficient of apparent acoustic absorption. In the ranges between 2500 and 4000 Hz and between 8000 and 12000 Hz the function assumes high values if a defect is present and low values for healthy points. Also the coherence graphs introduce values next to 1 in presence of defected zones. The graphs of the coefficients of absorption of the defected points strongly depend on the material used for simulating the void, but the maximum values are not directly proportional to the thickness. In conclusion, on the authors' opinion the applicability of this methodology in the diagnosis of antique frescoes is evident: the analysis of the coefficients of absorption for the identification of possible detachments between the state of plaster and the wall is effective. A great limitation is that this method doesn't give valid information for the knowledge of the extension and depth of the detached zones. It must be highlighted in fact that in these tests the position and the dimensions of the defects were known a priori; in this case it has been simple, by comparison between output data, to recognize some dynamic response as that of defected points. This must be absolutely considered when the methodology is applied to a real case: to have meaningful data on a structure with unknown dynamic behavior, it is necessary to analyze numerous points and to effect comparisons among them. However, the research is not concluded. The applicability of this diagnosis technique to real frescoes has to be experimented. The greatest efforts will turn to the development of the acoustic technique Figure 7. Coefficient of apparent acoustic absorption of healthy through the refinement of the measurement point II (blue line) and delaminated point B2 (red line). instrumentation characterized by the total lack of contact with the work of art.
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REFERENCES [1] AA.VV. Tecniche e linguaggio del restauro. La decorazione murale e la sua conservazione. Electa, Milano. 2007. (in italian) [2] Torraca G. I dipinti murali. Da: Problemi di conservazione, a cura di G. Urbani. Ed. Compositori, Bologna. 1973. (in italian) [3] Asmus J.F. Non-divestment laser applications in art conservation. Journal of Cultural Heritage. 2003;4:289s-293s. [4] Tornari V., et al. Laser multitask non destructive technology in conservation diagnostic procedures. 2005 Conference on Lasers and Electro-Optics Europe. Munich, Germany, 12-17 June 2005. [5] Castellini P., Esposito E., Marchetti B., Paone N., Tomasini E.P. New applications of Scanning Laser Doppler Vibrometry (SLDV) to non-destructive diagnostics of artworks: mosaics, ceramics, inlaid wood and easel painting. Journal of Cultural Heritage. 2003;4(S1):321-329. [6] Castellini P., Revel G.M. Damage Detection by Laser Vibration Measurement. Proceedings of 15th World Conference on Nondestructive Testing, Roma, 15-21 October 2000. [7] Esposito E., Del Conte A., Agnani A., Stocco R. Impiego integrato di strumentazione NDT nel parco archeologico di Pompei. In Conoscere Senza Danneggiare - Convegno Nazionale AIMAN, Ancona 6 giugno 2006. [8] Esposito E. Recent developments and applications of laser Doppler vibrometry to cultural heritage conservation. 2005 Conference on Lasers and Electro-Optics Europe. Munich, Germany, 12-17 June 2005. [9] Castellini P., Paone N., Tomasini E.P. The laser Doppler vibrometer as an instrument for nonintrusive diagnostic of works of art: application to fresco painting. Optics and Lasers in Engineering. 1996;25:227246. [10] Vignola J.F., et al. Locating faults in wall paintings at the U.S. Capitol by shaker based laser vibrometry. APT Bulletin: Journal of Preservation Technology. 2005;36(1):25-33. [11] Castellini P., Esposito E., Paone N., Tomasini E.P. Non-invasive measurement of damage of fresco paintings and icon by laser scanning vibrometer: experimental results on artificial samples and real works of art. Measurement. 2000;28(1):33-45. [12] Castellini P., et al.. On field validation of non-invasive laser scanning Vibrometer measurement of damaged frescoes: experiments on large walls artificially aged. Journal of Cultural Heritage. 2000;21:s349-s356. [13] Calicchia P., Cannelli, G.B. Detecting and mapping detachments in mural paintings by non-invasive acoustic technique: measurements in antique sites in Rome and Florence. Journal of Cultural Heritage. 2005;6:115-124. [14] Del Vescovo D., Fregolent A. Assessment of fresco detachments through a noninvasive acoustic method. Journal of Sound and Vibration. 2005;284:1015-1031. [15] Calicchia P., Cannelli, G.B. Revealing surface anomalies in structures by in situ measurement of acoustic energy absorption. Applied Acoustics. 2002;63:43-59.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
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ďƐƚƌĂĐƚ dŚŝƐƉĂƉĞƌƉƌĞƐĞŶƚƐĂŶĂƉƉƌŽĂĐŚĨŽƌƐƚƌƵĐƚƵƌĂůŚĞĂůƚŚŵŽŶŝƚŽƌŝŶŐ;^,DͿďLJƵƐŝŶŐĂĚĂƉƚŝǀĞĨŝůƚĞƌƐ͘ dŚĞ ĞdžƉĞƌŝŵĞŶƚĂů ƐŝŐŶĂůƐ ĨƌŽŵ ĚŝĨĨĞƌĞŶƚ ƐƚƌƵĐƚƵƌĂů ĐŽŶĚŝƚŝŽŶƐ ƉƌŽǀŝĚĞĚ ďLJ ƉŝĞnjŽĞůĞĐƚƌŝĐ ĂĐƚƵĂƚŽƌƐͬƐĞŶƐŽƌƐ ďŽŶĚĞĚ ŝŶ ƚŚĞ ƚĞƐƚ ƐƚƌƵĐƚƵƌĞ ĂƌĞ ŵŽĚĞůĞĚ ďLJ Ă ĚŝƐĐƌĞƚĞͲƚŝŵĞ ƌĞĐƵƌƐŝǀĞ ůĞĂƐƚ ƐƋƵĂƌĞ;Z>^ͿĨŝůƚĞƌ͘dŚĞďŝŐŐĞƐƚĂĚǀĂŶƚĂŐĞƚŽƵƐĞĂZ>^ĨŝůƚĞƌŝƐƚŚĞĐůĞĂƌƉŽƐƐŝďŝůŝƚLJƚŽƉĞƌĨŽƌŵĂŶ ŽŶůŝŶĞ^,DƉƌŽĐĞĚƵƌĞƐŝŶĐĞƚŚĂƚƚŚĞŝĚĞŶƚŝĨŝĐĂƚŝŽŶŝƐĂůƐŽǀĂůŝĚĨŽƌŶŽŶͲƐƚĂƚŝŽŶĂƌLJůŝŶĞĂƌƐLJƐƚĞŵƐ͘ Ŷ ŽŶůŝŶĞ ĚĂŵĂŐĞͲƐĞŶƐŝƚŝǀĞ ŝŶĚĞdž ĨĞĂƚƵƌĞ ŝƐ ĐŽŵƉƵƚĞĚ ďĂƐĞĚ ŽŶ ĂƵƚŽƌĞŐƌĞƐƐŝǀĞ ;ZͿ ƉŽƌƚŝŽŶ ŽĨ ĐŽĞĨĨŝĐŝĞŶƚƐ ŶŽƌŵĂůŝnjĞĚ ďLJ ƚŚĞ ƐƋƵĂƌĞ ƌŽŽƚ ŽĨ ƚŚĞ ƐƵŵ ŽĨ ƚŚĞ ƐƋƵĂƌĞ ŽĨ ƚŚĞŵ͘ dŚĞ ƉƌŽƉŽƐĞĚ ŵĞƚŚŽĚ ŝƐ ƚŚĞŶ ƵƚŝůŝnjĞĚ ŝŶ Ă ůĂďŽƌĂƚŽƌLJ ƚĞƐƚ ŝŶǀŽůǀŝŶŐ ĂŶ ĂĞƌŽŶĂƵƚŝĐĂů ƉĂŶĞů ĐŽƵƉůĞĚ ǁŝƚŚ ƉŝĞnjŽĞůĞĐƚƌŝĐƐĞŶƐŽƌƐͬĂĐƚƵĂƚŽƌƐ;WdƐͿŝŶĚŝĨĨĞƌĞŶƚƉŽƐŝƚŝŽŶƐ͘ŚLJƉŽƚŚĞƐŝƐƚĞƐƚĞŵƉůŽLJŝŶŐƚŚĞƚͲƚĞƐƚ ŝƐƵƐĞĚ ƚŽŽďƚĂŝŶƚŚĞĚĂŵĂŐĞĚĞĐŝƐŝŽŶ͘dŚĞƉƌŽƉŽƐĞĚĂůŐŽƌŝƚŚŵǁĂƐĂďůĞƚŽŝĚĞŶƚŝĨLJĂŶĚůŽĐĂůŝnjĞ ƚŚĞĚĂŵĂŐĞƐƐŝŵƵůĂƚĞĚŝŶƚŚĞƐƚƌƵĐƚƵƌĞ͘dŚĞƌĞƐƵůƚƐŚĂǀĞƐŚŽǁŶƚŚĞĂƉƉůŝĐĂďŝůŝƚLJĂŶĚĚƌĂǁďĂĐŬƐ ƚŚĞŵĞƚŚŽĚĂŶĚƚŚĞƉĂƉĞƌĐŽŶĐůƵĚĞƐǁŝƚŚƐƵŐŐĞƐƚŝŽŶƐƚŽŝŵƉƌŽǀĞŝƚ͘ <ĞLJǁŽƌĚƐ͗ ƐƚƌƵĐƚƵƌĂů ŚĞĂůƚŚ ŵŽŶŝƚŽƌŝŶŐ͕ ƐŵĂƌƚ ƐƚƌƵĐƚƵƌĞƐ͕ Z>^ ĨŝůƚĞƌ͕ ƚͲƚĞƐƚ͕ ŽŶůŝŶĞ ĚĂŵĂŐĞ ĚĞƚĞĐƚŝŽŶ͘ EŽŵĞŶĐůĂƚƵƌĞ ƌĞŐƌĞƐƐŝŽŶǀĞĐƚŽƌ ψ (k ) ˆ ƚŝŵĞͲǀĂƌLJŝŶŐƉĂƌĂŵĞƚĞƌǀĞĐƚŽƌ θ (k )
γ (k )
μγ
ref
μ γ unk λ ai bi d na nb P(k ) e(k ) y (k ) ˆy (k )
ŽŶůŝŶĞĚĂŵĂŐĞͲƐĞŶƐŝƚŝǀĞŝŶĚĞdž ƌĞĨĞƌĞŶĐĞŵĞĂŶǀĂůƵĞŽĨ γ (k ) ;ŚĞĂůƚŚLJĐŽŶĚŝƚŝŽŶͿ ƵŶŬŶŽǁŶŵĞĂŶǀĂůƵĞŽĨ γ (k ) ;ŚĞĂůƚŚLJŽƌĚĂŵĂŐĞĚĐŽŶĚŝƚŝŽŶͿ͘ ĨŽƌŐĞƚƚŝŶŐĨĂĐƚŽƌ ŝͲƚŚĂƵƚŽƌĞŐƌĞƐƐŝǀĞ;ZͿĐŽĞĨĨŝĐŝĞŶƚƐŝŶŬŝŶƐƚĂŶƚƐĂŵƉůĞ͘ ŝͲƚŚĞdžŽŐĞŶŽƵƐƉĂƌĂŵĞƚĞƌŝŶŬŝŶƐƚĂŶƚƐĂŵƉůĞ͘ ĚĂŵĂŐĞůŽĐĂƚŝŽŶŝŶĚĞdž ƚŚĞZŽƌĚĞƌ ƚŚĞĞdžŽŐĞŶŽƵƐƉĂƌĂŵĞƚĞƌŽƌĚĞƌ ĂĚĂƉƚŝǀĞĨŝůƚĞƌŐĂŝŶ ĞƌƌŽƌƉƌĞĚŝĐƚŝŽŶďĞƚǁĞĞŶƚŚĞŵĞĂƐƵƌĞŵĞŶƚĂŶĚƚŚĞƉƌĞĚŝĐƚĞĚƐŝŐŶĂů ŽƵƚƉƵƚŵĞĂƐƵƌĞĚƐŝŐŶĂů͘ ƉƌĞĚŝĐƚĞĚƐŝŐŶĂů
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_78, © The Society for Experimental Mechanics, Inc. 2011
875
BookID 214574_ChapID 78_Proof# 1 - 23/04/2011
876
ϭ͘/ŶƚƌŽĚƵĐƚŝŽŶ dŚĞ ƐƚŽĐŚĂƐƚŝĐ ĚŝƐĐƌĞƚĞ ƚŝŵĞ ŵŽĚĞůƐ ĨŽƌ ƐƚƌƵĐƚƵƌĂů ŚĞĂůƚŚ ŵŽŶŝƚŽƌŝŶŐ ;^,DͿ ŚĂǀĞ ďĞĞŶ ŚŝŐŚůLJŝŶǀĞƐƚŝŐĂƚĞĚŝŶůĂƐƚLJĞĂƌƐ͘^ĞǀĞƌĂůŝŶĚĞdžĞƐĨĞĂƚƵƌĞƐĨŽƌĚĂŵĂŐĞͲƐĞŶƐŝƚŝǀĞĐĂŶďĞǁĞůůƵƚŝůŝnjĞĚ ŝŶ ƉƌĂĐƚŝĐĂů ĞdžĂŵƉůĞƐ ďLJ ƵƐŝŶŐ ƚŚĞƐĞ ŵŽĚĞůƐ͘ KŶĞ ĐĂŶ ƵƐĞ ƚŚĞ ĞdžƚƌĂĐƚĞĚ ƉĂƌĂŵĞƚĞƌƐ ŝŶ ƚŚĞƐĞ ŵŽĚĞůƐ͕ĂƐĨŽƌŝŶƐƚĂŶĐĞŵĂĚĞďLJϭ͕ŽƌďLJƵƐŝŶŐƚŚĞŵŽŶŝƚŽƌŝŶŐŽĨƉƌĞĚŝĐƚŝŽŶĞƌƌŽƌƐ͕ƉĞƌĨŽƌŵĞĚĨŽƌ ĞdžĂŵƉůĞŝŶϮĞϯ͕ĂƐŝŶĚĞdžĚĂŵĂŐĞͲƐĞŶƐŝƚŝǀĞŝĚĞŶƚŝĨŝĞĚďLJŽĨĨůŝŶĞĚĂƚĂ͘ŽŵƉĂƌŝŶŐƚŚĞƌĞĨĞƌĞŶĐĞ ĂŶĚƵŶŬŶŽǁŶŝŶĚĞdžĞƐŝŶĂƐƚĂƚŝƐƚŝĐĂůǁĂLJŝƐƉŽƐƐŝďůĞƚŽĚŝĂŐŶŽƐƚŝĐƚŚĞƐƚƌƵĐƚƵƌĂůŚĞĂůƚŚƐƚĂƚĞŽĨƚŚĞ ƐLJƐƚĞŵ͘ ^ŵĂƌƚŵĂƚĞƌŝĂůƐŚĂǀĞďĞĞŶƐƵŝƚĂďůĞĨŽƌ^,DĂƉƉůŝĐĂƚŝŽŶƐĚƵĞƚŽŚŝŐŚĨƌĞƋƵĞŶĐLJĞdžĐŝƚĂƚŝŽŶ ĂŶĚ ůŽǁ ƉŽǁĞƌ ƌĞƋƵŝƌĞĚ ƚŚƌŽƵŐŚ WdƐ͘ WĂƌƚŝĐƵůĂƌůLJ ŝŶ ƐŵĂƌƚ ƐƚƌƵĐƚƵƌĞƐ ŚĂǀĞ ďĞĞŶ ĚĞƚĂĐŚĞĚ ƚŚĞ ŵŽŶŝƚŽƌŝŶŐƚŚƌŽƵŐŚĞůĞĐƚƌŝĐĂůŝŵƉĞĚĂŶĐĞƐŝŐŶĂůƐǁŝƚŚŚŝŐŚƐĂŵƉůŝŶŐƌĂƚĞ͕ƚLJƉŝĐĂůůLJďŝŐŐĞƌƚŚĂŶϭϬϬ Ŭ,nj͘dŚĞŝŶƉƵƚͲŽƵƚƉƵƚƐŝŐŶĂůƐĂƌĞŶŽƌŵĂůůLJƐƚŽƌĞĚƵƐŝŶŐĂƐŝŶĞͲƐǁĞĞƉŝŶƉƵƚǁŝƚŚĨƌĞƋƵĞŶĐLJǀĂƌLJŝŶŐ ŝŶ Ă ƌĂŶŐĞ ŽĨ ŝŶƚĞƌĞƐƚ͘ ^Ž͕ ƚŚĞ ƐŝŐŶĂůƐ ŝŶǀŽůǀĞĚ ĂƌĞ ĞƐƐĞŶƚŝĂůůLJ ƚƌĂŶƐŝĞŶƚƐ͘ ƉƌŝŽƌŝ͕ ƚŚĞ ƚŽŽůƐ ƚŽ ĂŶĂůLJnjĞ ƚŚĞ ŚĞĂůƚŚ ŽĨ ƚŚĞ ƐLJƐƚĞŵ ŝŶ ƚŝŵĞ ĚŽŵĂŝŶ ƐŚŽƵůĚ ďĞ ŶŽŶͲƐƚĂƚŝŽŶĂƌLJ ďĞĐĂƵƐĞ ƚŚĞ ƚŝŵĞͲ ĨƌĞƋƵĞŶĐLJĐŚĂƌĂĐƚĞƌŝƐƚŝĐƐĐŚĂŶŐĞƐŝŶĨƵŶĐƚŝŽŶŽĨƚŚĞƚŝŵĞ͘DŽƌĞŽǀĞƌ͕ĂŶĂďƌƵƉƚǀĂƌŝĂƚŝŽŶŽƌĂůŽĐĂů ĨƵŶĐƚŝŽŶǀĂŶŝƐŚŝŶŐŽƵƚƐŝĚĞĂƐŚŽƌƚƚŝŵĞŝŶƚĞƌǀĂůĐŽƵůĚďĞƌĞƉƌĞƐĞŶƚĞĚĂĚĂƉƚŝǀĞůLJ͘dŚĞŵŽƐƚƉĂƌƚŽĨ ƚŚĞ ^,D ŵĞƚŚŽĚƐ ŝŶ ƐŵĂƌƚ ƐƚƌƵĐƚƵƌĞƐ ƵƐĞ ƚŚĞ ĞŶƚŝƌĞ ĚĂƚĂ ƌĞĐŽƌĚ ĨŽƌ ƉŽƐƚͲĞǀĞŶƚ ĂŶĂůLJƐŝƐ ƌĂƚŚĞƌ ƚŚĂŶƌĞĂůͲƚŝŵĞĨĞĂƚƵƌĞŝĚĞŶƚŝĨŝĐĂƚŝŽŶĨŽƌĚĂŵĂŐĞĚĞĐŝƐŝŽŶ͘ dŚĞŽŶůŝŶĞŝĚĞŶƚŝĨŝĐĂƚŝŽŶĂŶĚƚƌĂĐŬŝŶŐƚŝŵĞͲǀĂƌLJŝŶŐƉĂƌĂŵĞƚĞƌƐĐĂŶďĞĚƌŝǀĞŶďLJƵƐŝŶŐŵĂŶLJ ƚŝŵĞͲĨƌĞƋƵĞŶĐLJ ŵĞƚŚŽĚƐ͘ /ƚ ĐĂŶ ďĞ ĐŝƚĞĚ ǁĂǀĞůĞƚ ĂŶĂůLJƐŝƐ ϰ͕ tŝŐŶĞƌͲsŝůůĞ ĚŝƐƚƌŝďƵƚŝŽŶƐ ϱ͕ ,ŝůďĞƌƚͬ'ĂďŽƌƚƌĂŶƐĨŽƌŵĂŶĂůLJƐŝƐƚĞĐŚŶŝƋƵĞƐϲ͕ƵƉĚĂƚŝŶŐĂůŐŽƌŝƚŚŵĨŽƌƐƵďƐƉĂĐĞƚƌĂĐŬŝŶŐϳ͕Žƌ ĂĚĂƉƚŝǀĞ ĨŝůƚĞƌƐ ĐŽŶƐƚƌƵĐƚĞĚ ďĂƐĞĚ ŽŶ ƚŚĞ <ĂůŵĂŶ ƚŚĞŽƌLJ͕ tŝĞŶĞƌ ƚŚĞŽƌLJ ƚŽ ĞdžƉĂŶĚ ĞĂĐŚ ƚŝŵĞͲ ǀĂƌLJŝŶŐ ƉĂƌĂŵĞƚĞƌ ĐŽĞĨĨŝĐŝĞŶƚ ŝŶƚŽ Ă ƐĞƚ ŽĨ ďĂƐŝƐ ƐĞƋƵĞŶĐĞ ĂŶĚ ǁŝƚŚ ƚŚĞ >ĞĂƐƚ ^ƋƵĂƌĞƐ ;>^Ϳ ĂůŐŽƌŝƚŚŵ͘ůůƚŚĞƐĞŵĞƚŚŽĚƐŚĂǀĞďĞĞŶƵƐĞĚĂŶĚĂĚĂƉƚĞĚĨŽƌ^,DƉƵƌƉŽƐĞƐŝŶƐĞǀĞƌĂůƌĞƐĞĂƌĐŚ ǁŽƌŬƐŝŶƚŚĞůŝƚĞƌĂƚƵƌĞ͘ ŵŽŶŐƚŚĞĂƉƉƌŽĂĐŚĞƐĨŽƌŶŽŶͲƐƚĂƚŝŽŶĂƌLJ ƐLJƐƚĞŵƐŝĚĞŶƚŝĨŝĐĂƚŝŽŶ͕ ƚŚĞŽŶůŝŶĞĚŝƐĐƌĞƚĞͲƚŝŵĞ ĂƉƉƌŽĂĐŚĞƐƚŚƌŽƵŐŚĂĚĂƉƚŝǀĞĨŝůƚĞƌƐĨŽƌƚƌĂĐŬŝŶŐƚŝŵĞǀĂƌLJŝŶŐƉĂƌĂŵĞƚĞƌĂƌĞǁĞůůĐŽŶƐŽůŝĚĂƚĞĚŝŶ ƚŚĞůŝƚĞƌĂƚƵƌĞǁŝƚŚĐůĂƐƐŝĐĂůƚĞdžƚŬƐ͕ĂƐĨŽƌĞdžĂŵƉůĞϴ͕ϵĂŶĚϭϬ͘ dŚĞ ƉƌĞƐĞŶƚ ƉĂƉĞƌ ĞdžƉĂŶĚƐ ƚŚĞ ƉƌŽĐĞĚƵƌĞ ďĂƐĞĚ ŽŶ Z>^ ĨŝůƚĞƌ ĨŽƌ ŝĚĞŶƚŝĨŝĐĂƚŝŽŶ ĂŶĚ ƐƚƌƵĐƚƵƌĂůŚĞĂůƚŚŵŽŶŝƚŽƌŝŶŐŝŶĂƌĞĂůĂĞƌŽŶĂƵƚŝĐĂůƉĂŶĞůǁŝƚŚƉŝĞnjŽĞůĞĐƚƌŝĐĐĞƌĂŵŝĐƐ;WdƐͿďŽŶĚĞĚ ŝŶ ƚŚĞ ĞdžƚĞƌŶĂů ƐƵƌĨĂĐĞ͘ dŚĞ ŝŶƉƵƚͲŽƵƚƉƵƚ ŵĞĂƐƵƌĞŵĞŶƚƐ ĨƌŽŵ WdƐ ǁĞƌĞ ƵƐĞĚ ƚŽ ĞdžƚƌĂĐƚ ƚŚĞ ĨĞĂƚƵƌĞ ŝŶĚĞdž ŝĚĞŶƚŝĨŝĞĚ ďLJ ƚŚĞ ĂƵƚŽͲƌĞŐƌĞƐƐŝǀĞ ƉŽƌƚŝŽŶ ŽĨ ƚŚĞ ŵŽĚĞů ŝŶ ĞĂĐŚ ƐĂŵƉůĞ ƚŝŵĞ͘ dŚĞ ĚĂŵĂŐĞĚĞĐŝƐŝŽŶƉƌŽĐĞĚƵƌĞĞŵƉůŽLJƐĂƐƚĂƚŝƐƚŝĐĂůŚLJƉŽƚŚĞƐŝƐƚĞƐƚƵƐŝŶŐƚŚŝƐŝŶĚĞdž͘dŚĞƌĞƐƵůƚƐŚĂǀĞ ƐŚŽǁŶƚŚĞĐŽƌƌĞĐƚŚĞĂůƚŚLJƐƚĂƚĞĂŶĚƚŚĞůŽĐĂůŝnjĂƚŝŽŶŽĨƚŚĞĨĂƵůƚŝŶƚŚĞƐLJƐƚĞŵ͘ Ϯ͘ZĞĐƵƌƐŝǀĞ>ĞĂƐƚ^ƋƵĂƌĞ&ŝůƚĞƌ dŚĞ Z>^ ĨŝůƚĞƌ ŚĂƐ ƚŚĞ ƐĂŵĞ ƐƚƌƵĐƚƵƌĞ ŽĨ ƚŚĞ >^ ŵĞƚŚŽĚ͕ ďƵƚ ŝƚ ŝƐ ŵŽĚŝĨŝĞĚ ďLJ ĐŽŶƐŝĚĞƌŝŶŐ ĚŝƐĐĂƌĚŝŶŐƚŚĞŽůĚŵĞĂƐƵƌĞŵĞŶƚƐŝŶŽƌĚĞƌƚŽƉƌŽǀŝĚĞĐŚĂŶŐĞƐŝŶƚŽƚŚĞŵŽĚĞůĚLJŶĂŵŝĐĂůůLJŝŶĞĂĐŚ k ƐĂŵƉůĞ͘ ƉŽǁĞƌĨƵů ĂůŐŽƌŝƚŚŵ ĨŽƌ ƚƌĂĐŬŝŶŐ ƐLJƐƚĞŵ ŝƐ ŽďƚĂŝŶĞĚ ďLJ ĞdžƉŽŶĞŶƚŝĂůůLJ ǁĞŝŐŚƚŝŶŐ ƚŚĞ ĚĂƚĂƚŽƌĞŵŽǀĞƚŚĞĞĨĨĞĐƚƐŽĨŽůĚĚĂƚĂ͘^Ž͕ƚŚĞƉĂƌĂŵĞƚĞƌǀĞĐƚŽƌ θˆ (k ) ƚŽŝĚĞŶƚŝĨLJŝƐŐŝǀĞŶďLJ͕ϭϭ͕ ϭϮĂŶĚϭϯ͗ ;ϭͿ θˆ (k ) = θˆ (k − 1) + P(k )ψ (k )e(k )
BookID 214574_ChapID 78_Proof# 1 - 23/04/2011
877
ǁŚĞƌĞ ψ (k ) ŝƐƚŚĞƌĞŐƌĞƐƐŝŽŶǀĞĐƚŽƌƚŚĂƚĐŽŶƚĂŝŶƐ ŽůĚǀĂůƵĞƐŽĨŽďƐĞƌǀĞĚŝŶƉƵƚƐĂŶĚŽƵƚƉƵƚƐ;ŝŶ ŽƵƌ ĞdžĂŵƉůĞ ǀŽůƚĂŐĞƐ ŝŶƉƵƚƐ ĂŶĚ ŽƵƚƉƵƚƐ ƉƌŽǀŝĚĞĚ ďLJ ƚŚĞ WdƐͿ͖ e(k ) = y (k ) − ˆy (k ) ŝƐ ƚŚĞ ĞƌƌŽƌ ƉƌĞĚŝĐƚŝŽŶďĞƚǁĞĞŶƚŚĞŵĞĂƐƵƌĞŵĞŶƚĂŶĚƚŚĞƉƌĞĚŝĐƚĞĚƐŝŐŶĂů ˆy (k ) ƚŚĂƚŝƐŐŝǀĞŶďLJ͗ ˆy (k ) = ψ T (k )θˆ (k − 1) ;ϮͿ ĂŶĚĨŝŶĂůůLJƚŚĞŐĂŝŶ P(k ) ŝƐŐŝǀĞŶďLJ͗ § P(k − 1)ψ T (k )ψ (k )P(k − 1) · ¸ ;ϯͿ P (k ) = λ−1 ¨¨ P (k − 1) − λ +ψ T (k )P(k − 1)ψ (k ) ¸¹ © ǁŚĞƌĞ ƚŚĞ ƌĞĂů ǀĂůƵĞ ƐĐĂůĂƌƐ ƚŚĂƚ ŵĂŬĞ ƵƉ P(k ) ŝŶ k ƐĂŵƉůĞ ĂƌĞ ƉĂƌƚ ŽĨ Ă ƌĞĐƵƌƐŝǀĞ ŵĞĂŶƐ ŽĨ
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−1
ĐŽŵƉƵƚŝŶŐƚŚĞŝŶǀĞƌƐĞŽĨƚŚĞŵĂƚƌŝdž ȥ T ȥ ͕ϭϮdŚĞƐĐĂůĂƌ λ ŝƐŬŶŽǁŶĂƐĨŽƌŐĞƚƚŝŶŐĨĂĐƚŽƌ͕ĂůƐŽ ĐĂůůĞĚĂƐǀĂƌŝĂŶĐĞŽĨŝŶŶŽǀĂƚŝŽŶƐ͘dLJƉŝĐĂůůLJƚŚĞĨŽƌŐĞƚƚŝŶŐĨĂĐƚŽƌŝƐŝŶƚŽƚŚĞƌĂŶŐĞŽĨϬ͘ϵϲƚŽϬ͘ϵϵϱ͕ ǁŚŝĐŚĂŵŽƵŶƚƐƚŽĂƉƉƌŽdžŝŵĂƚĞůLJƌĞŵĞŵďĞƌŝŶŐϯϯͲϮϬϬůĂƐƚŽďƐĞƌǀĂƚŝŽŶƐƌĞƐƉĞĐƚŝǀĞůLJŝŶĚĂƚĂƐĞƚ͘/Ĩ ƚŚĞ ĨŽƌŐĞƚƚŝŶŐ ĨĂĐƚŽƌ λ ŝƐ ĐŚŽƐĞŶ ĞƋƵĂů ϭ͘Ϭ͕ ƚŚĞ ĨŝůƚĞƌ ƌĞƐƵůƚĂŶƚ ĐŽƌƌĞƐƉŽŶĚƐ ƚŽ ƚŚĞ ƐƚĂŶĚĂƌĚ ĂĚĂƉƚŝǀĞůĞĂƐƚŵĞĂŶƐ;>D^ͿĨŝůƚĞƌ͘ /Ŷ ŽƵƌ ĞdžĂŵƉůĞ ŝƐ ĂƐƐƵŵĞĚ ƚŚĂƚ ƚŚĞ ŵŽĚĞů ƐƚƌƵĐƚƵƌĞ ĐŽƌƌĞƐƉŽŶĚ ƚŽ Ă ZĞĐƵƌƐŝǀĞ ƵƚŽͲ ZĞŐƌĞƐƐŝǀĞǁŝƚŚĞyŽŐĞŶŽƵƐŝŶƉƵƚ;ZZyͿŵŽĚĞů͕ŽŶĐĞƚŚĂƚĂƌĞŬŶŽǁŶƚŚĞŝŶƉƵƚĂŶĚŽƵƚƉƵƚƐƐŝŐŶĂůƐ ĨƌŽŵƚŚĞWdƐďŽŶĚĞĚŝŶƚŚĞƉĂŶĞů͘ZĞƌĐƵƌƐŝǀĞƵƚŽͲZĞŐƌĞƐƐŝǀĞǁŝƚŚDŽǀŝŶŐǀĞƌĂŐĞĞyŽŐĞŶŽƵƐ ŝŶƉƵƚ;ZDyͿŵŽĚĞůĐŽƵůĚďĞƵƐĞĚ͕ďƵƚŝŶƚŚĞƉƌĞƐĞŶƚĞdžĂŵƉůĞƚŚĞŶŽŝƐĞƐŽƵƌĐĞƉƌŽǀŝĚĞĚďLJƚŚĞ ƐLJƐƚĞŵŝƐůŽǁ͘ dŚĞƉĂƌĂŵĞƚĞƌƐƚŽďĞĞƐƚŝŵĂƚĞĚŝŶĂZZyŵŽĚĞůĂƌĞŐŝǀĞŶďLJ͗ ;ϰͿ θˆ (k ) = [a1 , a2 , , ana ,b1 ,b2 , ,bnb ] ǁŚĞƌĞ a i ŝƐƚŚĞŝͲƚŚĂƵƚŽƌĞŐƌĞƐƐŝǀĞ;ZͿĐŽĞĨĨŝĐŝĞŶƚƐ͕ bi ŝƐƚŚĞŝͲƚŚĞdžŽŐĞŶŽƵƐƉĂƌĂŵĞƚĞƌ͕ĂŶĚ na ĂŶĚ nb ĂƌĞƚŚĞŽƌĚĞƌŽĨƚŚĞĂƵƚŽƌĞŐƌĞƐƐŝǀĞĂŶĚĞdžŽŐĞŶŽƵƐƉĂƌĂŵĞƚĞƌƐ͕ƌĞƐƉĞĐƚŝǀĞůLJ͘dŚĞŽƌĚĞƌŝŶ ƚŚĞƉƌĞƐĞŶƚƉĂƉĞƌǁĂƐŽďƚĂŝŶĞĚďLJĂƐƚĂďŝůŝnjĂƚŝŽŶĐƌŝƚĞƌŝŽŶďĂƐĞĚŽŶƚŚĞŽďƚĂŝŶŝŶŐĂƐƚĂďůĞŵŽĚĞů ĂŶĚŵŝŶŝŵƵŵƌĞƐŝĚƵĂůĞƌƌŽƌďĞƚǁĞĞŶƚŚĞŵĞĂƐƵƌĞĚĂŶĚƉƌĞĚŝĐƚĞĚĚĂƚĂ͘ ϯ͘Z>^ĂŵĂŐĞͲ^ĞŶƐŝƚŝǀĞ/ŶĚĞdž dŚĞ Z ĐŽĞĨĨŝĐŝĞŶƚƐ ŝŶ θˆ (k ) ĂƌĞ ĚŝƌĞĐƚůLJ ĂƐƐŽĐŝĂƚĞĚ ǁŝƚŚ ƚŚĞ ƉŽůĞƐ ŝŶ ƚŚĞ ĚŝƐĐƌĞƚĞͲƚŝŵĞ ĐŽŵƉůĞdž ƉůĂŶĞ͘ dŚĞƌĞĨŽƌĞ͕ ƚŚĞ ƉƌĞƐĞŶĐĞ ŽĨ ĚĂŵĂŐĞƐ ĐĂŶ ďĞ ĚĞƚĞĐƚĞĚ ƉƌŝŵĂƌŝůLJ ďLJ ĐŚĞĐŬŝŶŐ ƚŚĞ ĐŚĂŶŐĞƐŽĨƉĂƌĂŵĞƚĞƌƐĚƵƌŝŶŐƚŚĞŽƉĞƌĂƚŝŽŶ͘dŚĞĐŽŵƉůĞdžƌŽŽƚƐŽĨƚŚĞZŵŽĚĞůƐĂƌĞƐĞŶƐŝƚŝǀĞƚŽ ƐƚƌƵĐƚƵƌĂůĚĂŵĂŐĞĐĂƵƐŝŶŐĂǀĂƌŝĂƚŝŽŶŝŶƚŚĞůŽĐĂƚŝŽŶŝŶƚŚĞnjͲĚŽŵĂŝŶ͕ĂƐǁĞůůĚŝƐĐƵƐƐĞĚŝŶϭϰ͘^Ž͕ ŽŶĞĐĂŶƉƌŽƉŽƐĞĂŶŽŶůŝŶĞĚĂŵĂŐĞͲƐĞŶƐŝƚŝǀĞŝŶĚĞdž γ (k ) ƚŽƋƵĂŶƚŝƚĂƚŝǀĞůLJŝĚĞŶƚŝĨLJƚŚĞŵŝŐƌĂƚŝŽŶŽĨ ƉŽůĞƐŝŶƚŚĞnjͲƉůĂŶ͘dŚŝƐŝŶĚĞdžŝƐĐŽŵƉƵƚĞĚďLJĐŽŶƐŝĚĞƌŝŶŐƚŚĞZĐŽĞĨĨŝĐŝĞŶƚƐŝŶƚŚĞƚŝŵĞͲǀĂƌLJŝŶŐ ƉĂƌĂŵĞƚĞƌǀĞĐƚŽƌ θˆ (k ) ŶŽƌŵĂůŝnjĞĚďLJƚŚĞƐƋƵĂƌĞƌŽŽƚŽĨƚŚĞZĐŽĞĨĨŝĐŝĞŶƚƐ͗ na ai ¦ i =1 ;ϱͿ γ (k ) = , k = 1,2 , , N na
¦a i =1
2
i
BookID 214574_ChapID 78_Proof# 1 - 23/04/2011
878
ǁŚĞƌĞEŝƐƚŚĞŶƵŵďĞƌŽĨƚŝŵĞƐĂŵƉůĞƐ͘ ref /Ĩ ƚŚĞ ƌĞĨĞƌĞŶĐĞ ŵĞĂŶ ǀĂůƵĞ ŽĨ γ (k ) ͕ μ γ ͕ ŚĂƐ Ă ƐŝŐŶŝĨŝĐĂŶƚ ĚŝĨĨĞƌĞŶĐĞ ďĞƚǁĞĞŶ Ă ŵĞĂŶ ǀĂůƵĞƐ ŽďƚĂŝŶĞĚ ǁŚĞŶ ƚŚĞ ƐLJƐƚĞŵ ŝƐ ŝŶ ƵŶŬŶŽǁŶ ĐŽŶĚŝƚŝŽŶ͕ μ γ
unk
͕ ƚŚĞ ĚĂŵĂŐĞ ĚĞĐŝƐŝŽŶ ƐŚŽƵůĚ
ŝŶĚŝĐĂƚĞƚŚĞƉƌĞƐĞŶĐĞŽĨƐƚƌƵĐƚƵƌĂůǀĂƌŝĂƚŝŽŶ͘ /Ŷ ŽƌĚĞƌ ƚŽ ŚĞůƉ ŝŶ ƚŚŝƐ ĚĞĐŝƐŝŽŶ Ă ŚLJƉŽƚŚĞƐŝƐ ƚĞƐƚ ŝŶǀŽůǀŝŶŐ ƚͲƚĞƐƚ ƉƌŽĐĞĚƵƌĞ ŵĂLJ ďĞ ƉĞƌĨŽƌŵĞĚ ƚŽĚĞƚĞƌŵŝŶĞŝĨƚŚĞƐĞ ĚŝĨĨĞƌĞŶĐĞƐĂƌĞƐŝŐŶŝĨŝĐĂŶƚ ϭϱ͘/ŶƚŚŝƐ ǁĂLJ͕ dŚĞ ŶƵůů ŚLJƉŽƚŚĞƐŝƐ H 0 ĂŶĚĂůƚĞƌŶĂƚŝǀĞŚLJƉŽƚŚĞƐŝƐ H 1 ĂƌĞŐŝǀĞŶďLJ͗ ref unk H 0 : μγ = μγ ;ϲͿ ref unk H1 : μγ ≠ μγ H 0 ŝƐĂŶŝŶĚŝĐĂƚŝǀĞŽĨŚĞĂůƚŚLJƐLJƐƚĞŵĂŶĚ H 1 ŝŶĚŝĐĂƚĞƐĚĂŵĂŐĞĐŽŶĚŝƚŝŽŶƐ͘dŚĞƐŝŐŶŝĨŝĐĂŶƚůĞǀĞůŽĨ ƚŚĞƚĞƐƚŝƐƐĞƚĂƚɲǀĂůƵĞ͘dŚĞƉͲǀĂůƵĞŝƐƚŚĞƉƌŽďĂďŝůŝƚLJƚŚĂƚƚŚĞŝŶĚĞdž γ (k ) ĚŽĞƐŶŽƚŐŝǀĞŝŶĚŝĐĂƚŝǀĞ ŽĨĚĂŵĂŐĞŝĨƚŚĞƌĞĂƌĞƐƚƌƵĐƚƵƌĂůĐŚĂŶŐĞƐŝŶƚŚĞƐLJƐƚĞŵ͘dŚĞƌƵůĞŝƐƚŽĂĐĐĞƉƚŽƌƌĞũĞĐƚ H 0 ďĂƐĞĚ ŽŶ ƚŚĞ ƉͲǀĂůƵĞ͘ /Ĩ ƚŚĞ ƉͲǀĂůƵĞ ŝƐ ůĞƐƐĞƌ ƚŚĂŶ ɲ͕ ƚŚĞŶ ƚŚĞ ŶƵůů ŚLJƉŽƚŚĞƐŝƐ H 0 ŝƐ ƌĞũĞĐƚĞĚ ĂŶĚ ƚŚĞ ĂůƚĞƌŶĂƚŝǀĞŚLJƉŽƚŚĞƐŝƐ H 1 ŝƐĂĐĐĞƉƚĞĚ;ĚĂŵĂŐĞĐŽŶĚŝƚŝŽŶͿ͘ YƵĂůŝƚĂƚŝǀĞůLJ ƚŚĞ ĚĂŵĂŐĞ ůŽĐĂƚŝŽŶ ĐĂŶ ďĞ ƉĞƌĨŽƌŵĞĚ ďLJ ĐŽŵƉĂƌŝŶŐ ƚŚĞ ĚŝƐƚĂŶĐĞ ďĞƚǁĞĞŶ ref unk ƚŚĞ ŵĞĂŶ ǀĂůƵĞƐ ŽĨ ƌĞĨĞƌĞŶĐĞ μ γ ĂŶĚ ĚĂŵĂŐĞ ĐŽŶĚŝƚŝŽŶ μ γ ĨŽƌ ĞĂĐŚ Wd͘ /Ĩ ƚŚĞ ĚĂŵĂŐĞ ŝƐ ĐůŽƐĞ ƚŽ ŽŶĞ Wd ƚŚĞ ĚŝĨĨĞƌĞŶĐĞ ďĞƚǁĞĞŶ ƚŚĞ ŵĞĂŶ ǀĂůƵĞƐ ŝŶ ƚŚŝƐ ƐĞŶƐŽƌ ƐŚŽƵůĚ ďĞ ďŝŐŐĞƌ ƚŚĂŶ ŽƚŚĞƌWdƐ͘dŚĞĚĂŵĂŐĞůŽĐĂƚŝŽŶŝŶĚĞdžƉƌŽƉŽƐĞĚŝŶƚŚŝƐƉĂƉĞƌŝƐĐŽŵƉƵƚĞĚďLJ͗ μ γ ref − μ γ unk d= ;ϳͿ ref
μγ
ϰ͘ƉƉůŝĐĂƚŝŽŶ džĂŵƉůĞ /ŶŽƌĚĞƌƚŽǀĞƌŝĨLJƚŚĞƉƌŽƉŽƐĞĚŵĞƚŚŽĚŽůŽŐLJ͕ĞdžƉĞƌŝŵĞŶƚĂůƚĞƐƚƐŝŶůĂďŽƌĂƚŽƌLJǁĞƌĞĚŽŶĞ ŝŶĂŶĂĞƌŽŶĂƵƚŝĐĂůƉĂŶĞůĐŽƵƉůĞĚǁŝƚŚƚŚƌĞĞWdƐƐŚŽǁŶŝŶĨŝŐ͘ϭ͘dŚĞWdƐĞůĞŵĞŶƚƐ͕ĐĂůůĞĚWdϭ͕ WdϮĂŶĚWdϯ͕ǁĞƌĞďŽŶĚĞĚŽŶƉĂŶĞůƐƵƌĨĂĐĞ͘dŚĞWdƐ͘dŚĞWdƐĂƌĞƚLJƉĞďƵnjnjĞƌηϳͲϮϳͲϰĨƌŽŵ DƵƌĂƚĂDĂŶƵĨĂĐƚƵƌŝŶŐ͘dŚĞWdϭŝƐƵƐĞĚĂƐĞdžĐŝƚĂƚŝŽŶŝŶƉƵƚĂŶĚWdϮĂŶĚWdϯĂƌĞƵƐĞĚĂƐƐĞŶƐŽƌ͘ dŚĞŝŶƉƵƚͲŽƵƚƉƵƚ ĚĂƚĂĂĐƋƵŝƐŝƚŝŽŶ ǁĂƐĐŽŶƚƌŽůůĞĚƵƐŝŶŐƚŚĞ>ĂďsŝĞǁ ǁŝƚŚĂEĂƚŝŽŶĂů/ŶƐƚƌƵŵĞŶƚƐ ĐĂƌĚďŽĂƌĚ͕ŵŽĚĞůE/Ͳh^ϲϮϭϭ͕ϭϲďŝƚƐ͘dŚĞƐŝŐŶĂůƐǁĞƌĞƐƚŽƌĞĚǁŝƚŚĂƐĂŵƉůĞƌĂƚĞŽĨϮϱϬŬ,njĂŶĚ ϲϱϱϯϲƐĂŵƉůĞƐǁĞƌĞƌĞĐŽƌĚĞĚŝŶĞĂĐŚĐŚĂŶŶĞů͘
BookID 214574_ChapID 78_Proof# 1 - 23/04/2011
879
ĂŵĂŐĞ
Wdϯ
Wdϭ
WdϮ
;ĂͿ
;ďͿ
&ŝŐƵƌĞϭ͘ĞƌŽŶĂƵƚŝĐĂůƉĂŶĞůƵƐĞĚŝŶƚŚĞĞdžƉĞƌŝŵĞŶƚĂůƚĞƐƚƐ͖;ĂͿ^ĐŚĞĚƵůĞƉĂŶĞů͖;ďͿĞƚĂŝůƐŽĨƚŚĞ WdƐĂŶĚƚŚĞĚĂŵĂŐĞƉŽƐŝƚŝŽŶ͘ dŚĞƐŝŐŶĂůŝŶƉƵƚĂƉƉůŝĞĚǁĂƐĂƐŝŶĞͲƐǁĞĞƉƚLJƉĞǁŝƚŚцϭϬsĂŶĚĨƌĞƋƵĞŶĐLJƌĂŶŐĞϬƚŽϲϱ͘ϱ Ŭ,nj͘^ŝŶĐĞƚŚĞƐŝŐŶĂůƐǁĞƌĞůŝŵŝƚĞĚŝŶƚŽƚŚŝƐƌĂŶŐĞĂĚŽǁŶƐĂŵƉůŝŶŐƉƌŽĐĞĚƵƌĞďLJϯǁĂƐƉĞƌĨŽƌŵĞĚ ƚŽ ŽďƚĂŝŶ Ă ŶĞǁ ƐĂŵƉůŝŶŐ ƌĂƚĞ ŽĨ ϴϯ͘ϯ Ŭ,nj͘ ůů ƚŚĞ ƐŝŐŶĂůƐ ǁĞƌĞ ƉƌĞͲĨŝůƚĞƌĞĚ ƚŽ ƌĞŵŽǀĞ ĐŽŵƉŽŶĞŶƚƐ͕ůŽǁĨƌĞƋƵĞŶĐLJĚŝƐƚŽƌƚŝŽŶƐĂŶĚƚŽĞůŝŵŝŶĂƚĞƉŽƐƐŝďůĞůŝŶĞĂƌƚƌĞŶĚƐ͘ &ŽƵƌŵĞĂƐƵƌĞŵĞŶƚƐǁĞƌĞƉĞƌĨŽƌŵĞĚŝŶƚŚĞŚĞĂůƚŚLJĐŽŶĚŝƚŝŽŶŝŶWdϮĂŶĚWdϯǁŚĞŶƚŚĞ ŝŶƉƵƚǁĂƐĂƉƉůŝĞĚŝŶWdϭ͘dŚƌĞĞŽĨƚŚĞƐĞĚĂƚĂǁĞƌĞƵƐĞĚĂƐƌĞĨĞƌĞŶĐĞďĂƐĞůŝŶĞĂŶĚŽŶĞĨŽƌƚĞƐƚ ĨĂůƐĞƉŽƐŝƚŝǀĞ͘dŚĞĚĂŵĂŐĞĐŽŶĚŝƚŝŽŶƐǁĞƌĞĐŽŶƐŝĚĞƌĞĚďLJĐŽŶƚƌŽůůŝŶŐƚŝŐŚƚĞŶŝŶŐĂŶĚůŽŽƐĞŶŝŶŐŽĨ ƚŚĞďŽůƚĐůŽƐĞƚŽWdϮ;ƐŚŽǁŶŝŶĨŝŐƐ͘ϭͿ͘dŚƌĞĞŵĞĂƐƵƌĞŵĞŶƚƐǁĞƌĞƉĞƌĨŽŵĞĚĨŽƌĞĂĐŚĐŽŶĚŝƚŝŽŶ͘ ĨƚĞƌŝŶƚƌŽĚƵĐŝŶŐƚŚĞĚĂŵĂŐĞƚŚĞďŽůƚǁĂƐŚĂŶĚŝůLJƌĞƚŝŐŚƚĞŶĞĚĨŽƌƚŚĞŝŶŝƚŝĂůĐŽŶĚŝƚŝŽŶ͕ĐĂůůĞĚŚĞƌĞ ďLJƌĞƉĂŝƌĞĚĐŽŶĚŝƚŝŽŶ͘dŚĞƐĞĂĐƋƵŝƐŝƚŝŽŶƐǁĞƌĞĐŽůůĞĐƚĞĚŝŶĚŝĨĨĞƌĞŶƚĚĂLJƐŝŶŽƌĚĞƌƚŽŝŶĐůƵĚĞƐŽŵĞ ǀĂƌŝĂďŝůŝƚLJ ŽĨ ƚŚĞ ĞŶǀŝƌŽŶŵĞŶƚĂů ŝŶƚŽ ƚŚĞ ĚĂƚĂ͘ dĂďůĞ ϭ ƐƵŵŵĂƌŝnjĞƐ ƚŚĞ ƐƚƌƵĐƚƵƌĂů ĐŽŶĚŝƚŝŽŶƐ ŝŶǀĞƐƚŝŐĂƚĞĚ͘ dĂďůĞϭ͘^ƚƌƵĐƚƵƌĂůĐŽŶĚŝƚŝŽŶƐŝŵƵůĂƚĞĚ͘ dĞƐƚ ĞƐĐƌŝƉƚŝŽŶ Ϭ ĂƐĞůŝŶĞʹ,ĞĂůƚŚLJ ϭ ĂŵĂŐĞʹ>ŽŽƐĞŶŝŶŐƚŚĞďŽůƚŶĞĂƌƚŽWdϮ Ϯ ZĞƉĂŝƌĞĚʹ,ĂŶĚŝůLJƚŝŐŚƚĞŶŝŶŐƚŚĞďŽůƚŶĞĂƌƚŽWdϮ dŚĞŽŶůŝŶĞĚĂŵĂŐĞŝŶĚĞdž γ (k ) ǁĂƐƉƌŽƉŽƐĞĚďĂƐĞĚŽŶZ>^ĨŝůƚĞƌƉĂƌĂŵĞƚĞƌĞƐƚŝŵĂƚĞĚ ďLJ ĐŽŶƐŝĚĞƌŝŶŐŽŶůLJƚŚĞŽƵƚƉƵƚƐŝŐŶĂůŝŶWdϮĂŶĚWdϯ͘dŚĞŝŶƉƵƚͲŽƵƚƉƵƚĚĂƚĂŝŶĞĂĐŚWdǁĞƌĞƵƐĞĚ ƚŽ ŝĚĞŶƚŝĨLJ ƚŚĞ ƉĂƌĂŵĞƚĞƌ ǀĞĐƚŽƌ θˆ(k ) ĂƐƐŽĐŝĂƚĞĚ ǁŝƚŚ ĂŶ ZZy ŵŽĚĞů͘ dŚĞ ŽƌĚĞƌƐ ŽĨ ƚŚĞ ƉŽůLJŶŽŵŝĂůƐ ĂƌĞ͕ ŝŶ ŐĞŶĞƌĂů͕ ƵŶŬŶŽǁŶ Ă ƉƌŝŽƌŝ͘ ,ŽǁĞǀĞƌ͕ ƚŚĞƌĞ ĂƌĞ ƐĞǀĞƌĂů ĐƌŝƚĞƌŝĂ ƚŽ ĞƐƚŝŵĂƚĞ ƚŚĞŵ͘,ĞƌĞ͕ŽŶĞĐŽŶƐŝĚĞƌƐƚŚĂƚƚŚĞŽƌĚĞƌŝƐƚŚĞƐĂŵĞŝŶĂůůĐŽŶĚŝƚŝŽŶƐĂŶĚĚƵƌŝŶŐĂůůƚŚĞƚŝŵĞ͘dŚĞ ŬĂŝŬĞ͛Ɛ ŝŶĨŽƌŵĂƚŝŽŶ ƚŚĞŽƌĞƚŝĐ ĐƌŝƚĞƌŝĂ ;/Ϳ ǁĂƐ ƵƐĞĚ ƚŽ ĐŚŽŽƐĞ ƚŚĞ ŽƌĚĞƌ͘ ƉƌĞǀŝŽƵƐ ĂŶĂůLJƐŝƐ ƐŚŽǁĞĚƚŚĂƚ na = 10 ĂŶĚ nb = 4 ǁĂƐƐƵŝƚĂďůĞĨŽƌƚŚĞƉƌĞƐĞŶƚĂƉƉůŝĐĂƚŝŽŶ͘dŚĞZZyŵŽĚĞůƐǁŝƚŚ ƚŚŝƐŽƌĚĞƌǁĞƌĞĐŽŵƉƵƚĞĚĨŽƌĞĂĐŚƐƚƌƵĐƚƵƌĂůĐŽŶĚŝƚŝŽŶ͘&ŝŐƵƌĞϭϬƐŚŽǁƐƚŚĞĐŽŵƉĂƌŝƐŽŶďĞƚǁĞĞŶ
BookID 214574_ChapID 78_Proof# 1 - 23/04/2011
880
ƚŚĞ ŵĞĂƐƵƌĞĚ ĚĂƚĂ ŝŶ WdϮ ĂŶĚ ƚŚĞ ƉƌĞĚŝĐƚĞĚ ƐŝŐŶĂů ďLJ ƚŚĞ ZZy ŝĚĞŶƚŝĨŝĐĂƚŝŽŶ ŝŶ ƚŚĞ ŚĞĂůƚŚ ĐŽŶĚŝƚŝŽŶ ;ďĂƐĞůŝŶĞͿ ƚŽ ŝůůƵƐƚƌĂƚĞ ƚŚĞ ĐĂƉĂďŝůŝƚLJ ĨŽƌ ƉƌĞĚŝĐƚŝŽŶ͘ dŚĞ ĨŽƌŐĞƚƚŝŶŐ ĨĂĐƚŽƌ ĞŵƉůŽLJĞĚ ŝŶ ƚŚŝƐZ>^ĨŝůƚĞƌǁĂƐĐŚŽƐĞŶĂƐ λ = 0.98 ͘ dŚĞĚĂŵĂŐĞͲƐĞŶƐŝƚŝǀĞĨĞĂƚƵƌĞ γ (k ) ŝŶƚŚĞWdϮĂŶĚWdϯĂƌĞƐŚŽǁŶŝŶ&ŝŐ͘Ϯ͕ďLJĂƐƐƵŵŝŶŐ ƚŚĞ ĨŝƌƐƚ ϭϬϬϬϬ ƐĂŵƉůĞƐ͘ /ƚ ŝƐ ǁŽƌƚŚLJ ƚŽ ŶŽƚĞ ƚŚĂƚ ƚŚĞ γ (k ) ǀĂůƵĞ ŝƐ ƋƵĂůŝƚĂƚŝǀĞůLJ ĚŝĨĨĞƌĞŶƚ ŝŶ ƚŚĞ ĚĂŵĂŐĞĚ ĐŽŶĚŝƚŝŽŶ ĂŶĚ ŝƚ ƌĞƚƵƌŶƐ ĐůŽƐĞ ƚŽ ƌĞĨĞƌĞŶĐĞ ĐŽŶĚŝƚŝŽŶ ĂĨƚĞƌ ƚŚĞ ƐƚƌƵĐƚƵƌĞƐ ŝƐ ƌĞƉĂŝƌĞĚ ;ƌĞƚŝŐŚƚĞŶŝŶŐ ƚŚĞ ďŽůƚͿ͘ ŶŽƚŚĞƌ ŝŵƉŽƌƚĂŶƚ ƚŚŝŶŐ ƚŽ ĚĞƚĂĐŚ ŝƐ ƚŚĂƚ ƚŚŝƐ ŵĞƚŚŽĚ ĐĂŶ ďĞ ƵƐĞĨƵů ƚŽ ŚĞůƉƚŚĞƵƐĞƌƚŽĐŚŽŽƐĞǁŚŝĐŚĨƌĞƋƵĞŶĐLJƌĂŶŐĞƚŚĞŝŶĚĞdžŝƐŵŽƌĞĂƉƉƌŽƉƌŝĂƚĞĨŽƌĚĂŵĂŐĞĚĞƚĞĐƚ ĚĞĐŝƐŝŽŶ͘/ŶƚŚĞƉƌĞƐĞŶƚƚĞƐƚƚŚĞ γ (k ) ŝƐŵŽƌĞĚŝƐƚŝŶŐƵŝƐŚĞĚŶĞĂƌƚŚĞƐĂŵƉůĞϰϯϬϬĨŽƌĂůůWdƐ͘dŚŝƐ ƐĂŵƉůĞĐŽƌƌĞƐƉŽŶĚƐƚŽƚŚĞŝŶƐƚĂŶƚŽĨϬ͘ϬϱϭϲƐĞĐŽŶĚƐ͕ǁŚĞƌĞƚŚĞĨƌĞƋƵĞŶĐLJŝƐĐůŽƐĞƚŽƚŚĞƌĞŐŝŽŶŽĨ ϭϭ͘ϱŬ,nj͘ 1
0.6 Undamaged Damaged Repaired
0.8
Undamaged Damaged Repaired
0.4
0.6 0.2
Damage Index
Damage Index
0.4 0.2 0 −0.2
0
−0.2
−0.4
−0.4 −0.6 −0.6 −0.8
−0.8 −1
1000
2000
3000
4000
5000 6000 Samples
;ĂͿ
7000
8000
9000
10000
−1
1000
2000
3000
4000
5000 6000 Samples
7000
8000
9000
10000
;ďͿ
&ŝŐƵƌĂϮ͘KŶůŝŶĞĚĂŵĂŐĞͲƐĞŶƐŝƚŝǀĞŝŶĚĞdž γ (k ) ͖;ĂͿĨŽƌŽƵƚƉƵƚŝŶWdϮ͖;ďͿĨŽƌŽƵƚƉƵƚŝŶWdϯ͘ dŚĞĚĂŵĂŐĞĚĞĐŝƐŝŽŶǁĂƐŽďƚĂŝŶĞĚďLJŚLJƉŽƚŚĞƐŝƐƚĞƐƚƚŚƌŽƵŐŚƚŚĞƚͲƚĞƐƚǁŝƚŚƐŝŐŶŝĨŝĐĂŶĐĞ ůĞǀĞůŽĨϱй͘dĂďůĞϮƐŚŽǁƐƚŚĞƌĞƐƵůƚƐŽĨĚĂŵĂŐĞĚĞĐŝƐŝŽŶĨŽƌĚĂŵĂŐĞƉĂƚƚĞƌŶĨƌŽŵƚĂďůĞϭ͘/ŶĂůů ĐĂƐĞƐƚŚĞŚLJƉŽƚŚĞƐŝƐĨŽƵŶĚǁĞƌĞĐŽƌƌĞĐƚůLJĐŽƌƌĞůĂƚĞĚǁŝƚŚƚŚĞŚĞĂůƚŚLJŽƌĚĂŵĂŐĞĚĐŽŶĚŝƚŝŽŶƐ͘/ƚ ƐŚŽƵůĚ ďĞ ŶŽƚĞĚ ƚŚĂƚ ƚŚĞ ƉͲǀĂůƵĞ ĨŽƌ ďŽƚŚ ĐĂƐĞ ĂƌĞ ǀĞƌLJ ĐůŽƐĞ ƚŽ Ϭ ŝŶ ŚLJƉŽƚŚĞƐŝƐ H 1 ;ĚĂŵĂŐĞ ĐŽŶĚŝƚŝŽŶͿ ĂŶĚ ĨĂƌ ĨƌŽŵ Ϭ͘Ϭϱ ŝŶ ŚLJƉŽƚŚĞƐŝƐ H 0 ;ŚĞĂůƚŚLJ ĐŽŶĚŝƚŝŽŶ Žƌ ƌĞƉĂŝƌĞĚͿ͘ dŚĞ ƉͲǀĂůƵĞ ŝƐ ďŝŐŐĞƌ ƚŽ ƚŚĞ ŝŶĚĞdž ƌĞůĂƚŝǀĞ ƚŽ WdϮ ;ĚĂŵĂŐĞ ůŽĐĂƚŝŽŶͿ͘ dŚĞ ŝŶĚĞdž ĨŽƌ ĚĂŵĂŐĞ ůŽĐĂƚŝŽŶ Ě ĂůƐŽ ŝŶĚŝĐĂƚĞƐƚŚĂƚƚŚĞŐƌĞĂƚĐŚĂŶŐĞŝŶƚŚĞŝŶĚĞdžĐƵƌǀĞ γ (k ) ŽĐĐƵƌƌĞĚŝŶƚŚĞŶĞĂƌŶĞƐƐŽĨWdϮ͘ dĂďůĞϮ͘ZĞƐƵůƚƐŽĨĚĂŵĂŐĞĚĞĐŝƐŝŽŶǁŝƚŚƐŝŐŶŝĨŝĐĂŶƚůĞǀĞůŽĨϱй͘ dĞƐƚ ^ĞŶƐŽƌ ĞĐŝƐŝŽŶ ƉͲǀĂůƵĞ ĂŵĂŐĞůŽĐĂƚŝŽŶŝŶĚĞdžĚ Ϭ;&ĂůƐĞWŽƐŝƚŝǀĞͿ WdϮ Ϭ͘ϲϮϴ EŽĚĂŵĂŐĞ H0
ϭ Ϯ
WdϮ WdϮ
H1 H0
Ϭ Ϭ͘ϴϯϰ
Ϭ͘ϲϬϭϯ EŽĚĂŵĂŐĞ
Ϭ;&ĂůƐĞͲWŽƐŝƚŝǀĞͿ
Wdϯ
H0
Ϭ͘ϴϭϮ
EŽĚĂŵĂŐĞ
ϭ Ϯ
Wdϯ Wdϯ
H1 H0
Ϭ͘ϬϬϴϲ Ϭ͘ϳϳϲ
Ϭ͘Ϭϭϲϲ EŽĚĂŵĂŐĞ
BookID 214574_ChapID 78_Proof# 1 - 23/04/2011
881
ϱ͘&ŝŶĂůZĞŵĂƌŬƐ dŚĞĞdžƉĞƌŝŵĞŶƚĂůĂƉƉƌŽĂĐŚĚŝƐĐƵƐƐĞĚŝŶƚŚŝƐƉĂƉĞƌƉƌŽǀĞĚƚŚĞĂƉƉůŝĐĂďŝůŝƚLJŽĨZ>^ĨŝůƚĞƌƚŽ ĞdžƚƌĂĐƚĂŶŝŶĚĞdžĨŽƌ^,DƉƵƌƉŽƐĞŝŶĐŽŵƉůĞdžƐƚƌƵĐƚƵƌĞƐ͘/ŶƚŚĞ ƉƌĞƐĞŶƚƉĂƉĞƌ͕ƚŚĞŝŶĚĞdžĞƐǁĞƌĞ ƌĞƉƌĞƐĞŶƚĂƚŝǀĞĂŶĚǁĞůůĐŽƌƌĞůĂƚĞĚǁŝƚŚƚŚĞĚĂŵĂŐĞĚŽƌŚĞĂůƚŚLJƐƚĂƚĞŝŶƚŚĞƐƚƌƵĐƚƵƌĞ͘ůƚŚŽƵŐŚ ƚŚĞƌĞƉĂŝƌĞĚĐŽŶĚŝƚŝŽŶǁĂƐŽďƚĂŝŶĞĚďLJŚĂŶĚŝůLJƌĞƚŝŐŚƚĞŶŝŶŐƚŚĞďŽůƚ͕ƚŚĞƌĞƐƵůƚƐǁĞƌĞǀĞƌLJƐŝŵŝůĂƌ ƚŽ ƚŚĞ ŝŶŝƚŝĂů ĐŽŶĚŝƚŝŽŶ͕ ŝ͘Ğ͕͘ ĨŽƌ ƚŚĞ ŚĞĂůƚŚLJ ƐƚƌƵĐƚƵƌĞ͘ dŚĞ ĂƉƉƌŽĂĐŚ ŝƐ ǀĞƌLJ ĂƚƚƌĂĐƚŝǀĞ ĨŽƌ ŝŵƉůĞŵĞŶƚĂƚŝŽŶŝŶĂƌĞĂůŽŶůŝŶĞŵŽŶŝƚŽƌŝŶŐƐLJƐƚĞŵǁŝƚŚŽƵƚŚƵŵĂŶƐƵƉĞƌǀŝƐŝŽŶ͕ƐŝŶĐĞƚŚĞZ>^ĨŝůƚĞƌ ŝƐĞĂƐŝůLJƉƌŽŐƌĂŵŵĞĚŝŶĂĚŝŐŝƚĂůƐŝŐŶĂůƉƌŽĐĞƐƐŝŶŐĂŶĚŝƚŝƐĂďůĞƚŽǁŽƌŬŝŶƚƌĂŶƐŝĞŶƚĂŶĚĚLJŶĂŵŝĐ ĐŽŶĚŝƚŝŽŶ ;ĨůŝŐŚƚͲƚĞƐƚ ĐŽŶĚŝƚŝŽŶͿ͘ ĞƐŝĚĞƐ͕ ƚŚĞ ŝŶĚĞdž γ (k ) ĐĂŶ ďĞ ĞŵƉůŽLJĞĚ ĨŽƌ ƚŚĞ ƐƚƌƵĐƚƵƌĞ ŽƉĞƌĂƚŝŶŐŝŶŶŽŶͲƐƚĂƚŝŽŶĂƌLJĐŽŶĚŝƚŝŽŶ͘ ĐŬŶŽǁůĞĚŐŵĞŶƚƐ ůů ĂƵƚŚŽƌƐ ǁŽƵůĚ ůŝŬĞ ƚŽ ƚŚĂŶŬ ƚŚĞ /ůŚĂ ^ŽůƚĞŝƌĂ ZĞƐĞĂƌĐŚ &ŽƵŶĚĂƚŝŽŶ ;& W/^͕ ƌĂnjŝůͿ ĨŽƌ ƚŚĞ ĨŝŶĂŶĐŝĂůƐƵƉƉŽƌƚ͘WƌŽĨ͘ƌ͘>ŽƉĞƐ:ƌ͘ĂŶĚƚŚĞDƐ'ŽŶƐĂůĞnjƚŚĂŶŬƚŚĞEWƋƚŚĂŶŬƚŚĞEWƋƚŚƌŽƵŐŚ /ŶƐƚŝƚƵƚŽEĂĐŝŽŶĂůĚĞŝġŶĐŝĂĞdĞĐŶŽůŽŐŝĂͲ/Ed͘͘WƌŽĨ͘ƌ͘^ĂŵƵĞůĚĂ^ŝůǀĂŝƐƚŚĂŶŬĨƵůƚŽƚŚĞ/ƚĂŝƉƵ dĞĐŚŶŽůŽŐŝĐĂůWĂƌŬ&ŽƵŶĚĂƚŝŽŶ;&Wd/͕ƌĂnjŝůͿĂŶĚ ^ͬƌĂnjŝůĨŽƌƚŚĞĨŝŶĂŶĐŝĂůƐƵƉƉŽƌƚƉƌŽǀŝĚĞĚ͘ ZĞĨĞƌĞŶĐĞƐ ϭ EĂŝƌ͕ <͘ <͕͘ <ŝƌĞŵŝĚũŝĂŶ͕ ͘ ^͘ ĂŶĚ >Ăǁ͕ <͘ ,͕͘ dŝŵĞ ƐĞƌŝĞƐͲďĂƐĞĚ ĚĂŵĂŐĞ ĚĞƚĞĐƚŝŽŶ ĂŶĚ ůŽĐĂůŝnjĂƚŝŽŶĂůŐŽƌŝƚŚŵƚŽƚŚĞ^ ďĞŶĐŚŵĂƌŬƐƚƌƵĐƚƵƌĞ͘:ŽƵƌŶĂůŽĨ^ŽƵŶĚĂŶĚsŝďƌĂƚŝŽŶƐ͕ǀ͘Ϯϵϭ͕Ɖ͘ ϯϰϵͲϯϲϴ͕ϮϬϬϲ͘ Ϯ ^ŝůǀĂ͕ ^͕͘ ŝĂƐ :ƌ͕͘ D͕͘͘ >ŽƉĞƐ :ƌ͕͘ s͕͘ ĂŶĚ ƌĞŶŶĂŶ͕ D͘ :͘ ^ƚƌƵĐƚƵƌĂů ĚĂŵĂŐĞ ĚĞƚĞĐƚŝŽŶ ďLJ ĨƵnjnjLJ ĐůƵƐƚĞƌŝŶŐ͘DĞĐŚĂŶŝĐĂů^LJƐƚĞŵƐĂŶĚ^ŝŐŶĂůWƌŽĐĞƐƐŝŶŐ͕ǀ͘ϮϮ͕Ɖ͘ϭϲϯϲͲϭϲϰϵ͕ϮϬϬϴĂ͘ ϯ^ŝůǀĂ͕^͕͘ŝĂƐ:ƌ͕͘D͘ĂŶĚ>ŽƉĞƐ:ƌ͕͘s͕͘^ƚƌƵĐƚƵƌĂůŚĞĂůƚŚŵŽŶŝƚŽƌŝŶŐŝŶƐŵĂƌƚƐƚƌƵĐƚƵƌĞƐƚŚƌŽƵŐŚ ƚŝŵĞƐĞƌŝĞƐĂŶĂůLJƐŝƐ͘^ƚƌƵĐƚƵƌĂů,ĞĂůƚŚDŽŶŝƚŽƌŝŶŐ͕ǀ͘ϳ͕Ɖ͘ϮϯϭͲϮϰϰ͕ϮϬϬϴď͘ ϰ 'ŚĂŶĞŵ͕ Z͘ ĂŶĚ ZŽŵĞŽ͕ &͘ ͕͘ ǁĂǀĞůĞƚ ďĂƐĞĚ ĂƉƉƌŽĂĐŚ ĨŽƌ ƚŚĞ ŝĚĞŶƚŝĨŝĐĂƚŝŽŶ ŽĨ ůŝŶĞĂƌ ƚŝŵĞͲ ǀĂƌLJŝŶŐĚLJŶĂŵŝĐĂůƐLJƐƚĞŵƐ͘:ŽƵƌŶĂůŽĨ^ŽƵŶĚĂŶĚsŝďƌĂƚŝŽŶ͕ǀ͘Ϯϯϰ͕Ŷ͘ϰ͕Ɖ͘ϱϱϱʹϱϳϲ͕ϮϬϬϬ͘ ϱ ,ĂŶ͕ z͘ ^ ĂŶĚ >ĞĞ͕ ͘ t͕͘ dŚĞ ĚŝƌĞĐƚŝŽŶĂů tŝŐŶĞƌ ĚŝƐƚƌŝďƵƚŝŽŶ ĂŶĚ ŝƚƐ ĂƉƉůŝĐĂƚŝŽŶ͕ :ŽƵƌŶĂů ŽĨ ^ŽƵŶĚĂŶĚsŝďƌĂƚŝŽŶ͕ǀ͘Ϯϭϲ͕Ɖ͘ϱϴϱͲϲϬϬ͕ϭϵϵϴ͘ ϲ ^ƉŝŶĂ͕ ͕͘ sĂůĞŶƚĞ͕ ͘ ĂŶĚ dŽŵůŝŶƐŽŶ͕ '͘Z͕͘ ŶĞǁ ƉƌŽĐĞĚƵƌĞ ĨŽƌ ĚĞƚĞĐƚŝŶŐ ŶŽŶͲůŝŶĞĂƌŝƚLJ ĨƌŽŵ ƚƌĂŶƐŝĞŶƚĚĂƚĂƵƐŝŶŐƚŚĞ'ĂďŽƌƚƌĂŶƐĨŽƌŵ͘EŽŶůŝŶĞĂƌLJŶĂŵŝĐƐ͕ǀ͘ϭϭ͕Ɖ͘ϮϯϱʹϮϱϰ͕ϭϵϵϲ͘ ϳ ^ƚĞǁĂƌƚ͕ '͘ t͕͘ Ŷ ƵƉĚĂƚŝŶŐ ĂůŐŽƌŝƚŚŵ ĨŽƌ ƐƵďƐƉĂĐĞ ƚƌĂĐŬŝŶŐ͕ / dƌĂŶƐĂĐƚŝŽŶƐ ŽŶ ^ŝŐŶĂů WƌŽĐĞƐƐŝŶŐ͕ǀ͘ϰϬ͕͘Ŷ͘ϲ͕ϭϵϵϮ͘ ϴ,ĂLJŬŝŶ͕^͘ĚĂƉƚŝǀĞ&ŝůƚĞƌdŚĞŽƌLJ͕ϮŶĚĞĚŝƚŝŽŶ͕h^͕WƌĞŶƚŝĐĞͲ,Ăůů͕ϭϵϵϭ͘ ϵ>ũƵŶŐ͕>͘^LJƐƚĞŵ/ĚĞŶƚŝĨŝĐĂƚŝŽŶ͗dŚĞŽƌLJĨŽƌƚŚĞhƐĞƌ͘WƌĞŶƚŝŶĐĞ,Ăůů͕Ϯ͘Ǒ ĚŝƚŝŽŶ͕ϭϵϵϴ͘ ϭϬ ŐƵŝƌƌĞ͕ >͘ ͕͘ /ŶƚƌŽĚƵĕĆŽ ă /ĚĞŶƚŝĨŝĐĂĕĆŽ ĚĞ ^ŝƐƚĞŵĂƐ ʹ dĠĐŶŝĐĂƐ >ŝŶĞĂƌĞƐ Ğ EĆŽͲ>ŝŶĞĂƌĞƐ ƉůŝĐĂĚĂƐĂ^ŝƐƚĞŵĂƐZĞĂŝƐ͘ ĚŝƚŽƌĂh&D'͕Ϯ͘Ǒ ĚŝĕĆŽ͕ϮϬϬϰ͘
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ϭϭ/ĨĞĂĐŚŽƌ͕ ͘͘ĂŶĚ:ĞƌǀŝĞǁ͕͘t͘ŝŐŝƚĂů^ŝŐŶĂůWƌŽĐĞƐƐŝŶŐ͕WƌĂĐƚŝĐĂůƉƉƌŽĂĐŚ͕WƌĞŶƚŝĐĞ,Ăůů͕ h^͕ϮϬϬϮ͘ ϭϮ ,ĂLJŬŝŶ͕^͘ĚĂƉƚŝǀĞ&ŝůƚĞƌdŚĞŽƌLJ͕ϮŶĚĞĚŝƚŝŽŶ͕h^͕WƌĞŶƚŝĐĞͲ,Ăůů͕ϭϵϵϭ͘ ϭϯ,ŽŐŐ͕Z͘s͘ĂŶĚ>ĞĚŽůƚĞƌ͕:͕͘ ŶŐŝŶĞĞƌŝŶŐ^ƚĂƚŝƐƚŝĐƐ͕DĂĐŵŝůůĂŶWƵďůŝƐŚŝŶŐŽŵƉĂŶLJ͕EĞǁzŽƌŬ͕ h^͕ϭϵϴϳ͘
BookID 214574_ChapID 79_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Prediction of Full Field Dynamic Stress/Strain from Limited Sets of Measured Data
Pawan Pingle, Peter Avitabile Structural Dynamics and Acoustic Systems Laboratory University of Massachusetts Lowell One University Avenue Lowell, Massachusetts 01854
ABSTRACT Transient events may cause degradation of a structure’s useful life. Often these events have an unknown effect on the structure. Some examples are helicopter hard landing, strong wind gusts on large turbine blades, and other similar occasional events. In many cases, only a handful of sensors are available for the evaluation of these rare, short-term events. This limited data does not always provide the detail necessary in order to determine the effects of the events that occur; this is very important especially in terms of the overall fatigue usage for the system. While a finite element model may be available, use of this limited data does not lend itself to easily integrate into the identification of the dynamic stress and dynamic strain. Often times, estimates of the forcing function can be approximated and used to identify the response. But these techniques are very approximate. A newly developed real time operating data expansion technique has been successfully used for the identification of more detailed information from limited sets of data. This expansion has mainly been used to augment the limited data acquired and allows for better overall visualization of these transient types of events. This technique has mainly been used for general response characteristics for structures. In this work, expansion techniques are developed in conjunction with the finite element mass and stiffness matrices to provide information at all of the finite element degrees of freedom from these limited sets of measured data points. With this approach, dynamic stress and dynamic strain can be identified from measured transient events thereby enabling estimation of fatigue accumulation or usage. Several cases are described to illustrate the overall process and results that are obtained from the expansion technique; these are compared to full reference solution results to verify the accuracy of the technique. NOMENCLATURE Symbols:
^X n ` ^X a `
>M a @ >M n @ >K a @ >K n @ >U a @
full set displacement vector reduced set displacement vector reduced mass matrix expanded mass matrix reduced stiffness matrix expanded stiffness matrix reduced shape matrix
>U a @g generalized inverse of reduced shape matrix >U n @ expanded shape matrix >T @ transformation matrix. >TU @ SEREP transformation matrix. >REFn @ reference data at all degrees of freedom (dofs) >RTO a @ real time operating data at measured dofs >ERTO n @ expanded real time operating data at all dofs
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_79, © The Society for Experimental Mechanics, Inc. 2011
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INTRODUCTION Fatigue is one of the most significant failure mechanisms. A structure has to sustain various dynamic loadings during its operation which decrease its expected fatigue life. Assumed dynamic loadings on a structure are mostly accounted for in the design process. The problem arises when unknown and sudden dynamic loadings affect the structure’s integrity and fatigue life. Such an unexpected loading is very difficult, if not impossible to account for in the traditional design process of the structure. Examples of events that cause such fatigue loading would be helicopter hard landing, strong unexpected wind gust on a wind turbine blade, barge-bridge strike, to name a few. As an example, the stresses that the body of a helicopter would undergo during hard-landing are difficult for a finite element model to accurately predict because the dynamic forces the structure experiences cannot be measured or determined accurately. Similarly a sudden gust of wind on the blade of a wind turbine or collision of a barge to a bridge can cause severe damage to the structure consequently decreasing the fatigue life which was unaccounted for in the design process. There are primarily two approaches traditionally used to evaluate such transient stresses in structures. First approach would be to set up a variety of transducers at certain locations and collect the transient response information. Such a process although predicts the real-time response of the structure under severe dynamic loadings, but does so at only limited locations. Massive structures such as bridges, ships, wind-turbines would need thousands of sensors to get the required data, which is highly improbable and unfeasible. Second approach would be to predict the transient forces that the structure experiences and then using the analytical model find the dynamic stresses the structure experiences. The problem herein lies with the force estimation process which is highly sensitive to the number of chosen degrees of freedom and their distribution on the structure. Apart from this, the other issues such as approximations while developing a finite element model invariably induce errors in estimation of dynamic stresses on the structure. The approach used in this paper is based on real time operating data expansion theory recently studied by Chipman [2] to expand the limited sensor location information at places equivalent to the finite element nodes. The limited set of sensors would provide displacement data and the expansion algorithm would expand the limited data set to a full-field displacement solution thereby completely eliminating the force estimation step required in the previous approach. The proposed approach is explained schematically in the next section highlighting the advantages of this approach. The sections after briefly discuss the expansion theory used in the approach and also several test cases to validate the approach. The displacement solutions have been shown in this paper; however the stress and strain solutions can consequently be obtained easily from the displacement solutions using constitutive relationships and hence have not been presented. THEORY The concept proposed in this work relies on the general development of a finite element model, the assembly of the system matrices, the application of loads and boundary conditions followed by the solution of the set of equations and the recovery of the stress-strain solution. For many situations, the actual loading is not known and the actual boundary conditions are not well understood. This procedure is shown schematically in Figure 1 for a wind turbine blade where significant loadings may occur while the system is in normal operation (i.e., rotating). For the approach considered in this work, the difference is that the actual application of the loads and boundary conditions and the solution of the system set of equations are not specifically performed. Rather, the sparse set of displacements, measured from an actual operating event, is used with a set of orthogonal expansion functions to obtain the full field displacement solution. This displacement solution is then used with the normal recovery of the stress-strain solution in the finite element modeling process. This procedure is shown schematically in Figure 2 where the limited set of measurement degrees of freedom are used with expansion processes to obtain the full field displacement for the system. With this approach, the finite element modeling solution process is intercepted and replaced with the expansion of the limited set of measured degrees of freedom. The actual expansion process has been presented in [2], [3], and [4]. The basic theoretical approach is summarized below and utilizes concepts from model reduction and model expansion as the underlying methodology for the expansion approach used for this work.
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DISCRETIZATION
ASSEMBLY
BC & LOADS
SOLVE FOR DISPLACEMENT
SOLVE FOR DYNAMIC STRESS-STRAIN
FE MODEL
INCORRECT VIRTUAL MODEL PREDICTION
UNKNOWN OPERATING LOADS
ADS D LO E WIN L B A T EDIC UNPR
INTERIOR MESH
DYNAMIC STRESS-STRAIN NOT ACCURATE
UNCERTAIN BOUNDARY CONDITIONS
TRADITIONAL MODELS DO NOT INCORPORATE REAL LOADING AND TRUE IN-OPERATION DEFORMATIONS
Figure 1:
DISCRETIZATION
Schematic Showing Normal Finite Element Model Development
BC & LOADS
ASSEMBLY
MEASURE DISLACEMENT IN-SITU WHILE ROTATING WITH DIC
2
IMAGING SYSTEM
DEVELOP EXPANSION
>TU @ >E n @>E a @
g
t1
SOLVE FOR DISPLACEMENT
SOLVE FOR DYNAMIC STRESS-STRAIN
t2
FULL NODAL DISP. SOLUTION SURFACE POINT FROM REAL-TIME OPERATING DATA EXPANSION WHILE PREDICTED INTERIOR POINT ROTATING
EXPAND OPERATING DATA
>RTO n @ >T @>RTO a @
ARBITRARY TIME 1
t1 FULLY PREDICTED INTERIOR AND SURFACE STRESS-STRAIN ARBITRARY TIME 2
t2
OPERATING EXPANSION APPROACH ACCOUNTS FOR OPERATING DEFORMATIONS AND ACTUAL LOADS
Figure 2:
Schematic Showing Alternation Expansion/Solution Sequence
MODEL REDUCTION Model reduction is necessary in order to develop expansion approaches for modal data for the unmeasured translational DOF as well as for rotational DOF. For this work, the expansion is needed for augmenting the limited set real-time operating data to provide a full field displacement solution. The reduction techniques are the basis of the expansion discussed in this work. These techniques have been presented in earlier work cited in the references; only summarizing equations are presented below. Several model reduction methods have commonly been used for expansion of measured data. Four common methods are Guyan [5], Dynamic Condensation [6], SEREP [7], and a Hybrid method [8]. In these methods, the relationship between the full set of degrees of freedom and a reduced set of degrees of freedom can be written as
{X n }=[T]{Xa }
(1)
All of these methods require the formation of a transformation matrix that can project the full mass and stiffness matrices to a smaller size. The reduced matrices can be formulated as
>M a @ >T @T >M n @>T@ >K a @ >T@T >K n @>T@ For the specific work in this paper, only the SEREP method has been used for the expansion of mode shapes.
(2) (3)
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The System Equivalent Reduction Expansion Process (SEREP) produces reduced matrices for mass and stiffness that yield the exact frequencies and mode shapes as those obtained from the eigensolution of the full size matrix. The SEREP transformation is formed as
>TU @ >U n @>U a @g
(4)
The SEREP transformation is developed with analytical mode shapes for the structure (but can also be evaluated using measured modal vectors as well as done in Chipman’s work [2]). Equation 1 is used for expansion of real-time operating data and is written as
>ERTO n @ >T@>RTO a @
(5)
CORRELATION TOOLS The correlation tools utilized to compare the results of the expanded RTO [ERTOn] and "reference time solution" [REFn] are briefly discussed here in order to help clarify the differences between the techniques. Correlation tools such as, the Modal Assurance Criterion (MAC) [9] and the Time Response Assurance Criterion (TRAC) [10] will be used to verify the accuracy of the expanded RTO in each case. These functions are summarized here with details found in [2]. The MAC as used for this work will identify the correlation of the expanded real time operating displacement solution obtained with the reference solution. The MAC can be computed at each time step t to compare the transient displacement solution with time. The MAC is written as
>^REF ` ^ERTO `@ >^REF ` ^REF `@>^ERTO ` ^ERTO `@ 2
T
MAC _ RTO
n
n
T
(6)
T
n
n
n
n
Similar to the MAC, the TRAC is a tool used to determine the degree of correlation between two time traces. For the cases presented here, the TRAC is the correlation for one DOF over all time for the expanded time data [ERTOn] compared to the actual measured data [REFn]. The TRAC is written as
>^REF (t)` ^ERTO (t)`@ >^REF (t)` ^REF (t)`@>^ERTO (t)` ^ERTO 2
T
TRAC _ RTO
n
n
T
n
T
n
n
@
n ( t )`
(7)
The values produced by both the MAC and TRAC will range from 0 to 1; values approaching 1 indicate good correlation. Also used for comparing the [ERTOn] with reference time solution is an absolute difference between the two solutions at each time step. The magnitude of the difference is a measure of how much the [ERTOn] deviates from the reference time solution. MODEL DESCRIPTION
To demonstrate the expansion technique to be used for determining displacement for dynamic stress and strain from limited sensor locations, an analytical study was performed on a model that resembles a blade of a wind turbine. The box-beam model used for analysis is shown in Figure 3a. The dimensions of the model are: 1. Length (l) = 60 inches 2. Breadth (b) = 12 inches 3. Width (w) = 6 inches 4. Thickness of top and bottom plates (t1) = 0.5 inches 5. Thickness of internal ribs (t2) = 0.25 inches 6. Material - Aluminum
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SENSOR LOCATION
Mode 2: 45 Hz
b l
Mode 1: 12.7 Hz
FORCE LOCATION TOP PLATE
t2
Mode 3: 71.2 Hz
INTERNAL RIBS
w t1
(a)
Mode 4: 90.4 Hz
(b)
BOTTOM PLATE
FIXED END
Figure 3: (a) Box-beam model used for analytical study of the expansion algorithm along with typical set of sensor locations; (b) First four mode shapes of the structure. The finite element model of the box-shaped structure contains a total of 434 nodes. The natural frequencies and mode shapes of the beam are shown in Figure 3b. The frequencies typically are well separated. The model has an assortment of modes containing transverse and lateral bending and torsion modes up to the first four modes which will be excited using a time pulse. Modal damping typically seen in these types of structures was added to the model. The excitation used was a time pulse applied vertically downwards, 10 inches from the bottom right corner of the box-beam. The input time pulse is shown in Figure 4a. The time pulse is a combination of two triangular functions which when seen in frequency domain show a very uniform excitation over the frequency range of interest as shown in Figure 4b. The time pulse primarily excites the first four modes of the structure as seen in FFT of the output response as shown in Figure 4c. -80
Input Pulse
1
-2 -3 0
0.02
FFT of Input Pulse
0 Magnitude(dB)
0.005 0.01 0.015 Time (sec)
-20
DISPLACEMENT MAGNITUDE (dB)
Amplitude
-1
-4
Mode 1: 12.7 Hz
-90
0
Mode 2: 45 Hz
-100 -110 Mode 4: 90.4 Hz
-120 Mode 3: 71.2 Hz
-130 -140
FFT of Output Response at one sensor location
-150 -40
-160
-60 0
100 200 Frequency(Hz)
300
-170 0
100
200 300 FREQUENCY (Hz)
Figure 4: Details of the excitation force and response of the structure.
400
500
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CASES STUDIED
The comparison between reference solution and expansion solution is made in this section. Several different analytical cases have been studied to evaluate the effectiveness of the proposed expansion algorithm. The cases studied are based on some important factors that affect the implementation of the technique. Foremost among the prominent factors affecting the expansion solution is the number of modes used in the expansion algorithm, specifically, the number of columns in the Un and Ua matrices in Equation (4). Theoretically, there is no upper limit for the number of modes to be used for expansion, but from a practical standpoint only a limited number of modes can be used for expansion. Case 1 deals with the study of number of modes used for expansion. Noise is another important issue that could affect the successful implementation of the expansion algorithm. Experimental noise could be due to a variety of reasons. Different noise levels have been studied in Case 2 to gage the performance of the expansion algorithm in the presence of noise. The sensor locations, from where the real-time data is obtained and expanded using the expansion algorithm, invariably plays an important role in accuracy of the solution. Case 3 discusses the effect of a variety of sensor locations on the solution obtained through expansion algorithm. CASE 1 To obtain accurate dynamic stress-strain data from the proposed expansion technique, the number of modes used for expansion plays an important role. Figure 5 shows the displacement response at the interior of the beam (on the left half of the page) where no sensor was placed, obtained through the expansion technique overlaid on the response predicted through the reference solution. Figure 5 (right half) shows response of a location on the top plate (where no sensor was placed) obtained through expansion technique compared with the reference solution. The force pulse primarily excites the first four modes of the system. Only 14 y (vertical) direction points of a total of 434 finite element model nodes were used as sensor locations measuring vibration response as shown in Figure 5a (left and right half of page). The expansion technique predicted the response at one of the candidate interior rib or top plate locations to be the same as that predicted by the finite element solution using the first six modes of the system for expansion as shown in Figure 5d (left and right half of page). Because the first four modes of the structure are primarily excited by the force pulse, using fewer modes has a tendency to degrade the resulting expansion results; this is shown in Figure 5b (left and right side) using only one mode and in Figure 5c (left and right side) using only three modes. This clearly shows that as long as the number of modes used for the expansion is greater than the actual number of modes that participate in the actual response, then the expansion produces accurate results; using fewer modes will not produce accurate results as also seen in previous studies [2]. Figure 6a shows the MAC between the reference solution and the solution obtained through expansion process at each time step using 3, 4 and 6 modes for expansion. The MAC is plotted from 0.9 to 1.0 to show how accurate the results are when using 3, 4 or 6 modes. There are at least 4 modes excited by the pulse so that using less than 4 modes is expected to produce unacceptable results. (Recall that MAC compares the vector so that all DOF are evaluated.) Using more than the number of modes excited by the pulse produces very good results overall. Figure 6b and 6c also show that when six modes were used for expansion, then the absolute error between the expanded and the finite element solution decreased to a large extent. Clearly, the number of modes used for expansion must be at least equal to (or greater than) the number of modes excited.
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Figure 5: (a) Box-beam model shown along with typical set of sensor locations and response location in the rib (left half of the page) and on the top plate (right half of the page); Overlay of the web-response predicted by FEA and expansion algorithm (b) using only one mode for expansion, (c) using first three modes for expansion, (d) using first six modes for expansion.
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Figure 6: (a) MAC between response vectors of FEA and expansion algorithm solutions with different number of modes used for expansion, (b) absolute error between FEA and expanded solution with different number of modes used for expansion, (c) Typical maximum absolute error at each time step for different number of modes used for expansion.
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CASE 2 Experimental noise is a very common phenomenon which corrupts the data. While collecting experimental data, the noise component is always inevitable. To study the expansion technique realistically, some random noise is added to the reduced mode shape using
^U a _ noise ` ^U a ` r^U a _ max `
(8)
where, r is random set of numbers ranging from -1% to +1% of the maximum shape. Figure 7 shows the noise induced in the model to simulate experimental conditions.
(a)
(b)
Figure 7: Typical noise induced (a) on reduced shape (b) individually at each sensor location. Case 1 showed the expansion algorithm to be an accurate tool when an appropriate number of modes are used for expansion. However, the expansion algorithm is also affected by noise as shown in Figure 8. A varying percentage of noise was added on the model used in Case 1 to simulate the experimental conditions of real-time data acquisition; noise levels ranging from typical levels of noise (1% and 5%) and excessive levels of noise (15% and 20%) were studied. As expected, using 6 modes produced good results when considering the lower levels of noise (1% and 5%); while the results degraded at excessive levels of noise (15% and 20%), the results were reasonable even at these high noise levels. Again the MAC is plotted from 0.9 to 1.0 to show how accurate the results are. An assortment of different cases were studied (not shown here) which clearly showed that the lack of good MAC values at the early part of the transient response was due to absence of use of higher order modes in the expansion algorithm. The higher order modes mattered for the beginning response because they were only slightly excited by the input time pulse as shown in Figure 4c. Addition of higher frequency modes will improve the solution but are omitted here to show their effect. The TRAC values were good for those cases where a sufficient number of modes were used for the expansion as shown in Table 1. When four or more than four modes were used for expansion, the TRAC values were greater than 0.8 for all the levels of noise cases.
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Figure 8: A plot of MAC values between response vectors of FEA and expansion algorithm solutions for all time steps using six modes for expansion, (a) with 1% noise in modes used for expansion, (b) with 5% noise in modes used for expansion, (c) with 15% of noise in modes used for expansion, (d) with 20% noise in modes used for expansion.
Table 1: TRAC values shown when different number of modes were used for expansion for different noise levels # of Modes used for expansion 1 % Noise 5 % Noise 15 % Noise 20 % Noise 1 mode
0.3449
0.3471
0.3459
0.3519
2 modes
0.7229
0.7208
0.7126
0.7068
3 modes
0.7266
0.7244
0.7195
0.7097
4 modes
0.8758
0.8748
0.8665
0.8618
5 modes
0.872
0.8729
0.8624
0.8567
6 modes
0.8728
0.8673
0.8603
0.8603
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CASE 3 Selection of sensor location is of primary importance because measurements at these locations form the basis of the expansion technique. The purpose of this case is to study the effect of use of different number and location of sensors on the expansion technique. Four different sensor set-ups are studied as shown in Figure 9. The web and plate response at a typical response location for these four different types of sensor set-ups as shown in Figure 10 and 11, clearly indicate that the greater the number of sensors, the more accurate the response predicted by the expansion technique. The expansion technique predicted the response accurately (with or without noise) when using only 14 sensors out of a total of 434 nodes. After reducing the number of sensors to 7, the expansion technique did not predict the response as well. The set of 7 DOFs (sensors) does not provide a linearly independent set of vectors and hence is not sufficient to properly describe the modes excited. However, even after using only 7 sensors, with 10% noise, the MAC and TRAC values did not drop significantly when the number of modes used for expansion (six) was greater than the number of modes excited; MAC and TRAC not shown here for brevity.
SENSOR LOCATION
(a)
(b) 42 sensors
(c)
24 sensors
(d) 14 sensors
7 sensors
Figure 9: Different sensor locations used to determine the response of the structure using expansion algorithm at all 434 FEA node locations, (a) 42 sensors, (b) 24 sensors, (c) 14 sensors, (d) 7 sensors.
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x 10
-4
42 Sensors
3
3
2
2
1 0 -1
-2 -3 -4 0
0.02
x 10
5
0.04
0.06
0.08
0.1 0.12 Time (s)
0.14
0.16
0.18
-5
0.2
(b) 0
-4
6 FEA Expansion
14 Sensors
4
Displacement (in)
1 0 -1 -2
0.06
0.08
0.1 0.12 Time (s)
0.14
0.16
0.18
0.2
-4
FEA Expansion
7 Sensors
0 -2 -4 -6
-3
-8
-4
-10
-5
x 10
0.04
2
2
(c)
0.02
4
3
Displacement (in)
0 -1
-3 -4
FEA Expansion
24 Sensors
1
-2
(a)
-4
4
Displacement (in)
Displacement (in)
4
-5
x 10
5 FEA Expansion
0
0.02
0.04
0.06
0.08
0.1 0.12 Time (s)
0.14
0.16
0.18
0.2
-12
(d)
0
0.02
0.04
0.06
0.08
0.1 0.12 Time (s)
0.14
0.16
0.18
0.2
Figure 10: Web-response as predicted by FEA and expansion algorithm solutions using first six modes for expansion, (a) with 42 sensors, (b) with 24 sensors, (c) with 14 sensors, and (d) with 7 sensors.
x 10
5
-4
5
42 Sensors
4
3
2
2
1
1
0
0
-1
-1
-2
-2
-3
-3
-4
(a)
-4
24 Sensors
4
3
-5
x 10
-4 0
5
x 10
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
-5
(b) 0
10 % Noise
-4
5
4
x 10
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.14
0.16
0.18
0.2
-4
7 Sensors
14 Sensors
3 2 0 1 0 -1 -5
-2 -3 -4 -5
(c)
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
-10 0
(d)
0.02
0.04
0.06
0.08
0.1
0.12
Figure 11: Plate-response as predicted by FEA and expansion algorithm solutions using first six modes for expansion, (a) with 42 sensors, (b) with 24 sensors, (c) with 14 sensors, and (d) with 7 sensors.
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CONCLUSION
Using a limited set of measured responses, full field displacement can be obtained using an expansion procedure. The measured response can then be used in the finite element recovery process to obtain full field dynamic stress-strain due to any measured loading condition such as helicopter hard landing, strong wind gusts on turbine blades and bridge-barge strikes to name a few scenarios. Several cases were investigated to show the accuracy and robustness of the expansion procedure. The expansion procedure will produce accurate results provided that there are a sufficient set of orthogonal mode shapes to span the space for the solution; this requires that a sufficient number of modes that participate in the response be included in the expansion process. In terms of the limited set of measurement points, there must be sufficient number of measurement locations available for the least squares minimization process; a linearly independent set of vectors for the required number of modes must be available from the measurement set. Noise presents variation in the results but from the cases studied, significant noise did not seriously contaminate the results obtained. Future work will study and explore this technique in more depth but these initial results show that this is a very good approach for determining the full field displacement solution along with the full field dynamic stress-strain resulting from occasional loading events using limited sets of measurement data. REFERENCES
[1]
Avitabile,P., Piergentili,F., Lown,K., “Identifying Dynamic Loadings from Measured Response”, Sound & Vibration, August 1999 [2] Chipman,C., “Expansion of Real Time Operating Data”, Master’s Thesis, University of Massachusetts Lowell, May 2009 [3] C.Chipman, P.Avitabile, “Expansion of Real Time Operating Data for Improved Visualization”, Proceedings of the Twenty-Sixth International Modal Analysis Conference, Orlando, FL, Feb 2008 [4] C.Chipman, P.Avitabile, “Expansion of Transient Operating Data”, Proceedings of the Twenty-Seventh International Modal Analysis Conference, Orlando, FL, Feb 2009 [5] Guyan, R.J., "Reduction of Stiffness and Mass Matrices", AIAA Journal, Vol. 3, No 2, 1965 [6] Paz, M, "Dynamic Condensation", AIAA Journal, Vol22, No 5, May 1984 [7] O'Callahan,J.C., Avitabile,P., Riemer,R., "System Equivalent Reduction Expansion Process", Seventh International Modal Analysis Conference, Las Vegas, Nevada, February 1989 [8] Kammer, DC, "A Hybrid Approach to Test Analysis Model Development for Large Space Structures", Journal of Vibration and Acoustics, Vol 113, July 1991 [9] Allemang,R.J. and Brown,D.L., "A Correlation Coefficient for Modal Vector Analysis", First International Modal Analysis Conference, Orlando, Florida, November 1982, pp. 110-116 [10] Van Zandt, T., “Development of Efficient Reduced Models for Multi-Body Dynamics Simulations of Helicopter Wing Missile Configurations,” Master’s Thesis, University of Massachusetts Lowell, April 2006. [11] MATLAB R2008b – The MathWorks, Natick, Massachusetts [12] Femap – Finite Element Modeling And Postprocessing, Version 9.3.1, Copyright © 2007 UGS Corp.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Origins and History of Shock & Vibration (S&V) Requirements (To be presented by James E. Howell, III P.E. of NSWCCD)
A presentation titled "Origins and History of Shock & Vibration (S&V) Requirements" will be given to provide a brief overview of the origins and history of the Navy's underwater explosion (UNDEX) shock and shipboard vibration testing requirements found in test specifications and standards. These requirements originated during World War II (WW II) and eventually evolved into MIL-S-901 for UNDEX shock testing and MIL-STD-167 for shipboard vibration testing. The presentation begins with a brief introduction and background followed by a brief review of definitions for mechanical shock and vibration including an illustration of these excitation sources. The presentation will then expound on the origins of shock and vibration requirements that go back to the early years of WWII based on experiences by the United Kingdom, United States of America (US) and Japan. More detailed information will be presented on documented US experiences throughout WWII and the results from those experiences. Following the origins discussion, a historical review of S&V test specifications and standards as they evolved throughout the years will be discussed starting from WWII through the present day. Some discussion will be on the development of shock test methods such as the Lightweight Shock Machine (LWSM), the Medium Weight Shock Machine (MWSM) and the various heavyweight test methods including the Floating Shock Platform (FSP), Intermediate Floating Shock Platform (IFSP) and the Large Floating Shock Platform (LFSP). During the vibration discussion, a review of frequencies and displacement amplitudes observed from various ship classes will be presented. A brief overview of the MIL-STD-167 vibration test method will also be given. Rationale for various aspects of the S&V test specifications and standards will be
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_80, © The Society for Experimental Mechanics, Inc. 2011
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discussed at various points throughout the presentation. Finally, the presentation will conclude by summarizing the main points of the presentation.
BookID 214574_ChapID 81_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Spinning Projectile Wing Deployment Structural Dynamic Modeling and Analysis
J. Edward Alexander BAE Systems, US Combat Systems 4800 East River Road Minneapolis, MN 55421 [email protected] Abstract Two structural dynamic models were developed to evaluate wing deployment on an in-flight, spinning, multi-wing projectile. A single degree-of-freedom model for each wing's opening angular displacement, velocity and acceleration relative to the projectile body was formulated. Projectile variable spin rate due to the projectile's changing polar moment of inertial as the wings' open was included. The model of the projectile includes provisions for a compression springs under each wing to facilitate opening into the air stream immediately after wing release. Dynamic forces included in the wing opening model are those arising from the spring, aerodynamic drag, inertial forces, and centrifugal forces from both the wings' opening angular velocity and the angular velocity due to projectile spin. The central-difference numerical integration method was used to solve the wing opening nonlinear dynamic equation of motion. A second model was also formulated to asses the wing's impact on a stop surface. This model was used to determine maximum reaction forces for cases where the wing's opening angle was limited by a stop surface on the projectile body. Two notional projectile/wing concepts were studied to demonstrate the utility of the dynamic models. Maximum dynamic forces and corresponding stresses were determined for each concept and compared to allowable material stress limits to evaluate feasibility of the concepts. I. Introduction Gun or mortar launched guided projectiles typically have fins or wings that are deployed after the body is in flight and spinning. Depending on the spin rate, the forces on the wings due to centrifugal acceleration can be high. Additionally, when the wing opening angular momentum becomes large, the forces needed to stop the wing rotation can be even higher if the stop surface is relatively stiff. Figure 1 shows a typical example of such a flight vehicle where the wings are deployed during flight while the projectile body is spinning. In this case the wings are latched near the rear of the body and pivot about a pinned connection near the ogive. As such, the wings open forward. Figure 2 is a solid model of a notional wing which illustrates the cylindrical shape of the wing needed to fit closely around the body for launch from a cylindrical tube. The two lugs at the top of the wing each have a small hole to receive a pivot pin that was secured to the body. To estimate the magnitude of these forces on each wing, relatively simple single degree-of-freedom (DOF) wing opening and wing stop models were developed. The wing opening model included a pre-compressed small spring under the wing to facilitate rapid opening of the wing immediately after it is released. This transient model begins (time=0) when the rear of the wing is released from the body while the body is spinning with angular velocity of Ȧ(0). The body spin velocity is not constant throughout the transient. As with wings open, the polar moment of inertial of the entire round increases and grows to a maximum when ș(t) = 90 degrees. The variable polar moment of inertial causes the spin velocity of the projectile to change due to the angular equivalent of the impulse and momentum principle. This variable spin rate was accounted for during the wing opening transient. A central difference explicit numerical integration scheme was used to march forward in time through the transient.
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_81, © The Society for Experimental Mechanics, Inc. 2011
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Figure 1 - Typical Projectile with Wings Open
Figure 2 - Solid Model of a Typical Wing Design
During certain combinations of projectile body spin rate, flight velocity and aerodynamic drag forces, the wing will continue to open until it contacts a stop surface on the body. At this point, a second linear one DOF model was used to estimate the maximum reaction forces on the wing as a result of the contact with the stop surface. When the wing contacts the stop surface, the wing angular velocity at that instance was used as an initial condition for the for the wing stop model. The reaction forces at the pinned joint and the stop surface were estimated by this model. Equivalent distributed, shear and moment wing loads were determined using these reaction forces. Stresses in the wing were determined from these loads. If necessary, a finite element model could be used for more detailed stresses results using the distributed inertial load applied to the wing. II. Models Overview A schematic of the wing and body is shown in Figure 3. A corresponding single DOF model was used to
determine the wing angle, angular velocity, and angular acceleration as a function of time; T (t), T t and T t , respectively. The resulting dynamic equation of motion was solved numerically using the central difference numerical integration method.
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Figure 3 - Schematic of One DOF Wing Opening Model The equation of motion was developed to determine the forces acting on a tapered wing as it opens due to the spin rate of the body Ȧ. The total wing forces were calculated by determining the forces acting on a differential element volume of mass, dm, and then integrating over the length from r = 0 to r = L. The incremental mass is given by dm UA(r )dr , where the cross sectional area of the wing decreases linearly with r according to the relationship,
A(r )
Ao mr
(1)
A linear pre-compressed spring between the body and the wing was included in the model. The function of the spring is to provide an initial opening moment when the wing is released. The projectile is spinning with an initial value of Ȧ (0) when the wing is released. The spin rate decreases as ș increases due to an increasing polar moment of inertia of the projectile as the wings open. The model was used to predict the tensile force in the wing during opening.
T t T max , another single DOF "wing stop model" was The wing hub has a physical stop surface if the wing angle T (t) exceeds T max . If
In cases where the wings contact the wing stop when formulated to study this event.
the wing contacts the stop surface, reaction forces at the pin and the wing stop result. The wing stop model was used to predict the maximum reaction forces, which were in turn used to determine the maximum wing bending stress. IIA. Wing Opening Model A single DOF rigid body model of the wing opening nonlinear dynamics was developed to determine T , T , and T as a function of time after the wings are released. As shown in Figure 4, multiple forces act on the wing during deployment. The forces acting on a differential volume of mass, dm, were developed and the total forces were determined by integration from r = 0 to r = L. These forces, shown on dm in Figure 4, are discussed in the next section. A spring force was included for a linear spring between the body and the wing that is used to “push” the wing into the airflow. The equation of motion for T t was developed using the forces in Figure 4.
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External forces and acceleration-dependent inertial forces were required to keep the wing in dynamic equilibrium. A resulting second order nonlinear ordinary differential equation was solved using the central-difference numerical integration method, which was coded into a FORTRAN program. The total tensile force in the wing was determined from the model.
Figure 4- Wing Opening Forces Acting on Differential Mass, dm
Forces on the Wing Figure 4 shows the forces acting on the differential volume of mass dm. Inertial forces are due to the angular velocity of the body, Ȧ, and the angular velocity and acceleration of the wing, T and T , respectively. External applied forces were the drag force due to aerodynamic flow, Fd and a spring force, FS . A tensile reaction force at the pivot was required to keep the wing attached to the hub. These forces are discussed individually below. x
dFZ is a centrifugal acceleration dependent inertial force due to the spin rate of the body, Ȧ. dFZ
(rb r sin T )Z 2 dm
UZ 2 (rb r sin T )( Ao mr ) dr x
(2)
dFT is a centrifugal acceleration dependent inertial force due to the angular velocity of the wing, T . dFT
U T r ( A
2 r T dm 2
o
mr )dr
(3)
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x
dFT is an inertial force due to the angular acceleration of the wing, T rTdm UTr ( Ao mr )dr
dFT x
(4)
dFd is an externally applied force due to aerodynamic drag on the wing [1]. dFd
1 C d U airV 2 dAP 2gc
(5)
1 Cd U airV 2t sin T dr 2gc Where : U air
air density
V velocity of the wing reative to the air C d drag coefficent dAP t
projected differential area of the top surface of the wing width of the wing
Figure 5 - Drag Force Projected Area, dAp The area of the top surface of the elemental volume is tdr , and the projected area normal to the flow V is tdr sin ș . In the program, the drag coefficient was set to C d 0.96 . This corresponds to an estimate of a subsonic flow for air spilling over one end of a 2D half cylinder with an aspect ratio of 3.9:1. For supersonic flow, the drag coefficient was reduced by a factor based on the Mach number M.
C d ( supersonic)
C d (subsonic) M 2 1
The Mach number was calculated from
M
(6)
V , where M1, the speed of sound, was determined from, M1
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M1 49.02 Trankine . Trankine was determined from a table of U.S. Standard Atmosphere air temperatures at different altitudes. The air density was set to 0.0765
lbm
ft 3
from the U.S. Standard Atmosphere density at sea level with a
density correction factor based on altitude. x
FS is an externally applied spring force. The spring acts on the wing at a distance L SF from the wing pivot (See Figure 6). The spring force was calculated using,
FS
k S' S
(7)
where : k S
spring constant
'S
LF LSF tan T dh
LF
spring free length
dh depth of spring hole in hub As the wing opens ( ș increases), at some point,
' S d 0 . When ' S is not positive, FS
0.
Wing Opening Equation of Motion Figure 6 shows the differential volume of mass dm with the differential forces transformed into r and ș directions.
Figure 6 - Differential Forces on Mass dm in r - ș directions The governing equation of motion was determined from a moment balance about the wing pivot axis at P required for dynamic equilibrium; ¦ 0 P
0 +
³ r cosTdFZ ³ rdFT ³ r sin TdF
d
FS LSF
0
(8)
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A second order nonlinear ordinary differential equation of motion was determined by making substitutions for the differential forces and integrating from r = 0 to r = L.
UL3 k1 UZ 2 L2 cos T 1 T (2rb k 2 Lk1 sin T ) k d sin 2 T ( L2 xa2 ) k S LSF ( LF LSF tan T dh) 12 12 2 1 where : k d Cd tU AIRV 2 2g c k1
4 Ao 3mL
k2
3 Ao 2mL
0
(9)
Numerical Integration Equation (9) was solved numerically using the explicit central difference algorithm [2]. The angular acceleration and angular velocity of the wing at time step (n) are approximated by,
Tn
1 (T n1 2T n T n1 ) 't 2
(10)
T n
1 (T n1 T n1 ) 2't
(11)
Where ǻt is the time step size and
T n { T (t n ) where t n
n't.
n Making the substitution for T in Equation (9) and solving for relationship,
T
n 1
T n 1
f (T n ,T n 1 ) yields the recursive
6k d sin 2 T n ( L2 xa2 ) 't 2 2 n n ^Z cos T (2rb k2 Lk1 sin T ) k1 L UL2 12k S LSF ( LF dh LSF tan T n )` 2T n T n1 2 UL
To start the procedure at time=0 (n = 0), a back step
T 1 T 0 't T 0 Since the wing initially had
T 1
T 1
was needed from (13)
't 0 T 2
T0
(12)
2
T 0
(13)
0 , the back step is:
't o T 2 2
(14)
0 where T was determined from Equation (9) at t = 0.
T0
2 Lk1
ª 2 6k S LSF ( LF d h ) º «Z rb k 2 » UL2 ¬ ¼
(15)
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Body Spin Model The model assumed that when the wings are released, that the projectile had an initial spin rate,
Z (0).
As the
wings open ( T increases), the polar moment inertia, I, of the projectile system increased, and the spin rate decreased. The governing equation to determine the change in Z n from time step (n) to time step (n+1) in the numerical integrated scheme is: n 1
I nZ n ¦ ³ 0 n 't
I n1Z n 1
(16)
n
Where
¦0
n
are the moments applied to the round at time step (n). In this model, the only applied moments to
the wing about its axis are the drag forces on the “edge” of the wings. Figure 7 shows the projected area on the edge of a differential area, with the assumption that the wing is a constant “height”, h. The drag force on the differential area is,
dFde
§ 1 · ¨¨ ¸¸C de dAe U AIRV (r ) 2 © 2gc ¹
where : dAe
hdr
V (r ) C de
(17)
(rb r sin T )Z 0.3
Figure 7- Differential Area used for Side-Acting Drag Force Resisting Projectile Body Spin It is noted that,
Cde
0.3 is the subsonic drag coefficient for the flow normal to a cylinder since the wing is an arc
segment of approximately 120 degrees. A conservative assumption in this case is to assume symmetric flow (conservative in the sense that the round spin rate would decrease less rapidly). Hence, C de 0.3 was used.
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Figure 8- Differential Drag Force acting on side of Wing Resisting Spin of Body Moment contribution of
dFde on the body from one wing is shown in Figure 8.
dM wing
(r sin T rb )dFde
dM wing
§ 1 · ¨¨ ¸¸Cde hU air Z 2 (rb r sin T )3 dr © 2 gc ¹
(18)
The total moment on one wing is:
M wing
L § 1 · ¨¨ ¸¸C de hU air Z 2 ³ (rb r sin T ) 3 dr ra © 2 gc ¹
(19)
Carrying out the integration yields the total moment contribution of one wing to be:
M wing
§ 1 · ¨¨ ¸¸Cde hU air Z 2 [rb3 ( L ra ) 1.5rb2 ( L2 ra2 ) sin T © 2 gc ¹ rb ( L3 ra3 ) sin 2 T 0.25( L4 ra4 ) sin 3 T ]
For a three wing projectile, the total moment is
M projectile
(20)
3M wing . From (16), the spin rate, Z , at time step
(n+1) is given by:
Z
n 1
n I nZ n 3M wing 't
I n 1
(21)
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The final value needed to solve for
Z n 1
from Equation (21) is the polar moment of inertia at time step (n+1). I
was computed from the updated wing angle projectile was computed from:
I n 1
ș n 1 computed in Equation (12). The total moment of inertia for the
n 1 I b 3I wing
where : I b
n 1
(22)
moment of inertia of the body
1 I nwing
moment of inertia of one wing
Figure 9 - Schematic of Wing Center of Gravity Relative to Pivot Location The moment of inertia of the wing relative to the centerline of the body, from Figure 9, is given by: n 1 I wing
1 ( I CG ) nwing M wing (rb YCG cosT n 1 rCG sin T n 1 ) 2
For this calculation, the small offset,
(23)
YCG , of the wing CG in the direction perpendicular to the radial direction was
considered, where: 1 (I CG ) nwing
inertia of the wing about wing CG at time step (n 1) M wing rCG YCG
mass of the wing radial distance of wing CG from pivot offset of wing CG normal to radial direction
The wing moments of inertia about the r-r axis and y-y axis can be obtained directly from a solid model of the conceptual wing design. These quantities are I rr and I yy , respectively. Using these quantities
(ICG ) wing was
calculated to be: 1 ( I CG ) nwing
I yy sin 2 T n 1 I rr cos 2 T n 1
(24)
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Wing Tensile Force The total tension in the wing at the pivot point, P (Figure 4), was determined from a dynamic force balance in the radial direction.
¦F
r
0
+
Tension ³ dFd cosT ³ dFZ sin T ³ dFT
0
(25)
The tensile force, T, was determined by integrating Equation (25) from r = 0 to r = L,
T
2 2 3 UL2 k2 2 T UZ 2 sin T §¨¨ Ao rb L Mrb L Ao sin TL M sin TL ·¸¸ kd sin T cosT ( L ra )
6
©
2
2
3
¹
(26)
where k d , k1 , k 2 were previously defined. The tensile force at time step n, Tn was solved directly from (26) using the values for numerical integration approximations,
T n and T n
from the
Tn
T (t n )
(27)
T n
1 (T n1 T n1 ) 2't
(28)
IIB. Wing Stop Model When the wing is released during the flight of the spinning projectile, depending on the parameters, it can open rapidly and travel through an arc until rotation is stopped by a wing stop surface on the body of the round. If the wing is traveling at a high angular velocity it may come in contact with the stop surface on the projectile body. This results in rapid deceleration and high inertial forces and reaction forces at the pinned connection and the wing stop surface. A one degree-of-freedom "wing stop" model was developed to determine these forces, shown in Figure 10. The stiffness of the body at the location where the wing is stopped was represented by a single linear spring of stiffness k. For the wing stop model, a new rotation angle ĭ(t) is defined with the origin corresponding to the instant that the wing contacts the body stop surface. At the instant of contact, the initial conditions are,
I 0 0 I 0 Tstop The wing angular velocity
Tstop
was determined from (11) at the location shown on Figure 4 where the wing
opening angle corresponds to the angle where wing contacts the body stop surface (T
T max ) .
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Figure 10 - Wing Stop Model Figure 11 is a free body diagram of the wing in dynamic equilibrium at the instant when it contacts the body stop surface. A moment balance at the pinned connection yields a linear second order differential equation of motion (29).
Figure 11 - Wing Stop Model Free Body Diagram
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I t Y 2I t 0 where Y 2
k §¨ rs m ¨© rcg
· ¸ ¸ ¹
2
(29)
Sinusoidal motion was assumed of the form:
I t C1 sin Yt C 2 cos Yt
(30)
Differentiating this equation and substituting the aforementioned initial conditions resulted in the following wing contact equations.
Tstop sin Yt Y
(31)
I t Tstop cos Yt
(32)
I t YTstop sin Yt
(33)
I t
A second equation was obtained by a force balance taken perpendicular to the wing. This equation was used to determine the reaction force, R(t), at the pinned connection.
R t mrCGI t krsI t
(34)
Substituting (32) and (33) into (34) (and performing a little math) provided the maximum reaction force at the pinned connection to the body.
R t
rCG rs
and,
R t max
mkTstop sin Yt
rCG rs
mkTstop
(35) (36)
The maximum force in the spring was determined (after a little math) in a similar way,
Fs t krsI t
§ Tstop · krs ¨ sin Yt ¸ ¨ Y ¸ © ¹
(37)
and,
Fs t max
rCG mkTstop
(38)
Equivalent Distributed Load on Wing at Stop Impact The wing stop model of Figure 10 was used to determine the maximum reaction loads R and Fs on the wing at stop impact using a lumped mass at the center of gravity representation for the wing inertial force. However, since the mass is distributed, the inertial load was distributed along the length of the wing. The distributed load
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was determined such that the reaction loads R and Fs were maintained. Since the mass per unit length of the wing was assumed to decrease linearly from the pinned end to the free end, Equation (1), the distributed inertial load w(r) in (39) on the wing was also assumed to also decrease linearly, as shown in Figure 12.
w r
ar b
(39)
Since R and Fs were known from the wing stop model, the distributed load in (39) can be determined. Constants a and b were determined using the two equations for zero balance of forces and moments. The distributed load w(r) could be applied to a finite element model of the wing, if necessary, with boundary conditions of zero displacement at the pinned connection to the body and the spring location. The reaction loads at the pin and the spring locations should be checked to ensure that they agree with theses same values from the wing stop model. The finite element model would be used to provide detailed stresses in the wing to check against allowable material property stresses. Figure 12 shows the shear and moment diagrams that results from the assumed distributed load w(r) and the reaction forces R and Fs. The maximum moment in the wing occurs at the location of the spring contact point (r=rs). The shear and moment force equations were determined from dV(r)/dr = w(r) and dM(r)/dr = V(r), respectively.
ar 2 br Rmax 2 ar 3 br 2 rRmax 6 2
V r
0 d r d rs
(40)
M r
0 d r d rs
(41)
Mmax corresponds to r = rs in the above moment equation.
Figure 12 - Shear and Moment Diagram of Wing due to Wing Stop Reaction Loads
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III. Results from Two Notional Projectile and Wing Concepts The results presented in this paper were from purely notional concepts, and were not associated with any specific or particular weapons system. These concepts have been termed the "T500 Wing" and the "D-Wing" concepts. IIIA. T500 Wing Concept In the T500 concept, the body initial spin rate was a relatively low 12 HZ. Figures 13 and 14 show typical outputs from the model for a T500 concept aluminum wing and body. In the case of Figure 13, the body was spinning at 12 HZ and traveling with a forward velocity of 712 feet per second (ft/sec) when the wing was released. Note that when the wing angle increased, the body's spin rate decreased in accordance to the relationship of Equation (21). Plotted also are the tensile force components for the wing as it opened. Note that the drag force component, due to the magnitude of the body forward velocity, was much larger than the other tensile force components. Since the drag force resists the opening of the wing, the maximum wing angle was only about 27 degrees, far short of the wing stop angle of 120 degrees for this concept. The maximum tensile force in the wing was approximately 22 lbf.
Figure 13 - Concept T500 - Wings Deployed at 712 ft/sec and 12 HZ Spin rate In Figure 14, two wing opening angles associated with body forward velocities of 160 and 230 ft/sec, respectively, are plotted. The projectile spin rate was 6.5HZ when the wing was released in both cases. This concept included a compression spring (see Figures 3 and 4), to provide an initial assist opening the wing against the aerodynamic drag force. For the higher velocity, 230 ft/sec, the wing reached a maximum angle of 78.48 degrees and thus did not impact the wing stop. This was due to higher drag forces associated with the higher body velocity. However, in the lower velocity case of 160 ft/sec, the wing impacted the stop at 120 degrees with an angular velocity of 33.98 rad/sec. The result of this impact was evaluated using the wing stop model. The body "spring stop" stiffness was determined to be 1.25 X 106 lbf /in from a finite element model of the body in a region around the wing impact location. Using this spring stiffness and the angular velocity at impact of 33.98 rad/sec the maximum reaction loads R and Fs were calculated to be 6411 lbf and 7500 lbf, respectively. Using Equation (41) the wing moment equation was determined to be,
M (r )
0.2495 r 3 47.78r 2 6411r ( in lb f )
(42)
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Figure 14 - Concept T500 - Wings Angles with Body Velocities of 160 & 230 FPS Using Equation (42), the wing moment was calculated to be 4,835 in-lbf at the impact location (r = 0.75 in) which resulted in a bending stress (Mc/I) of 84,492 psi. Since this stress is above the yield strength for common aluminum alloys, a more flexible interface at the impact location on the body was evaluated. Using a small 0.045 inch thick rubber pad (shore A 75) as a softer "spring," the reaction forces were reduced to 546 lbf and 652 lbf for R and Fs, respectively, reducing the maximum bending stress to an acceptable 7,364 psi. IIIB. D-Wing Concept A second "D-Wing" concept projectile with significantly higher spin was evaluated. This projectile and wing concept was intended to be fired from a rifled barrel resulting in a much higher spin rate than the T500 concept. When the wings were released in flight, the transient dynamics are more violent, resulting in significantly higher tensile forces. The D-Wing stop surface had a wing opening angle of 155 degrees maximum. Figures 15 is a plot of the body spin rate and the wing opening angle. The wing was released with the projectile moving 2,500 ft/sec and spinning at a rate of 218 HZ. The wing "flapping" frequency was approximately 70 HZ and the frequency of the body spin rate reversal was approximately 140 HZ. The maximum angle that the wing would achieve was 159.4 degrees had it not been stopped by the wing stop surface. Even though the wing angular velocity has decreased when it contacted the body stop surface, there was still a sudden collision with the projectile body at the wing stop location. Figure 16 is a plot of the tensile force components in the wing during the opening cycles. The maximum tensile for of 5,478 lbf was more than two orders of magnitude higher than that of the T500 wing concept where the spin rate was much lower. This tensile force was applied to a static finite element model of the wing as shown in Figure 17. A peak stresses of 108,900 psi occurred at a stress concentration location. Since this and other high stresses were quite localized, and since the projectile was intended to be a one-time munition, these localized stresses should be acceptable for a high strength aluminum alloy such as T6. A manual stress analysis of the pin resulted in an acceptable bending stress of 3200 psi.
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Figure 15 - Concept D - Wings Angles and Body Spin Rate
Figure 16 - Concept D - Tensile Force Components in Wing
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Figure 17 - Concept D Finite Element Model Stresses of Wing Lug due to Tensile Forces However, the most significant structural issue with the D concept projectile occurred when the wing hit the wing stop surface. Because of the high impact loads, the wing stop surface was positioned with șmax as large as possible, corresponding to a wing opening angle of 155 degrees. As shown in Figure 15, if the wing travels unrestrained, it's maximum opening angle was 159.4 degrees, which is only 4.4 degrees of interference. However, the angular velocity of the wing at 155 degree opening angle was still a high 200.7 rad/sec. This resulted in R and Fs being 35,855 lbf and 52,644 lbf, respectively. The resulting moment equation in the wing was,
M (r )
44.03 r 3 1496 r 2 35,855 r ( in lb f )
(43)
resulting in a bending stress in the wing cross section at the stop location (r = 1.18 in) of over 500,000 psi; an order of magnitude too high. If this concept were to be further explored, the bending stress would need to be reduced by either elimination of the wing stop completely and/or adding a shock mitigating material at the interface location to reduce the contact forces, as was done in the T-500 concept. IV. Conclusions and Recommendations Two single DOF models were developed to evaluate the dynamics of deploying wings on an in-flight spinning projectile. The wing opening model included external wing forces for aerodynamic drag and a spring force, as well as inertial forces arising from projectile spin and wing opening angular velocity and acceleration. Variable projectile spin rate was included due to changing moment of inertia as the wings open. The nonlinear wing opening equation of motion was readily solved by numerical integration. A second linear single DOF wing stop model was developed to determine the reaction loads on the wing due to impacts with a body stop surface. This equation had a closed form solution. Reaction forces from wing impact were used to determine maximum bending stresses in the wing due to contact with the body. Two projectile/wing concepts were evaluated to demonstrate the utility of the models Development of these models was straight forward and they were easily modified as necessary to accommodate concept changes such as the addition of a compressing spring to assist wing deployment. The power of numerical integration was demonstrated by a straightforward solution of a highly nonlinear equation of motion with minimal overhead computing requirements. V. References [1] Discussions with, and e-mail dated 8/20/03 from, Dr. Winston Chuck, BAE Systems Inc, USCS-Minneapolis Division. [2] Bathe, K.J., 1982, Finite Element Procedures in Engineering Analysis, Prentice-Hall, Inc., New Jersey
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Underwater Explosion Phenomena and Shock Physics Frederick A. Costanzo Naval Surface Warfare Center Carderock Division, UERD 9500 MacArthur Boulevard West Bethesda, MD 20817-5700 (301) 227-1650
NOMENCLATURE C0 D e E FT G/CC I IN lbs, LB, # M/SEC msec P P0 P(t) psi R sec t W Wi ȡ T
Nominal Sound Speed Depth of Charge Exponential Function Energy Flux Density Feet Grams per Cubic Centimeter Impulse per Unit Area Inches Pounds Meters per Second Milliseconds Pressure Peak Incident Pressure Pressure as a Function of Time Pounds per Square Inch Range Seconds Time Explosive Charge Weight Image Charge Nominal Mass Density Shock Wave Decay Constant
ABSTRACT This paper presents a brief introduction to the basic fundamentals of underwater explosions, including discussion of the features of explosive charge detonation, the formation and characterization of the associated shock wave, bulk cavitation effects, gas bubble formation and dynamics, surface effects and shock wave refraction characteristics. Illustrations of each of these fundamental aspects of underwater explosion (UNDEX) loadings are made with a set of videos from a variety of experimental testing events. In addition, analyses of associated measured loading and dynamic response data, as well as descriptions of supporting numerical simulations of these events are presented. At the conclusion of this paper, each of these UNDEX effects are tied together with a summary discussion and illustration.
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_82, © The Society for Experimental Mechanics, Inc. 2011
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INTRODUCTION The field of underwater explosions and shock physics is both complex and fascinating. There are many aspects of the underwater explosion event that must be studied in order to properly understand the development and propagation of the dynamic shock loading through the fluid. The following sections of this paper provide an introduction to the basic features associated with the underwater detonation of an explosive charge [1], dividing the discussion into a series of primary features. These features include: the explosion or detonation phase, the formation of the shock wave and its effects, the secondary loading effect known as bulk cavitation, the effects of the expanding and contracting gas bubble, observed surface effects, and shock wave refraction effects. Each of these features will be described in some level of detail, and then all of these features will be summarized as they collectively form a composite illustration of underwater explosion phenomena. Fig. 1 below illustrates a typical underwater explosion event against a full scale ship target. In these types of events, all of the features outlined above and described within this paper play an important role in defining the dynamic loadings that are imparted to the ship, as well as in the visual effects that are observed in the vicinity of the event. It is the objective of this paper to provide a detailed enough introduction of these phenomena that a deeper understanding of the basic physics involved will be attained.
Fig. 1 – Typical Underwater Explosion Test of a Full Scale Ship Target
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EXPLOSION PHASE The underwater detonation of an explosive charge can best be described as an exothermic chemical reaction that is self-sustaining after initiation. Forming throughout the detonation process are gaseous reactive components that are at an extremely high temperature (approximately 3000 degrees Celsius) and pressure (approximately 50000 atmospheres). The entire detonation process represents a rapidly propagating reaction, with propagation speeds in the neighborhood of 25000 feet per second. Shown below in Fig. 2 are examples of typical explosives. Presented for each explosive listed are its explosive name, its chemical composition, as well as its specific gravity and detonation velocity. The most common and well-known explosive type is TNT, which is most often used as the standard when comparing energy and impulse yields of the other types of explosives. RDX is another common type of explosive. Most of the other explosives listed are essentially compositions of TNT, RDX and other additives in order to produce the desired effects. The exception to this is PBXN-103, which is composed of a variety of different elements. Explosives such as COMP B, H-6, HBX-1 and HBX-3 all have aluminum added in order to enhance the late-time burn and thus generate greater bubble energy. HBX-1, due to its stability and availability, is the most common explosive used by the Navy for shock qualification purposes and full ship shock trials. As can be seen from the chart in Fig. 2, there is a range of densities and detonation velocities associated with this group of explosives.
EXPLOSIVE
FORMULA
DENSITY (G/CC)
DET. VELOCITY (M/SEC)
TNT
C7H5N3O6
1.60
6940
RDX
C3H6N6O6
1.57
8940
COMP B
RDX/TNT/WAX 59.4/39.6/1.0
1.68
7900
H-6
RDX/TNT/AL/WAX 45.1/29.2/21.0/4.7
1.74
7440
PBXN-103
AP/AL/PNC/MTN/RESOURCINOL/TEGDN 38.73/27.19/6.92/24.36/0.36/2.44
1.89
6130
HBX-1
RDX/TNT/AL/WAX
1.72
7310
HBX-3
PBX/TNT/AL/WAX 31/29/35/5
1.82
7310
Fig. 2 – Examples of Common Explosives
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FORMATION OF THE SHOCK WAVE Upon detonation of the explosive charge, a very steep-fronted shock wave develops at the source and propagates rapidly in the surrounding fluid. A schematic diagram illustrating this shock wave propagation is shown in Fig. 3. In this diagram, one can see the shock wave propagating spherically away from the source of detonation, while at the center of detonation a gas bubble is forming at a much slower rate. To put things in proper perspective, the shock wave propagation phase is on the order of milliseconds, while the bubble expansion and contraction phase is on the order of seconds. Thus, there is about a three order of magnitude time scale difference for these two phenomena. Such a large difference as this in the time scales does pose challenges for some computational methods that intend to include both phases. The shock wave propagation can be likened to the stress wave that propagates axially along an elastic rod that has been hit on one end with a hammer. Once the shock wave has formed and has propagated to distance beyond about 2-3 charge radii, the propagation speed remains constant and assumes linear acoustic behavior. Inside of 2-3 charge radii, however, the propagation is highly nonlinear. Most UNDEX applications involve intermediate or far field scenarios, and thus for these the acoustic propagation assumption is valid. For extremely closein or near contact scenarios, the nonlinear propagation characteristics must be considered.
Fig. 3 – Shock Wave from an Underwater Explosion
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In Fig. 4, the far field shock wave pressure for a conventional weapon is illustrated. As can be seen from the curve and expression for pressure as a function of time, the shock wave pressure varies as an exponential function. As a result, the pressure function starts out at a peak value of P0 at time zero, and decays to 1/e or about 37% of its original value in Tmilliseconds in time. Here T is referred to as the decay constant. For the example provided in Fig. 4, which is for the case of a 250# HBX-1 explosive charge detonated at a distance of 50 ft from the gage point, the resulting pressure function has a peak value of about 2500 psi and decays exponentially down to a value of about 850 psi in 0.62 milliseconds. This exponential behavior of the free-field shock wave is extremely convenient and lends itself to straightforward computations when evaluating shock wave impulse and energy, as will be shown later. This exponential variation of the incident shock wave pressure is accurate for at least about one decay constant. After that point, the incident shock wave pressure actually begins to decay at a slower rate in the tail of the shock wave.
Fig. 4 – Shock Wave Pressure History for a Conventional Weapon
Next, the variation of shock wave pressure with range will be briefly discussed. Shown in Fig. 5 is a plot of the shock wave peak pressure vs. shock severity, expressed in the non-dimensional form of W1/3/R, where W is the explosive charge weight and R is the standoff from the charge to the point of interest. From this curve, it can be observed that for a large range of standoff values, the variation of peak shock wave pressure with shock severity is linear. However, as one moves closer to the explosive charge source location, the peak pressure variation becomes nonlinear. This point was made earlier when describing the linear nature of the shock wave propagation for ranges greater than about 2-3 charge radii. Inside of this range, the acoustic approximation for shock wave propagation no longer is valid and nonlinear behavior begins to take place.
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Fig. 5 - Variation of Shock Wave Pressure with W1/3/R The description of the shock wave pressure variation with time is summarized below in Fig. 6. Here for TNT, the relationship of peak shock wave pressure, P0, and shock wave severity, W1/3/R, is given as a power function. A similar power relationship is provided for the shock wave decay constant, T. These relationships, referred to as shock wave similitude equations, were developed through a series of free-field experiments conducted by the Naval Ordnance Laboratory (NOL) in the 1950s and 1960s. In these experiments, a variety of explosive charge sizes were detonated in the vicinity of arrays of pressure transducers positioned at various ranges from the source. These resulting pressure measurements were then analyzed and regression analyses were conducted to determine the best curve fits relating peak shock wave pressure and decay constant to shock severity. The basic form of these power relations can be seen from Fig. 6 to be a coefficient multiplied by the non-dimensional term W1/3/R raised to a power. For the decay constant, T, the power curve fit contains an additional W1/3 term.
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Fig. 6 – Shock Wave Pressure Functional Form
When using these power relations, one must be aware of the limits of applicability of these curve fits. Each set of published similitude equations has associated with it the ranges of charge sizes and standoffs that were used in the experiments to obtain the pressure data. For values of explosive charge weight and standoff outside of these ranges the power relations may no longer be valid. Thus, it is extremely important for the user of these power relationships to fully understand the appropriate ranges of applicability. Shown in Fig. 7 are the expressions for shock wave energy and impulse. For the shock wave energy expression, the energy per unit area, also know as the energy flux density, is expressed as 1/(ȡ 0C0) times the integral of the square of the pressure with time. For the shock wave impulse, the impulse per unit area is expressed as simply the integral of the pressure function with time. Due to the exponential nature of the shock wave pressure function, these expressions for shock wave energy and impulse can easily be evaluated. Also, as was developed for the peak shock wave pressure and decay constant, power relationships for the both the shock wave energy and shock wave impulse were developed and are presented in Fig. 7. These empirical expressions were developed for specific values of the upper limit of integration of usually 3, 5 or 7 times the shock wave decay constant. As a result, the analyst is cautioned when using these to properly identify the appropriate version of the power fit for the particular problem at hand. Finally, the concept of surface cutoff will now be described. From the diagram in Fig. 8, the direct path of the shock wave from the explosive source to the target is illustrated with the vector labeled R. This is the shortest path to the target and thus at the point of impingement, assuming time is dated from first arrival of the shock wave to the target and ignoring for the moment any shock wave reflections, the incident pressure would appear as shown in the lower of the two pressure-time curves given in the figure. However, due to the presence of the free surface and the fact that the shock wave propagates spherically away from the source, there is a second path that the incident shock wave takes that intersects the water surface. Due to the significant
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Fig. 7 – Shock Wave Energy and Impulse
Surface Cutoff
P t
R
P Shock Wave t
W
Fig. 8 – Illustration of Surface Cutoff
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impedance mismatch between water and air, this reflection of the compressive shock wave with the free surface results in a tensile wave that is reflected back into the fluid. This reflected tensile wave subsequently propagates towards the target and arrives at the point of impingement at some finite time after the arrival of the incident shock wave. Upon arrival at the target, this tensile wave has the effect of suddenly reducing the amplitude of the loading pressure at the target, as shown in the upper pressure-time history shown in Fig. 8. This modification of the incident shock wave pressure is known as surface cutoff. The time delay associated with this surface cutoff can be computed by simply considering the respective paths of the direct and surface reflected shock waves, and is referred to as the surface cutoff time. As can be seen from the surface cutoff illustration in Fig.8, reflections of the incident shock wave are extremely important and must all be considered as these reflections can significantly modify the dynamic loading that is imparted to the target. A more detailed illustration of shock wave reflections is presented in Fig. 9, where the effects of both a free surface and a reflecting ocean bottom can be observed. In this example, the pressure at a selected point in the fluid, P, which could represent a point on the target, is illustrated in the time history curve shown. Initially, the incident shock wave compressive pressure arrives at the point of interest and results in a sharp rise in pressure from zero gage pressure to the peak shock wave pressure. This pressure then begins to decay exponentially in time, as shown. At the same time this is occurring, another path that the incident shock wave travels is to the surface, and is then reflected back into the fluid as a tensile wave. This tensile wave arrives at the point of interest delayed in time from the direct shock wave and abruptly reduces the decayed incident shock wave pressure to some negative value. In addition to these two loading effects, there is a third path that the shock wave can travel and that is to the ocean bottom. Depending on the nature of the bottom material, the resulting reflected wave can vary between a strong compressive reflection to a weak tensile reflection. Normally, a reinforcing compressive wave is reflected and when this arrives at the point of interest, an enhancement of the current pressure amplitude will occur, as shown in the time history illustration in Fig, 8. This example illustrates the complex nature of identifying the dynamic loadings associated with an UNDEX event and the importance of considering all potential reflective sources present in the particular problem of interest.
Fig. 9 – Shock Wave Reflections
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BULK CAVITATION PHENOMENA The next concept to be discussed is that of bulk cavitation [2]. Bulk cavitation occurs when a compressive shock wave travels to the surface and is reflected back into the fluid as a tensile wave. Since water cannot normally sustain a large amount of tension, it cavitates and thus is transformed from a continuous, homogeneous liquid into a non-homogeneous, vaporous region. This cavitated zone is thus incapable of further transmitting any shock disturbances in its current state. An illustration of the region of fluid that is ultimately affected by bulk cavitation is provided in Fig. 10. The region shown is not a snapshot in time, but an envelope showing the maximum extent of cavitated fluid over all time. The reason that bulk cavitation is of interest is that when the cavitated zone actually closes, due to the effects of gravity and atmospheric pressure from above and the flow from the expanding gas bubble from below, two significant fluid masses collide creating a water hammer effect and producing a secondary shock in the form of a compressive pulse. This compressive pulse, known as the bulk cavitation closure pulse, depending on the circumstances can represent a significant reloading of the target. A somewhat exaggerated depiction of the bulk cavitation closure pulse is shown with respect to the incident pressure and first bubble pulse. In most cases, the compressive bulk cavitation closure pulse will be detectable but is usually of reduced magnitude with respect to the primary shock and bubble pulses. However, certain explosive geometry scenarios can exist that bring about a more significant and possibly damaging bulk cavitation closure pressure pulse. As a result, all UNDEX test scenarios should explore the bulk cavitation dynamics in order to identify the possibility for a significant target reloading.
Fig. 10 – Effects of Bulk Cavitation of the Surface Layer
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Illustrated in Fig. 11 are the propagation of the direct incident shock wave and its surface reflected counterpart. As the spherical compressive shock front moves through the fluid, it loads any structures that are within its path. Trailing behind this shock front is the spherical surface reflected tensile wave that sweeps through and tries to reduce the total pressure in the fluid to below the vapor pressure. As a result, a state of cavitation is produced in the fluid as this “relief” wave sweeps through. The maximum extent of the bulk cavitation region that occurs over time is depicted by the boundaries in Fig. 11 that separate the white cavitated region from the blue uncavitated fluid. As mentioned earlier, this is not a snapshot in time.
TARGET SHIP
SURFACE
SURFACE REFLECTIONÆ
EXPLOSION
SHOCK FRONTÆ
Fig. 11 – Illustration of the Formation of the Bulk Cavitation Region The illustration in Fig. 12, however, with the arrows indicating the direction that the boundaries are moving, does represent a snaphot in time of the opening and closure of the bulk cavitation region. Here it is convenient to depict the surface reflection as emanating from an image charge located at a distance, D, above the water surface. As the shock front and trailing surface reflection front are passing through the region towards the right hand side of this sketch, the forces of atmospheric pressure and gravity, along with the flow from the expanding gas bubble, are forcing the two surfaces to collide. This point of collision, referred to as the point of first closure, normally occurs at a point that is about ¼ of the total extent of the cavitated region. This closure impact then propagates like a zipper going in the directions away from and towards the charge location. The result of these cavitation closure dynamics is to produce compressive shock impulses that reload the target. The process continues until the cavitation zone is completely closed. As the cavitation closure pulses propagate to the surface and are subsequently reflected back into the fluid as tensile waves, additional cavitations can occur until all of the available energy is dissipated.
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Wi D
D
W Fig. 12 – Closure of the Bulk Cavitation Region
Finally, a summary of the UNDEX shock environment associated with the incident shock wave, surface reflected wave, bottom reflected wave, and the opening and closure of the bulk cavitation region is illustrated by the collection of snapshots in time and associated pressure-time history shown in Fig. 13. The snapshots shown are animations from a free-field analysis of a shock scenario using a hydrocode. For this application, the explosive charge is placed approximately midway between the surface and a hard reflecting ocean bottom. Shortly after the explosive charge is detonated, the snapshot shown is the upper left was generated. This schematic shows the incident pressure wave emanating directly from the charge at the left. A black dot at the edge of the compressive shock front represents a pressure reference point. Also seen from this schematic is the compressive wave reflection off of the sea bottom and its propagation throughout the fluid and the bottom material. In the upper left hand portion of this same snapshot one can also see the formation of the bulk cavitation region (white area) as the surface reflected wave propagates through the fluid from upper left to lower right. By analyzing the pressure at the point of interest (black dot) associated with the snapshots at the top and bottom left of the figure, one can see in chronological order the arrival of incident shock wave pressure and its subsequent exponential decay, the arrival of the bottom reflected compressive wave that occurs an instant before the arrival of the surface reflected wave, and the effect of the surface reflected wave that reduces the pressure at this point down to the cavitation pressure. During the next phase of response, a state of cavitation exists at the gage point as is manifested by the constant slightly negative pressure in the corresponding time history. This is associated with the snapshot shown in the upper right hand side of Fig. 13. This state of cavitation exists until the atmospheric pressure, gravity and gas bubble flow work to close this region.
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Finally, as the cavitated region closes, as depicted in the snapshot in the lower right hand side of Fig. 13, the associated pressure-time history indicates the effects of the compressive closure pulse. Although the closure pulse in this example has a low amplitude when compared with the incident shock wave peak pressure, it does have a rather significant impulse associated with it. This is the kind of later time impulse that can significantly reload a structure and once again points to the importance of considering all of the aspect of the UNDEX event.
DIRECT SHOCK WAVE
CAVITATION
CAVITATION CLOSURE
BOTTOM REFLECTION
Fig. 13 – UNDEX Shock Environment
GAS BUBBLE DYNAMICS In the earlier discussion describing the detonation of an explosive charge, two primary physical aspects of the UNDEX event were shown to develop. The first aspect of the UNDEX event, the formation and propagation of the shock wave, was described in detail, and was observed to last on the order of milliseconds. The second primary aspect of the UNDEX event, the expansion, contraction and migration of the gas bubble [3], will now be described. This aspect of response occurs within a time frame that is on the order of seconds. To introduce this aspect of response, the diagram shown in Fig. 14 will now be explained in detail. This diagram illustrates different phases of bubble growth, contraction and migration, and associates them with corresponding phases of a pressure-time history.
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Fig. 14 – Explosion Bubble Phases and Pressure-Time History
The initial part of the time history show in Fig. 14 shows the incident shock wave pressure and its exponential decay phase. As mentioned earlier, after the shock front propagates rapidly throughout the medium, a gas bubble begins to expand radially outward due to the high temperature and pressure of the explosive byproducts at its center. The gas bubble continues to expand radially outward as the pressure inside the bubble is greater than the pressure outside of the bubble. At some point in time, the bubble grows to the point to where the pressures inside and outside the bubble are the same, but due to its significant outward momentum, the bubble continues to expand radially outward. Eventually the momentum of the bubble expansion is overcome by the imbalance between the pressure outside of the bubble and that inside of the bubble. At this instant the bubble has reached its first bubble maximum, and there is very low pressure inside the bubble. This is manifested in the associated pressure-time history as the long duration negative pressure phase that exists for most of the duration of the bubble oscillation. The bubble now begins its contraction phase, rapidly passing through the point of pressure equilibrium and continuing on to recompress the bubble gasses. Bubble contraction continues until the bubble cannot contract any more due to the compressibility of the gasses inside. Here the inward contraction of the bubble is rapidly reversed causing the first bubble pulse, which is evident in the corresponding pressure-time history. In addition to the interplay of the dynamic forces associated with pressure imbalances and fluid momentum, the forces of gravity and buoyancy also affect the gas bubble dynamics. As the gas bubble expands and becomes larger in diameter, it becomes more buoyant. At the same time, as it tries to move vertically upward, fluid drag forces resist its upward migration. Eventually, after
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the first bubble maximum occurs and the bubble is contracting, it begins to migrate vertically upward as the combination of bubble inward flow and buoyancy overcome the drag forces. This is depicted in the diagram in Fig. 14 as a vertical movement of the center of the gas bubble to a shallower depth. Next, the process continues with subsequent bubble expansions, contractions and pulsations, and migrations until either the gas bubble vents to the water surface or, for extremely deep detonations, all of the gas bubble energy is expended. For each of these subsequent oscillations, the maximum bubble diameters are becoming progressively smaller while the minimum bubble diameters at pulsation are becoming progressively larger. These subsequent phases are likewise accompanied with the corresponding negative pressure and bubble pulse aspects in the associated pressure-time history. Each of these pressures is diminishing in amplitude with each successive pulsation. The primary events associated with the bubble dynamics described above and illustrated by the schematic and time-history plot of Fig. 14 will now be briefly summarized in the list below: •
Summary of General Features of Explosive Gas Bubble Dynamics: – – – – – – – –
Gaseous products expand outward Water has large outward velocity, bubble diameter increases Internal gas pressure decreases — water inertia outward Outward flow eventually stops Bubble contracts — water flow inward Compressibility of gas stops inward motion abruptly Bubble pulse occurs Process of general features repeats until bubble vents to surface or energy = 0
A photograph of an underwater explosion gas bubble that has reached its maximum radius is presented in Fig. 15. From this view it can be observed that the bubble is nearly perfectly spherical in shape and has not moved much vertically since its generation.
Fig. 15 – Underwater Explosion Bubble at Maximum Radius
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In Fig. 16, a plot of the displacement of the gas sphere as a function of time is shown for a 300-lb TNT charge fired 50 ft below the surface. As can be observed from this plot, at the point in time where the first bubble maximum is reached, the center of the bubble has moved very little in the vertical direction. It’s only during the contraction phase and first bubble pulse phase that the migration of the center of the bubble becomes significant. The small series of photographs to the right of the plot in Fig. 16 clearly illustrate the shapes and sizes of the bubble during the various phases. Moving from top left to top right, and then from bottom left to bottom right one can see the progression of bubble geometries throughout the initial expansion, contraction and pulsation phases, as well as visualize the amount of vertical migration that has taken place. Also interesting to note is the fact that during the contraction phase the bubble does not remain spherical but instead assumes a more toroidal shape as the bottom of the bubble folds inward to create a re-entrant jet during the bubble pulse. This geometric shape change coupled with the associated flow assist the bubble in its vertical migration, as mentioned previously.
300-lb TNT Charge Fired 50 ft Below the Surface Scaled Version of Large Charge, Shallow Explosion
Fig. 16 – Displacement of the Gas Sphere
Measured pressure histories from a free-field UNDEX test are presented in Fig. 17. From the long-time playback of the recorded pressure-time history shown in the lower plot, one can observe the initial shock wave pressure, followed by the first and second bubble pulses. For this explosive scenario, these events all occur within one second. Above this plot are two expanded plots that more clearly illustrate the shock wave and first bubble pulse. In these plots, one can see the exponentially decaying nature of the incident shock wave, as well as the bell-shaped nature of the first bubble pulse. For this explosive scenario, the duration of the incident shock wave pressure pulse is on the order of a few milliseconds, whereas the duration of the first bubble pulse is on the order of 100 milliseconds.
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Expanded 1st Bubble Pulse
SHOCK WAVE 2nd BUBBLE PULSE 1st BUBBLE PULSE
Fig. 17 – Measured Pressure History The final aspect of bubble dynamics that will be addressed in this paper is that of bubble attraction. Fig. 18 presents a series of snapshots from a boundary element code simulation of an explosive gas bubble interacting with a rigid wall. From this series of snapshots, one can observe that the solid boundary acts as a flow obstruction and influences the motion of the gas bubble. In the first snapshot, the boundary is seen not to have much effect during the initial expansion phase. However, as the bubble begins to contract, flow on the left side of the bubble is impeded due to the presence of the rigid boundary. Thus, as the bubble contracts its left side remains virtually stationary while its right side begins moving towards the left. This significant effect results in the figure shown in the second snapshot from the left, where the bubble’s shape becomes distorted and its center moves closer to the rigid plate. As the flow continues to rush around the contracting bubble, the apparent attraction to the rigid plate becomes more evident and the right side of the bubble begins to turn inward and jet towards the plate, as seen in the third and fourth snapshots, respectively. The degree of attraction depends on the charge standoff, maximum bubble radius, as well as on the size and curvature of the obstruction. Also observed from experiments is the fact that for UNDEX generated gas bubbles, the depth and orientation of the obstruction relative to gravity have a significant effect on bubble jet direction.
Fig. 18 – Illustration of Bubble Attraction
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SURFACE EFFECTS The next feature of UNDEX phenomena to be discussed is that of observed surface effects. To illustrate these effects, the surface phenomena observed for the detonation of a 250-lb HBX-1 explosive charge at a depth of 50-ft are sketched in Fig. 19 below. The first sketch shows the spray dome, which is caused by the interaction of the compressive shock wave with the free surface. As the shock wave is reflected back into the fluid as a tensile wave, the water particles near the surface are launched vertically upward forming a parabolic shaped spray dome. This column of water takes on a more conical shape and grows in vertical extent as the gas bubble progresses through its first expansion phase and causes a radial outward flow. Later in time after the water column has reached its maximum size, the first gas bubble pulse occurs and results in a sharp protrusion through this column of a plume of water that is moving both vertically upward and radially outward. This radial breakout of the first plume is shown in the second sketch of Fig. 19. As the gas bubble continues to pulsate, additional radial plumes, such as the one shown in the third sketch of Fig. 19, will be observed. In the event that the bubble migration has occurred to the point that during one of its pulsations it vents to the water surface, then the observed plume associated with that pulsation will tend to be dark in color as a result of the explosive byproducts now being released above the water surface.
Fig. 19 – Surface Phenomena for 250-lb HBX-1 at 50-ft Depth
The surface effects associated with an underwater explosion are further illustrated with the series of photographs from an actual shock test and corresponding animations from hydrocode simulations of the same event, presented in Fig. 20. In this series of illustrations as one moves from left to right, the progression of gas bubble states is correlated with the observed surface effects. In the first pair of illustrations, the early expansion of the gas bubble is shown in the simulation, and the corresponding photograph of the surface shows the spray dome caused by the shock wave.
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Bubble Maximum
Bubble Minimum
Radial Plume Breakout
Fig. 20 – UNDEX Plume Above-Surface Effects
The second pair of illustrations shows the bubble at its maximum, corresponding to photo above it of the surface state showing a conical-shaped water column. The water column has progressed in vertical extent as can be observed in the third set of illustrations, where the gas bubble is now shown to be at its first minimum. The last two sets of illustrations show the corresponding simulation results and above water photographs of the early and late phases, respectively, of the radial breakthrough of the first plume. These surface states occur following the first bubble pulse, as was described earlier. The extent and characteristics of the observed surface effects will vary with the explosive charge size, depth of charge detonation, and proximity of reflecting boundaries such as the ocean bottom. The illustration presented above, however, should enable the reader to make a general connection between the observed events above the water surface and the shock wave and bubble dynamics that are occurring below the surface. SHOCK WAVE REFRACTION EFFECTS The final element of UNDEX phenomena to be addressed is that of shock wave refraction. For an UNDEX event, the influence of refraction on the shock wave propagation becomes of great interest for scenarios that involve large standoff ranges where the fluid medium can have varying thermal conditions. In this situation, the assumptions of linear acoustic propagation of the incident shock wave begin to break down in that the thermal gradients bring about changes in the propagation speed and thus have the effect of causing the shock wave to bend along its path from the charge source to the target. As a result, the propagation of the incident shock wave will be modified in terms of both speed and direction. These shock wave refraction effects can be best described by studying the series of plots and diagrams presented in Fig. 21. In this figure, a series of ray tracing plots generated by the REFMS code [4] for five UNDEX tests involving long standoff ranges at decreasing standoff are depicted. Alongside these plots are the corresponding measured sound velocity profiles from the ocean environment at the test site. These sound velocity profiles are represented by the vertical plots to the left of each ray tracing plot and show the variation of sound velocity in the medium as a function of depth. From these curves, it is evident that the sound velocity values fluctuate significantly for each test. Each ray tracing plot, on the other hand, shows the various paths that
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Five Shot Sequence vs. Shallow Depth Target
Fig. 21 – Observed Shock Wave Refraction Effects the shock wave takes as it propagates from the charge source on the left side of the plot to the target represented by the black dot to the right. The plot set on the top left of Fig. 21 represents the condition for Shot 1, where the explosive charge was the furthest from the target. In this case, from the ray tracing curves it can be observed that all of the paths that the shock wave takes from the charge location bend in such a wave that by the time they reach the range where the target is located, they completely fall below the target. Thus, for this particular test, significantly lower shock wave effects were experienced at the target than had been estimated assuming iso-velocity water. As one progresses through the test series and examines the ray tracing plots associated with Shots 2 and 3, where the target is located increasingly closer to the charge, the effects of shock wave refraction are still evident, but these are diminishing as the standoff distance decreases. For Shot 4, however, one can notice the fact that the bending shock wave paths actually converge at the position of the target, focusing the shock wave effects at this location. In the ray tracing plot for Shot 5, however, the focusing effect is less pronounced as it was for Shot 4. The characteristics of the sound velocity profiles for these two tests were also different. The overall effect of these variations was that even though the charge standoff for Shot 4 was larger than that of Shot 5, the shock wave focusing effect brought about by refraction caused the levels of shock wave energy experienced by the target to be nearly the same for these two tests. Thus, from the example presented above in Fig. 21, refraction effects can significantly alter the propagation of the shock wave and the energy that eventually is imparted to the target. This is especially significant for test scenarios involving large standoff ranges and varying sound velocity profiles. These effects must be accounted for, especially if one desires to apply scaling techniques to the resulting dynamic loading or target response data.
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SUMMARY This report introduced and briefly described the basic elements of underwater explosions and shock physics. From the details and examples that are presented, it is clear that UNDEX phenomena are indeed both fascinating and complex. The illustration presented in Fig. 22 below summarizes each of these basic features in a single diagram. From this figure one can clearly see the various paths of the direct shock waves, the surface reflected wave, the bottom reflected waves traveling directly through the fluid and traveling partially through the bottom material, the bulk cavitation region, the gas bubble migration and pulsation, and the various associated surface phenomena. All of these phenomena, along with the effects of shock wave refraction, are important to accurately defining the dynamic loading environment that occurs throughout the fluid medium and can affect any structure that is present.
Surface Reflected Wave
Plume
SZ Water Bubble at first Minimum
Air
Spray Dome
Bulk Cavitation Region
First Bubble Pulse Burst
Water Surface
Direct Shock Waves
Bottom Reflected Wave (Traveling entirely through water) Bottom Reflected Wave (Traveling partially through bottom material)
Ocean Bottom Fig. 22 – Summary of Underwater Explosion Phenomena
Finally, the information presented within this report provides a basic introduction to the basic phenomena associated with underwater explosions. The objective was to provide a detailed enough introduction of these phenomena that the reader would obtain both a deeper understanding of the basic physics involved and a stimulated interest in the subject.
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REFERENCES
1. Cole, Robert H., “Underwater Explosions,” Princeton University Press, 1 st Edition, 1948. 2. Costanzo, Frederick A., and John D. Gordon, “A Solution to the Axisymmetric Bulk Cavitation Problem,” 53 rd Shock and Vibration Bulletin, Shock and Vibration Information Center, Naval Research Laboratory, Washington, D.C., May 1983. 3. Snay, H. G., “Hydrodynamics of Underwater Explosions,” Naval Hydrodynamics Publication 515, National Academy of Sciences – National Research Council, 1957. 4. Britt, J. R., R. J. Eubanks and M. G. Lumsden, “Underwater Shock Wave Reflection and in Deep and Shallow Water: Volume I – A User’s Manual for the REFMS Code (Version 4.0),” Science Applications International Corporation, St. Joseph, LA., DNR-TR-91-15-VI, 1991.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Some Aspects of using Measured Data as the Basis of a Multi-Exciter Vibration Test Marcos Underwood Russ Ayres Tony Keller Spectral Dynamics, Inc. Abstract: With the unrestricted release of MIL-STD-810G, which includes Method 527 – Multi-Exciter Testing, there is more interest than ever in performing vibration tests, which involve using more than one shaker. Also, in the tradition of earlier releases of MIL-STD-810, the use of actual measured (field) data is encouraged wherever possible. While it is acknowledged that Tailoring of the measured data will most probably be required and a Modal survey of the proposed test setup is desirable, many of the steps required to use field data have not been fully reported. This paper will examine some of the detailed requirements for using measured data to perform a multi-exciter random vibration test in the laboratory. Using MIL-STD-810G and IEST committee DTE-022 as background, measurements that can be transformed into a 4-shaker test will be used as an example. While both Time Waveform Replication and Random multi-shaker tests can be established from measured field data, this paper will concentrate on the methodology associated with such Random tests. Using power and crossspectral densities, from measured field data, the entire system Spectral Density Matrix is filled, to completely define the field vibration environment. Test results will be compared with field measurements and suggestions will be made as to the potential recommended steps needed to assure a successful field simulation. Introduction: As the use of Multiple Shakers to excite a single structure increases, there has been much discussion about optimum ways to configure such tests and provide the best possible control.1 Also, as the number and size of shakers increases, we find ourselves conducting Multiple-Input-Multiple-Output (MIMO) tests with incredibly large Force levels at our disposal.1 Yet we know that when all is said and done, we cannot force a structure to move in ways that are unnatural to it, no matter how hard we try.1 So no matter how we arrive at the test requirements, we need to be sure both that the test setup represents an achievable configuration, hopefully related to field conditions, and that the test control philosophy is based on a rigorous, mathematically correct approach.1 As the repercussions of MIL-STD-810G2 spread around the world, and more testing organizations begin to consider using multiple shakers for the first time, we need guidance from the most experienced Test Engineers. Many of these Engineers are contributing to a Recommended Practice (RP), which is being developed under the auspices of IEST Committee DTE-022. One of the key premises included in this RP is that experience has shown that the entire test system including shakers, amplifiers, fixtures, test articles, transducers, cables T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_83, © The Society for Experimental Mechanics, Inc. 2011
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etc. must be taken into account in order to achieve a successful MIMO test. The only sure way to achieve this is to define the entire system as a Frequency Response Matrix (FRM), measure this FRM before the test begins, and continuously control and update this Matrix, on a Real-Time basis, as the test progresses.3,4 This is the only technically robust way to assure that Magnitude, Phase, Coherence and Crosscoupling compensation are achieved in a correct, logical and mathematically rigorous way.1 One way of describing a 4-shaker test is to visualize the relationships between the desired Control vectors, the Drive vectors necessary to produce this control and the matrix of Frequency Response Functions, which is the FRM of the system under-test, that relate all the Drive inputs and Control responses in our test system. A Matrix representation of this 4-shaker system, with the use of the FRM: [H(f)], looks like: c1(f) h11(f) c (f) h (f) 2 21
= c3(f) h31(f) c4(f) h41(f)
h12(f) h13(f) h14(f) d1(f) h22(f) h23(f) h24(f) d 2(f)
h32(f) h33(f) h34(f) d 3(f)
h42(f) h43(f) h44(f) d 4(f)
{C(f)}= [H(f)]{D(f)}
(Eq. 1)
(Eq. 2)
where : {C(f)} is the vector of Control Fourier spectra {D(f)} is the vector of Drive Fourier spectra [H(f)] is the matrix of Frequency Response Functions between control i and drive j Important Concepts: • This paper presents results obtained with the use of data forming a measured Spectral Density Matrix (SDM) as the reference for a multi-shaker test.5 • The SDM is meant to represent measured field data, which describes the vibration present on a structure due to its service environment.1,2,5 • By using this SDM as a test’s reference, a MIMO Random approach can be used to reproduce this measured field environment in the lab.2,5 • The major advantage of using this methodology, vs. Replication, for example, is that this test will visit all the vibration states, according to a Gaussian distribution and in an ergodic sense.1,5 Defining the Test: In the example to be used here, a plate driven by 4electrodynamic shakers will be employed. In order to create the most straightforward control situation, 4 control points will be used creating a “Square” Control configuration. If we are to use measured, continuous, “field” data as the
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basis for the test, the Spectral Density Matrix (SDM) for this test will have the form shown in figure 1.
Fig. 1; 4 X 4 Spectral Density Matrix, Magnitude; Note 0.75 g test level
In this example the 4 PSD terms of the major diagonal are highlighted. If a test is being defined by separately entering PSD profiles, associated Phase and desired Coherence, then after entering the PSD’s, which have Magnitude values only, along the major diagonal, additional data must be entered. For example, the required Phase profiles would be entered as shown in figure 2.
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Fig. 2; 4 X 4 Spectral Density Matrix, Phase, with Zero specified
The profiles for the test Phase has the form seen in figure 2. This fills in the upper off-diagonal SDM elements, with the required Hermitian symmetry between the upper and lower off-diagonal elements enforced.1,5 So for a 4exciter system, we will need to define 4 PSD’s and 6 profiles for Phase and 6 profiles for Coherence. In this example, where we start by defining and entering reference profiles for Magnitude, Phase and Coherence, we would let the control system calculate the reference Cross-Spectral Densities.
Fig. 3; 4 X 4 Spectral Density Matrix, Coherence, with 0.95 specified
So we actually set up the References for the test as:1,5
g11(f) g (f) [GRR(f)] = 21 g31(f) g 41(f)
g12(f) g 22(f) g 32(f) g 42(f)
g13(f) g 23(f) g33(f) g 43(f)
g14(f) g 24(f) g 34(f) g 44(f)
(Eq. 3)
where : [GRR(f)] is the Reference Spectral Density matrix g nn(f)
is the Reference Power Spectral Density :
g nm(f)
is the Reference Cross Spectral Density : CSD = mn2 g mm g nn eiphasemn
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For the current test the first 4-shaker test setup looks like:
Fig. 4; Test plate shown with shakers, stingers and Control & Measurement accelerometers
Fig. 5; Plate and accelerometer orientation for first Test setup
For the first part of this experiment the 4-Control accelerometer locations were those shown on the plate, vertically, nearest the shaker attachment points. For response measurements, 4 additional vertical accelerometers were placed, using generally non-symmetrical locations in the central portion of the plate.
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Two basic premises concerning measured field data: In the tests to be described here, where we wish to use measured “field” data as the references for a series of tests, our approach will be dictated by the amount of and completeness of the actual measured data. Of primary concern will be how much accurate information on the off diagonal terms of the vibration environment’s SDM we can glean from the field measurements. At least 2 sets of conditions can exist: A. Full SDM information is known Simultaneously sampled time histories Phase and Coherence properties between measurements is known or can be extracted B. Full SDM information is unknown Non-simultaneously sampled time histories or Only PSD data was requested Previous presentations6 have shown that by changing only the Phase and/or Coherence between control points, with no change to reference Magnitude values, response magnitudes at non-control points can change dramatically, often by 100% or more. With this in mind, a series of tests was set up to explore the different possibilities that can exist. The sequence of these tests was as follows: Test #1 – Create “field” data: • Create and run a MIMO Random test, controlling the 4-corners of the test plate. Use a Coherence of 0.95 and in-Phase conditions between the 4 control accelerometers. At the same time, measure and record 4-Auxiliary acceleration channels at other, non-symmetrical, locations on the plate. • Save the time histories of the Auxiliary channels and use this data as the field data for subsequent tests. • Process these time histories using a general purpose Signal Analysis routine to create a full Spectral Density Matrix (SDM) which includes cross-spectral densities between control responses; that is full major diagonal PSD terms and off diagonal CSD terms.1,5 • Create a SDM with PSD terms only and no cross-spectral density information. That is, set the reference SDM’s relative coherence and phase between control responses to zero. Test #2 – Run tests using field data with Cross Spectral Density terms: • Perform a MIMO Random test using new control locations, which correspond to the Auxiliary measurement locations from Test #1.
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• •
Compare the control and response data to the original Test #1 data. Check for Dynamic Reciprocity between Test #1 and Test #2.
Test #3 – Run tests using field data without Cross Spectral Density terms: • Perform a MIMO Random test using the new control locations.
•
Compare the control and response data to the original Test #1 data.
Finally, compare the results with and without cross-spectral density information and control system compensation. TEST 1;
A. Monitoring the Control and Response data.
Fig. 6; Control Mag, Ph, Coh and 4 response readouts [1A.]
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Figure 6 shows the result of the Square control of the 4 plate corners. Each of the control points have the same test Grms level, within less than 2% and the same general PSD shape. The Auxiliary response channels, which show some general plate resonant activity near 180 Hz, still show Grms levels within 8% of the 4-corner test levels. These results are for high Coherence and zero Phase test conditions. B. Recording the raw time records from all active control and response channels. The results from recording the data and processing all channels for their PSD characteristics are shown in figure 7. The independent signal processing shows virtually identical results to the data from the controlled test. The rms levels for Control (channel) #1 each read 0.752 Grms and the channel 11 rms levels show 0.727 vs. 0.729 Grms with virtually identical shapes as expected.
Fig. 7; Time histories recorded during the MIMO test and their corresponding PSD’s [1B.]
C. Processing the raw time records to fill in the SDM. Figure 8 shows some of the processed time records in terms of the Cross Spectra, off-diagonal, terms, which are required to fill in the entire Spectral
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Density Matrix. Once the Matrix is completely filled, the UUT can be attached to the shakers and testing can proceed.
Fig. 8; Cross Spectra calculated from the raw time histories [1C.]
To complete the required information for the field data Spectral Density Matrix, figure 8 shows the measured Cross Spectral Densities at the auxiliary response channel locations.
TEST 2; A. Using the previous auxiliary response locations for Control. Now that a complete Spectral Density Matrix has been created from “field” data recorded at 4 non-symmetrical plate locations, these new locations can be used as control locations for a 4X4 MIMO Random test. In essence we will use the variable shaped PSD’s, which peak near 180 Hz, from the previous TEST 1, as our new “Square” (same number of shakers and control locations) control points. Recall however, that these channels are not arranged in a square symmetric configuration, but are non-symmetrically placed in the central portion of the plate.
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Fig. 9; Control and measurement results for “swapped” control points. [2A.]
B. Compare the control and response data to the original Test #1 data. In the original Test #1 setup, we used accelerometer locations 1, 2, 3 and 4 for Control and locations 11, 12, 13 and 14 for Response measurements. Now we are reversing these locations. Accelerometers 11 through 14 will now be used as the Square control locations and Accelerometers 1 through 4 will be used for response measurements. The control profiles for Test #2, using locations 11 through 14, will be identical to the measured “field” responses from Test #1. If the new Control is perfect and our structure is somewhat linear, we would expect the original control locations, 1, 2, 3 and 4 to show responses, which match the original Test #1 control profiles. If the system is truly linear, the system‘s Dynamic Reciprocity will cause this to happen. However, most mechanical systems exhibit an amount of nonlinear behavior and so we don’t expect this match to be “perfect.”
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In Test #1, using control locations 1 through 4 our Control Matrix could be described by the following system response equations:1,5
[G
CC (14 )
( f )] = [ H ( f )][GDD ( f )][ H ( f )]
*
(Eq. 4)
By changing the Control locations to 11 through 14, we have a new definition of the Control Matrix:
[G
CC (1114 )
( f )] = [ H ' ( f )][GDD ( f )][ H ' ( f )]
*
(Eq. 5)
Note that the Drive Spectral Density Matrices are the same for both cases, although the Control Spectral Density Matrices are different, since the energy traverses different paths through the structure, which is why the FRM’s [H(f)] and [H’(f)] are different in general, as shown by (Eqs. 4 and 5).
Fig. 10; Overlay of Tests #1 and #2, showing excellent Dynamic Reciprocity. [2B.]
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Figure 10 clearly shows several things. First, using the measured field data as a set of new Reference values, and including all Cross Spectral Density terms, results in a test which exactly reproduces those PSD shapes, as seen in the traces for Control 1,1 with Auxiliary 11 and Control 3,3 with Auxiliary 13, the top traces. In these 2 cases, Auxiliary 11 and 13 are from stored data, taken during the operation of Test #1. In both cases the test RMS levels are within 1% of the original measured response levels. This shows that the field data was “replicated” faithfully, in a MIMO random sense. Second, “looking back” at the original Control locations, 1 and 3, the reciprocal measurements for these locations during Test #2, show an RMS value that is within 5% of the original PSD data. And the Drive Spectral Density Matrices for the 2 tests are virtually identical, indicating fairly linear system behavior over the test frequency range, further supporting the fact that the field data was reproduced faithfully. In this case the Drives look like:
Fig. 11; Example of Drives from Test #1 and Test #2
Fig. 11 shows that by controlling not only the Reference PSD Magnitudes, but also the test Phase and Coherence, allows us to achieve virtually repeatable, reciprocal conditions at non-control locations. This indicates that it is possible to create the needed vibration environment at hard to reach locations by instead controlling more accessible locations, as long as the measured data is
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completely specified, i.e. the needed cross-spectral density information is available and the dynamic system paths are sufficiently linear, as described by (Eqs. 4 and 5). TEST 3. A. Run tests using field data without Cross Spectral Density terms. To better understand the critical importance of defining a complete test including the off-diagonal terms of the Spectral Density Matrix, TEST #2 was repeated with the reference SDM’s off-diagonal terms set to zero. This is TEST #3. Reference 6 gives some examples of how responses at non-control locations on a test structure can vary widely if even slight changes are made to the Phase and/or Coherence between control locations, which can also be inferred from (Eqs. 4 and 5). This will be the case, since as these equations show, the drive SDM can no longer be the same and still satisfy (Eqs. 4 and 5). Such results thus will occur on a highly resonant systems when the relative Coherence is arbitrarily set to zero, as in this case, which creates in essence a free-Phase situation that is different than what existed as part of the measured field environment. The results of just such a test are seen in figure 12. Note that by specifying PSD magnitudes only in the Reference parameters for the TEST #3, the (shaped) control PSD’s virtually overlay with the (shaped) auxiliary measurements from TEST #1 which had complete Spectral Density Matrix definition. However, by not defining Phase and Coherence, the response or auxiliary measurements, at non-control points, vary by as much as 161%! Thus, these results show the high importance of these off-diagonal terms in completely specifying a field environment with MIMO random.1,5 This result also shows how important it is for the controller to be capable of forcing particular coherence and phase relationships between control responses. Our results show that this controller was in fact able to accomplish this important task. The result also underscores the importance of specifying a particular vibration environment’s cross-spectral densities between control points and not only the PSD’s at control points. Thus, these results show that accomplishing these two tasks: 1) completely specifying the environment’s Spectral Density Matrix and 2) having a capable controller conduct the MIMO random test, are key in properly simulating a random vibration in the lab.
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Fig. 12; Controls (top) and Responses (bottom) for PSD only control
B.
Compare the control and response data to the original Test #1
data. The Grms levels for each Control and Response, which were measured during the discussed tests, are shown in Table 1.
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Test #1
Test #2
Test #3
Location
Control
Location
Response
Error (%)
1
0.749
11
0.728
2
0.735
12
0.774
3
0.751
13
0.713
4
0.74
14
0.805
11
0.728
1
0.787
5.07
12
0.776
2
0.78
6.12
13
0.717
3
0.77
2.53
14
0.807
4
0.76
2.70
11
0.713
1
1.617
115.89
12
0.75
2
1.919
161.09
13
0.705
3
1.008
34.22
14
0.802
4
1.424
92.43
Table 1. Measured Grms for 3 test conditions
As Table 1 clearly shows, the greatest divergence in response values for non-control locations on the test article can take place when an absence of data for Coherence and/or Phase between measured data channels forces us to take an alternate approach. If we arbitrarily pick a set of Coherence values, or decide to set the Coherences to zero, it is most likely that our Multi-Shaker test using “field” data inputs will not create nearly the same dynamic motion in the laboratory, which was observed under field conditions. In order to really simulate in the laboratory what is happening in the field, many conditions must be satisfied. One of the most important of these conditions is to obtain as much information as possible from measured data to be able to fill in the entire system Spectral Density Matrix, including both the major diagonal and off-diagonal terms. Conclusions: 1. Multi-exciter, MIMO, tests can be developed either from measured field data, which may have to be tailored, or by defining physically realizable control profiles. For the case of a Random test, the test definition should include supportable values of Magnitude, Phase and Coherence. The latter spectra are contained in the measured crossspectral densities between response quantities. 2. If the measured field data includes the Cross Spectral Densities between control points (Coherence and Phase), and if the system under test is approximately linear, then other response (non control) points on the structure that were present when the data was gathered, will respond with good agreement to what they were when the field data was acquired for the chosen control points.
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3.
4.
5.
This allows locations that are not located conveniently, for instrumentation or control purposes, to be so controlled indirectly, by instead controlling other related locations and using Dynamic Reciprocity to cause these non-conveniently located response points to also respond as they do in the field. Of course, this is only possible if the system under test is sufficiently linear. If the measured field data does not include the cross-spectral densities between the control points, then even if the system under test is perfectly linear, the other response (non control) points on the structure that were present when the data was gathered, will not respond with good agreement to what they were when the field data was acquired for the chosen control points. If only the PSD’s from measured field data are used to define a multishaker test, then non-control response points may exhibit much higher levels in the laboratory simulation than were present in the actual field conditions.
References: 1. Underwood, Marcos A., Multi-exciter Testing Applications: Theory and Practice, Proceedings - Institute of Environmental Sciences and Technology, April 2002 2. DOD; MIL-STD-810G - Test Method Standard for Environmental Engineering Considerations and Laboratory Tests; http://www.dtc.army.mil/publications/MIL-STD-810G.pdf; 31 Oct 08 3. Underwood, Marcos A.; US Patent # 5,299,459, “Adaptive Multi-Exciter Control System”; April, 1994. 4. Underwood, Marcos A.; US Patent # 5,517,426, “Apparatus and Method for Adaptive Closed Loop Control of Shock Testing Systems, May, 1996. 5. Underwood, Marcos A. and Keller, Tony., “Understanding and using the Spectral Density Matrix,” Proceedings of the 76th Shock & Vibration Symposium, October, 2005, Destin, Florida, USA 6. Lamparelli M., Underwood M., Ayres R., and Keller T.; “An Application of ED Shakers to High Kurtosis Replication”; Presented at ESTECH2009, May 4-7, 2009, Schaumberg, IL, USA.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
On the New American National Standard for Shock Testing Equipment
Brian W. Lang Mechanical Engineer Naval Surface Warfare Center Carderock Division Building 19, Room A230 9500 MacArthur Blvd West Bethesda, MD 20817 NOMENCLATURE Symbol h g Ȟ1 Ȟ2 A D hr eres
Definition no-rebound drop height gravitational acceleration velocity before impact velocity after impact threshold pulse amplitude threshold pulse width (pulse duration) drop height with rebound coefficient of restitution
ABSTRACT This paper presents an overview of a recently published American National Standard [1] to be used for testing equipment that will be subjected to shock. This standard provides shock test parameters for testing a broad range of equipment, which will ensure inherent levels of shock resistance. It defines test requirements and severity thresholds for a large range of shock environments, including but not limited to shipping, transport, and rugged operational environments. The severity thresholds herein can be associated with specific shock environments and should be chosen for a given application on a case-by-case basis. The intent of the standard’s requirements is to outline those elements necessary for verification of a successful and accurate shock test, but not the specific test method. This standard will allow vendors to better market, and users to more easily identify, equipment that will operate or simply survive in rugged shock environments. INTRODUCTION This American National Standard defines graduated thresholds of shock severity for equipment, referred to as unit under test (UUT), whose normal use subjects it to some amount of shock. The shock severity thresholds are defined by drop height and shock pulse (type and width) or, alternatively, by velocity change and pseudo velocity shock response spectra (PVSRS). This standard requires that data be measured during a test to verify that a test shock is intense enough to meet a given shock severity threshold. This standard includes test requirements and guidelines to ensure adequate and accurate test results. INPUT CHARACTERISTICS The shock test should be performed using a drop shock machine with a simple half-sine pulse. An example of a vertical drop shock machine is shown in Figure 1. The severity threshold defines a shock magnitude applied to T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_84, © The Society for Experimental Mechanics, Inc. 2011
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the UUT. There are ten severity thresholds listed in Table 1, which are intended to be guidelines, not boundaries. Guidelines for testing above the listed thresholds are included in the standard. A digital time history in ASCII format shall be included in the shock test report to verify test results.
Figure 1. Example vertical drop shock machine Table 1. No-rebound severity threshold drop heights Severity Velocity Change Drop Height Threshold (m/s) (mm) 1 1.0 51 2 2.0 204 3 3.0 459 4 4.0 816 5 5.0 1275 6 6.0 1835 7 7.0 2498 8 8.0 3263 9 9.0 4130 10 10.0 5099 The test pulse shall be compared to a threshold (ideal) half-sine pulse at the specified severity threshold and pulse width. A no-rebound severity threshold half-sine pulse shall be generated for the specific severity threshold and pulse width chosen by the tester. A half-sine pulse can be generated using equations (1), (2), and (3).
h /Q
A
'Q 2 2g
Q 2 Q 1 S'Q 2D
(1) (2) (3)
To simulate a drop shock test with only one impact, acceleration data shall be added before and after the pulse as listed in Table 2. Table 2. Data added to threshold (ideal) half-sine pulse Description Time Range (s) Added Data (g) Beginning of Pulse 0.0 - 0.4D 0.0 UUT stops and remains at rest End of pulse - 1.0s 0.0
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An example of a half-sine pulse can be seen in Figure 2 and parameters for this example pulse are shown in Table 3. Table 3. Parameters for example half sine pulse in Figure 2 Threshold h (mm) ǻȞ (m/s) D (ms) A (g) Level 2 203.9 2.0 40.0 8.01
Figure 2. Half-sine pulse example: Level 2, 40 ms Before the test pulse, the measured acceleration amplitude should be within ±0.05 × A at t = 0 ms and ±0.15 × A at t = 0.4D ms. Between t = 0 ms and t = 0.4D ms the tolerance varies linearly. During and after the test pulse, the amplitude of the measured acceleration should be within ±0.15 × A, the threshold ideal half-sine pulse’s maximum amplitude. Figure 3 shows a threshold ideal half-sine pulse (black line) with acceleration tolerance bands (blue and green lines).
Figure 3. Threshold half-sine pulse (ideal) with tolerance bands
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Table 1 lists the no-rebound drop height for each severity threshold. A true “no-rebound” drop shock test is not possible, because it would require a perfectly inelastic collision (coefficient of restitution equal to zero). However, rebounds can be greatly reduced through cushioning or soft landing pads. For a drop shock test with considerable rebound (high coefficient of restitution), the drop heights listed in Table 1 will produce a more severe test (higher velocity change, ǻv) than the threshold requirement. A lower drop height may be used for a drop shock test; however, the test velocity change, ǻv, shall be greater than or equal to the velocity change, ǻv, listed in Table 1. The drop height for a surface with a specific coefficient of restitution can be calculated using equation 4. A comparison of threshold-equivalent drop heights for tests with 0% and 100% rebound (coefficient of restitution equal to 0 and 1, respectively) can be seen in Table 4. Drop heights for actual drop shock tests should be somewhere between the two values listed for each threshold, depending on the amount of rebound in the test.
hr
§ 'Q ¨¨ © 1 eres 2g
· ¸¸ ¹
2
(4)
Table 4. Drop heights for 0% and 100% rebound Severity Drop Height for Drop Height for Threshold eres=0% (mm) eres=100% (mm) 1 51 13 2 204 51 3 459 115 4 816 204 5 1275 319 6 1835 459 7 2498 625 8 3263 816 9 4130 1032 10 5099 1275
OTHER SHOCK TESTS The simple drop shock test may not be appropriate for some equipment shock environments. Therefore, procedures for other test methods are included in the standard and are described briefly below. A multi-bounce drop test is included for testing certain shock characteristics, such as load reversal or successive impacts. These types of shock can be found in earthquakes, automotive environments, and ships exposed to underwater explosions. Some equipment may see shock loads from multiple directions, either in service, during transport, or when the shock load occurs. There are a variety of methods for testing equipment in multiple axes, one of which is performing an angled drop test. In an angled drop test, equipment is mounted at an angle so that it will receive a shock load in multiple directions during one test.
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Figure 4. Schematic of a 30° angled drop test For any test other than a drop shock test using a half-sine pulse, the test shall fulfill the four criteria listed below. This includes pulse shapes other than half-sine, such as step, saw-tooth, and trapezoidal, as well as other alternative shock tests, such as hammer machines, floating shock platforms, and impact tests. 1. The shock test velocity change (ǻv) shall be greater than or equal to the severity threshold ǻv listed in Table 1, and shall be verified through the shock test time history. 2. At all frequencies, the test PVSRS shall be greater than or equal to the severity threshold PVSRS at the specified half-sine pulse width, both at 5% and 25% damping. At minimum, a frequency range of 3-250 Hz shall be considered. 3. Characteristics of the alternative shock test shall be listed in the shock test report. This includes the pulse shape, type of shock test, the time history, and all details of the shock test. 4. The words “alternative shock test” shall be listed along with the severity threshold level and pulse width. As noted above, a threshold level with a specific half-sine pulse width shall be chosen. The pulse width listed shall be for the half-sine pulse used to generate the severity threshold PVSRS, not the pulse width of an alternative pulse shape. TEST FIXTURES The UUT shall be mounted to the test apparatus in a configuration that simulates the intended service installation. Shock mitigation devices shall not be used, unless they are part of the as built production item. Packaging, extra housing, or any other types of added shock protection shall not be used unless they will be part of the UUT in its normal shock environment (i.e., shipping, production line, etc.). INSTRUMENTATION Instrumentation shall be used on all shock tests conducted for use with this standard. A transient time history of the entire shock event shall be recorded for verification that the shock input meets the specified severity threshold. All transducers used to record the shock event shall survive the specified severity threshold. The measurement(s) should be of the shock input to the equipment rather than the response of the equipment. Transducer(s) shall be placed as close as possible to the load path or attachment point of the UUT to the fixture (shock table, deck, rigid mount, etc.). The tester may choose, for their information, to install transducers to record internal responses of the UUT; however, any internal measurements shall not be used to satisfy the requirements of this standard.
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Instrumentation shall be calibrated in accordance with ISO 16063-1:1998, ISO 18431-4:2007, or by the instrumentation manufacturer’s guidelines. Instrumentation manufacturer’s calibration guidelines should be traceable back to the National Institute of Standards and Technology (NIST), otherwise specific details of how the instrumentation was calibrated shall be included in the shock test report. ACCEPTANCE CRITERIA There are two acceptance criteria that shall be met in order for the UUT to be considered shock certified to the chosen severity threshold. 1. All functional and operability requirements shall be successfully met and no failure criterion shall occur. 2. The shock test performed shall be at or above the specified severity threshold. SUMMARY A new American National Standard has been published for use in testing equipment that will be subjected to shock. The standard is based on test methods employing a simple half-sine pulse input to the equipment. Test requirements and severity thresholds are presented in addition to procedures to conduct and verify test results. Procedures are included for multi-bounce and angled drop tests. Procedures are also included for alternative tests where either a half-sine pulse input or drop test are not used or both. The standard summarized in this paper contains more information, including detailed instructions and example test reports. To obtain a copy of the new American standard described in this paper, go to http://asastore.aip.org. ACKNOWLEDGEMENTS The author is grateful to all the working group members for their technical contributions and the Acoustical Society of America for all their assistance in publishing this standard. REFERENCES [1] ANSI/ASA S2.62-2009, Shock Test Requirements for Equipment in a Rugged Shock Environment, American National Standards Institute, Inc. and American Society of Acoustics, Melville, NY, 2009.
BookID 214574_ChapID 85_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Shock Data Filtering Consequences By Howard A. Gaberson, Ph.D., P.E. Consultant, (Retired Civilian Navy) 234 Corsicana Dr.; Oxnard, CA 93036; (805) 485-5307 [email protected]
ABSTRACT The plateau of the Pseudo Velocity Shock Spectrum (PVSS) plotted on Four Coordinate Paper (4CP) shows the severe frequency range of the shock. Peak modal stress is proportional to PV [1, 2, 3, 4]; hence filtering effects can be quantified according to changes in the plateau. Maximum acceleration usually defines the high frequency extent of the plateau and low pass filtering reduces the peak acceleration levels of shock data. Thus low pass filtering hides the high frequency content of the shock in the shock analysis. This is demonstrated in both the time history as well as the PVSS. Both Butterworth and Bessel filters are compared to try and see if the linear phase attribute of the Bessel filter causes any changes in the PVSS. INTRODUCTION I am helping in a major change of shock analysis technology that is moving from emphasis on the acceleration shock spectrum, the SRS, to rely on the pseudo velocity shock spectrum plotted on four coordinate paper, the PVSS on 4CP. The change has many advantages, but mainly it specifically shows the damage capacity of the shock. It allows a better way to quantify the effects of filtering on shock data. This report presents a brief examination of some of the effects. I have not done an exhaustive study of the mechanical shock filtering literature. Two documents seem to summarize results from many authors: Piersol's 1992 Sound and Vibration article [5], and the IEST Recommended Practice Handbook [6]. Some filtering recommendations for pyroshock (high frequency shock) from both of these documents are: (a) Low pass filtering should not be used with cutoff frequencies of less than 20 kHz, without a thorough analysis indicating why. The low pass cutoff should always be 1.5 times the highest frequency for later data analysis. [5, 6] (b) For high pass filters to remove "electrical offsets or drift in the transducer instrumentation", the cutoff frequency should be less than 20 Hz or 0.1% of the highest frequency of subsequent data analysis computations." Cutoff frequencies higher than this might remove as temporary zero shift indicating invalid data. My experience is that 75-100 Hz high pass will easily hide an invalid data zero shift, and make the data look gorgeous. Dave Smallwood shows examples of this in [12]. A general rule for low pass filters repeated many times is that the cutoff frequency must be at least 1.5 times the highest analysis frequency. That is, that the shock spectrum of a filtered shock is only accurate to two-thirds of the filter cutoff frequency. The testing I'm reporting here is that it can be below a half of the cutoff frequency. Matlab became buyable in 1988 or so, and it makes manipulating and handling and plotting digital data quite easy. All of the calculations and plots for this document were made with Matlab Release 12 [7]. About 1988 it became easy for people to filter digital data on their PC's, but before that time, it took a serious programmer or electronic technician or one skilled in electronics to test filtering of shock data. I have books on writing filter routines in C and FORTRAN which are dated 1991 and 1993, so people were still writing C programs then. They refer to "designing" filters, which seems nuts; I use filters and assume Bessel and Butterworth did the designing years ago. Filtering was trusted to instrument makers who were trying to sell a product. Now, because we know the PVSS-4CP plateau is the severe shock region, we can look at filtering effects from a much more sensible point of view and this changes things. We can evaluate filtering by what it does to the plateau. The point being that papers on filtering dated before 1987 or so weren't done with Matlab and were much more difficult to do. In the early to mid 1970's time frame I was testing no-name low pass digital filters on shock data. The filters were FORTRAN programs, programmed by Dan Carlton [8] from WES; I remember him instructing me to run them forward and backward to remove any phase shift, so I did. I didn't want any phase shift; would you? That's terrible. (Matlab's "filtfilt" function does the forward backward filtering.) However I tested the results by calculating the PVSS with the files filtered both ways and could detect no difference. T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_85, © The Society for Experimental Mechanics, Inc. 2011
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Now with Matlab it has become very easy to filter data, high pass and low pass, with Butterworth, Bessel, Chebyshev, Elliptical, of any order you desire, forward or fore and back. It's time for more testing of these filters on shock data. I'll start that ball rolling with this paper, and I invite your comments, corrections, or better ideas. This work will be easier to understand for those who are now convinced that the pseudo velocity shock spectrum plotted on four coordinate paper, PVSS on 4CP, is the only way you can evaluate severe shock. We believe that stress is proportional to modal velocity, that high modal velocities are around 100 ips, 2.5 meters/sec (modal velocities at the elastic limit of materials range from 100 for mild steel on up 1000 ips, 25 meters/sec, for ultra strong materials) and that the modal velocity a shock can deliver to equipment is given by the 5% damped PVSS on 4CP, So we are gong to look at filtering shock data according to what it does to the 5% damped PVSS on 4CP of the shock data. Low pass filtering is a high frequency information erasing operation, and high pass filtering is a low frequency erasing operation. I'll demonstrate and explain the information erased by low pass. Low pass filtering cuts the peak acceleration, which in turn erases the high frequency portion of the plateau, and the plateau shows the frequencies where the shock is strong. DROP TABLE SIMPLE SHOCKS I'll start with a simple theoretical (algebraically specified), half sine drop table shock. Then I'll test an explosive shock, and finally move on to a real simple shock. The equation specified simple shock I'll use is a high frequency half sine shock of 2000 g and a 0.0004 sec duration, including the drop and a rebound with a coefficient of restitution of 0.65. Figure 1, shows its time history and integrals.
Figure 1. Acceleration, velocity and displacement of the severe high frequency half sine shock
Figure 2 shows its 5% damped PVSS. The plateau comes out to be 185.3 ips. Stopping the analysis at 10,000 Hz barely lets us see that it hits the 200 g asymptote.
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185.3 ips
200 g 18.4 in
Figure 2. The 5% damped PVSS plotted on four coordinate paper for the high frequency shock of Figure 1. It has a long plateau going from 2 Hz out to about 800 Hz. I call this is a high frequency shock because it has a high PV content near 200 ips (5 meters/sec) out close to 1000 Hz. Since PV indicates stress, this shock is severe for equipment with modal frequencies from 2 to 1000 Hz. Figure 3 shows the effect of low pass filtering the shock with 2 pole Butterworth filters with cut off frequencies of 1000, 500, and 250 Hz.
Peaks: 2000, 1259.6, 702.1, 360.5
Peaks: 2000, 1259.6, 702.1, 360.5
Figure 3. This shows a comparison of filtered time histories to the unfiltered shock. Notice the drastic effect on the peak acceleration. Black is unfiltered, green is 1000 Hz low passed, red 500, and blue 250 Hz low passed. The maxima of the filtered halfsines are 1000: 1259.6, 500: 702.1, 250: 360.5.
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Figure 4. PVSS of the unfiltered, and Butterworth, 2-pole, low pass filtered high frequency simple drop table shocks. Notice thin blue line at 90% of plateau, 162.40 ips. This is where the stress has dropped to 90%. Filtering obscures part of severe plateau. Figure 4, shows the effect of low pass filtering on the 5% damped PVSS of our high frequency half sine simple shock. On this shock the flat portion or plateau is at a PV of 180.54 ips. Recall that this is proportional to stress. I have drawn a thin blue line at 90% of this value or 162.40 ips. Where this line intersects the PVSS's one might consider the high and low frequency limits of the shock; it's where the stress has dropped to 90% of its peak value. Thus the thin line intersection with the four PVSS plots shows the high frequency limits of the shock. I estimate these intersections to be at 900, 550, 300, 150, and 1.9 Hz. The unfiltered shock has high PV content from 1.9 Hz out to 900 Hz; 1000 Hz low pass cuts plateau upper frequency to 550 Hz, 500 Hz LP cuts it to 350 Hz, and 250 Hz LP cuts it to 150 Hz. The unfiltered data is the shock felt by the equipment. The filtered shock is what we might show in a report acknowledging that the data had been filtered, but probably misleading the reader about the extent of this effect. One would certainly expect a 1000 Hz low pass filter to leave the plateau untouched until after 1000 Hz. It is shown here that a 1000 Hz low pass filter hides high frequency data beyond 550 Hz. The conclusion is that low pass filtering hides the high frequency damage potential of the shock.
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Peaks: 2000, 1058.2, 557.2, 281.9
Figure 5 This shows a comparison of the 6-pole Butterworth filtered time histories to the unfiltered shock. Notice the drastic effect on the peak acceleration. Black is unfiltered, green is 1000 Hz low passed, red 500, and blue 250 Hz low passed. Notice also the ringing or decaying waviness of the 6-pole filter. The maxima of the filtered half sines are 1000: 1058.2, 500: 557.2, 250: 281.9 Figure 5 shows a 6-pole Butterworth filtering of the shock. Six poles means the cut off is sharper by 6 dB/octave and per pole. (When searching for filter cutoff rates, I found many references stating that Butterworth filters provide 6 dB/Octave per pole. e.g. [10], a very nice short article.) Thus this filter rolls off at 36 dB/octave. The sharp cutoff causes a ringing which can be seen as undulations in Figure 5. Notice also that the peak g levels are also reduced greatly. Figure 6 shows the PVSS of these 6-pole filtered shocks. I defined a simple shock PVSS characteristic I call the droop zone in [11]. The droop zone is where peak acceleration exceeds the peak acceleration asymptote. In comparing 2 and 6 pole Butterworth filtering effects, the 'droop zone' duration is reduced. Look at the blue PVSS's of Figures 4 and 6. On Figure 4 it ends at 1700 Hz, whereas on Figure 6 it ends at 550 Hz. Notice the blue curve in Figure 6 has an acceleration asymptote of 300 g, and the droop zone, which is between 150 and 600 Hz, rises to about 450 g.
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Peaks: 2000, 975.4, 507.3, 255.8
Figure 5a. Forward and backward 6 pole Butterworth filterings of the half sine. Notice now a precursor, and peak value reduction. Filtfilt is a Matlab option; I don't like the precursor. Figure 5a shows the effect of the forward backward 6-pole Butterworth filtering. SRS, and I assume PVSS analysis may be adversely affected by phase errors in the antialiasing low pass filter. Linear phase, constant time delay filters are theoretically desirable for antialiasing. Bessel filters have this, but they are no good because of their low cutoff rate. Any filter can be made to have a zero phase shift by forward backward filtering. Then without saying if this is a good idea, they jump to "nonlinear phase characteristics of the antialiasing filter can be avoided by simply limiting the analysis to two-thirds of the cut off frequency." [6, p 139]
Figure 6. The effect of using a 6 pole Butterworth is essentially the same as using a 2 pole.
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Bessel filters are reported to be best for low pass filtering because of their linear phase characteristics I'll examine the two and six pole Bessel's to see how they compare with the two and six pole Butterworth filters. I won't use the forward backward, filtering, because it introduces a precursor. See Figure 5a.
Peaks: 2000, 1181.1, 621.6, 319.1
Figure 7 This shows a comparison of filtered time histories to the unfiltered shock. Notice the drastic effect on the peak acceleration. Black is unfiltered, green is 1000 Hz low passed, red 500, and blue 250 Hz low passed. Maxima of the filtered halfsines: 1000: 1181.1, 500: 621.6, 250: 319.1.
Figure 8. PVSS of 2 pole Bessel filtered high frequency half sine shock. Smoother droop zone.
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Figures 7 and 8 show the time histories and the PVSSs for the 2-pole Bessel filtered half sine. The time history shows essentially no overshoot, or ringing waviness, and the droop zones in the PVSSs are very smooth. Figures 9 and 10 show the same thing for the 6-pole Bessel filterings. Smoother droop zone; still about the same high frequency plateau hiding.
Peaks: 2000, 786.1, 408.6, 205.9
Figure 9. This shows a comparison of 6-pole Bessel filtered time histories to the unfiltered shock. Notice the drastic effect on the peak acceleration. Black is unfiltered, green is 1000 Hz low passed, red 500, and blue 250 Hz low passed. Smoother droop zone; still about the same high frequency plateau hiding. Maxima of the filtered halfsines: 2000: 786.1, 500: 408.6, 250: 205.9. Shocks are delayed more, flattened, and look symmetrical. No steep rise and gradual tail off. Very slight ringing.
Figure 10. PVSS of 6 Pole Bessel low pass filtered 0.4 ms, 2000 g half sine with cutoffs at 1000, 500, and 250 Hz.
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In Figure 10, I've dropped the plateau cut off line to 20 % just to see the change in the plateau limiting frequencies. The upshot is 1000 Hz LP cuts things off at 450 Hz; the 500 Hz LP cuts it off at 250 Hz, and the 250 LP at 120 Hz. Table 1. Estimated Intersection values of PVSS with depressed plateau by 10% Frequency Intersect Butter 2 pole, 10 % Butter 6 pole, 10 % Bessel 2 pole, 10 % Bessel 6 pole, 20%
Low 1.9 1.9 1.9 1.6
unf 850 850 850 1300
1000 Hz 520, 52% 510, 51% 410, 41% 430, 43%
500 Hz 300, 60% 290, 58% 230, 46% 230, 46%
250 Hz 150, 60% 140, 56% 110, 44% 120, 48%
Table 1 gives a good summary of the drastic effects of filtering. We had kind of thought that a 250 Hz low pass would not distort meaningful content below 250 Hz, and that's simply not true. Butterworth filters hide the plateau at 50-60% of cutoff; Bessel filters hide the plateau at 40-50% of cutoff. Table 2 lists the peak accelerations for the different filters. Table 2. Maximum values of filtered half sines. Butter 2 pole Butter 6 pole Butter 6 pole FF Bessel 2 Pole Bessel 6 Pole
Unfiltered 2000 2000 2000 2000 2000
1000 Hz 1259.6 1058.2 975.4 1181.1 786.1
500 Hz 702.1 557.2 507.3 621.6 408.6
250 Hz 360.5 281.9 255.8 319.1 205.9
EXPLOSIVE MULTICYCLE SHOCK Now let's examine the effects of filtering on a non simple explosive shock motion. The results are similar. Figure 11 shows Navy Mil-S-901 heavyweight shock test acceleration and its integrals. The test is done by mounting the equipment in a barge and setting off an underwater explosion nearby to simulate ship shock motions. It makes a good example for this analysis. To filter this shock I used a 2-pole Bessel filter because its linear phase will not affect the time history. Figure 11 shows its time history and integrals.
Figure 11. Navy heavy weight shock test example. This is a multicycle real data shock for test of low pass filtering effects. These are often filtered.
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Figure 12 Filtered acceleration time histories. Unfiltered on bottom, 1000 next one up, 500 second from top, and 250 on the top.
In Figure 12, I tried to show the filter effect on peak acceleration, but it is not as clear as I like. The graphs are auto-ranged so you have to look at the scale on the ordinates. What I did in Figure 13 is to repeat the above, but only for the high intensity first 10 ms and not auto-range to show the dramatic effect on the acceleration. It is interesting to see how the peak accelerations are reduced and yet the PVSSs of Figure 14 are unaffected at low frequency and gracefully reduced at high frequencies.
Figure 13. I think the thing to notice is that the peak acceleration is surprisingly reduced by low pass filtering,
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Figure 14 Estimating the frequency at which the filter has reduced the PV by 10%, the 250 Hz low pass reduces the PV by 10% at about 90 Hz; the 500 Hz low pass at about 200 Hz, and the 1000 Hz low pass at about 300 Hz. Notice in Figure 14, How the high frequency plateau is successively reduced by the filtering, while the low frequency is unaffected. The filtering was done with a Bessel 2 pole filter which has a linear phase and is not supposed to affect the time history or the PVSS.
190 Hz 300 Hz
95 Hz
10% 20%
Figure 14a. Expanded view of high frequency plateau region as it is affected by the three different low pass filters. Notice the 4 vertical arrows that show the height or distance representing 10 and 20%. Green is a 1000 Hz low pass, red: 500, and blue 250 Hz.
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Figure 14a, is an expanded view of the affected high frequency plateau region portion so we can estimate the frequencies at which the filterings cause a 10% reduction on the plateau. The blue or the 250 Hz low pass reduces the PV by 10% at about 95 Hz; the red or 500 Hz low pass causes a 10% reduction at about 190 Hz, and the green or 1000 Hz low pass at about 300 Hz. The upshot is that a 500 Hz low pass does not mean you are only cutting content above 500 Hz at all. It's much worse. The content appears in the spectrum but at deceptively low levels. I want to emphasize this; the: 1000 Hz 2-pole Bessel low pass causes a 10% plateau depression at 300 Hz, 30% of 1000 Hz 500 Hz 2-pole Bessel low pass causes a 10% plateau depression at 190 Hz, 38% of 500 250 Hz 2-pole Bessel low pass causes a 10% plateau depression at 95 Hz, 38% of 250 So rather than the PVSS being affected at 67% of the cutoff frequency indicated in [5,6], it is affected by 10% at about 35% of the cutoff filter frequency. DROP TABLE SHOCK MACHINE TEST SHOCK
Figure 15 This is an unfiltered 60 g, trapezoidal shock. These are used a lot in package cushioning work. This is a different problem entirely. Figure 15, shows an acceleration time plot of an actual 60 g ASTM D3332 [9] package cushion testing trapezoidal shock. I filtered this shock many times because I suspected filtering was being used to hide important information. It wasn't, but that testing was instructive to me, and I think it will be interesting to you. In Figure 16, I added a sufficient 1 g drop to the front of the shock so the impact of the shock brings the final velocity to zero. The specification calls for the shock to be “faired” to hide the lumpiness. I'm thinking, if you don’t like it or it raises questions, hide it. Is this a case of deceptive filtering?
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Figure 16. Add a 1 g drop to the beginning to bring the final velocity to zero. Notice this requires a 47.7 inch drop, and the impact velocity change is about 200 ips, so this is a severe shock.
Figure 17 Shows the PVSS of this shock in the red curve, and we see it is quite severe from 1 to 40 Hz.
Figure 17. PVSS of the Bessel 2-pole-pole 300 Hz low pass filtered 60 g trapezoid and the red unfiltered shock. The ASTM Spec [9] talks about a "faired pulse and offers the picture shown in Figure 18. Lansmont Corporation [13] in their "Damage Boundary presentation" shows Figure 18, of a "faired" trapezoidal shock. It is amusing that in the presentation used to develop the Damage Boundary concept, a trapezoidal shock is chosen because its theoretical concept has an almost instantaneous rise time which gives the shock a doubling of the peak
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acceleration on the normalized SRS that they use to explain the theory. Figure 18 shows an exaggerated slope rise time and completely ignores the instant rise time used in the arguments to develop the theory..
Figure 18. Diagram illustrating a faired trapezoidal pulse. Figure 19. Shows an illustration, again from the Lansmont Corporation presentation, of what they consider a faired trapezoidal shock. You can see why my suspicions concerning filtering were aroused. (Available on the Lansmont website, www.lansmont.com.)
Figure 19. Illustration from Lansmont Damage boundary Presentation illustrating concept of a "faired" trapezoidal shock. The question I was asking myself was, "does this fairing hide shock severity from the observers?" If you have a need to hide shock severity from an observer, maybe filtering is a good way. Also I presume "fairing" is low pass filtering, so I'll try it. I'll filter the raw pulse to see if I end up with anything like Figure 19.
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Figure 20 Expanded portion of Lansmont trapezoidal shock of Figure 15. I count 10 valleys between the first and last peak. Time interval is from 12.5 to 21 ms. Frequency is about 10/(21-12.5) = 1.1765 cycles/ms = 1176 Hz. In Figure 20, I show an expanded view of the top of the trapezoid so we can look at its frequency content. I explain in the figure caption that I estimate that frequency to be about 1170 Hz. I thought maybe to eliminate this I should low pass at 1000, and probably should use a Bessel filter because we don't want to distort the trapezoidal appearance. I used a 2-pole 1000 Hz Bessel low pass, just on the pulse alone and found the result of Figure 21.
Figure 21. The shows a Bessel 2-pole-pole 1000 Hz, low pass filtering of the shock of Figure 15
I continued to try lower frequency low pass filters until I found a satisfactory result at 300 Hz, and this is shown in Figure 22 superimposed on the unfiltered shock.
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Figure 22. Bessel 2-pole-pole 300 Hz low pass filtering of the shock superimposed on the original shock. I think this does it. That’s faired. This seems good enough Also note the filter definitely changes initial rise time, so it will affect the droop zone. In Figure 23, I present the time history and integrals of this faired shock and it seems indistinguishable from unfiltered the plot of Figure 16.
Figure 23 Bessel 2-pole-pole 300 Hz low pass filtered 60 g trapezoid shock with 1 g drop so final velocity is 0. Figure 17 shows the PVSS of this shock in black and the original unfiltered shock in red. Notice the little 1100 Hz blip in the unfiltered PVSS. That's all the effect that the lumpiness on the trapezoid top causes. The droop zone is considerably reduced, due to the increased shock rise time. The black curve ends up at its 60 g peak asymptote. The red curve has its high frequency asymptote at 70 g where the tops of the lumpiness are. The only change due to filtering is down in the very low PVSS range; the "who cares" region. Fairing and filtering here does no harm and hides nothing bad. My suspicions were unfounded. After having gone through all of this, it finally occurred to me that the severe region of the shock, the high plateau region, is from 1 to 40 Hz as seen of Figure 17. I should not have been surprised that the 300 Hz low pass filter does not alter that severe region.
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CONCLUSIONS: The conclusion has to be certainly that low pass filtering of shock data has a more drastic affect on the shock severity analysis of the pseudo velocity shock spectrum than one is led to believe by the filter cutoff frequency. The PVSS plateau is reduced by 10% at frequencies of about one-third to 60% of the filter cutoff frequency. Standard guidance [5,6] has been that the SRS is good to 2/3 the cutoff frequency. Similarly, antialiasing filters are going to hide true plateau levels at about one third to 60% of their cutoff frequencies. An anitialiasing filter with a cutoff frequency of 20 kHz is likely to reduce the calculated level of the plateau at frequencies above 6.7 12 kHz. REFERENCES: 1. Gaberson, H.A,, "Pseudo Velocity Shock Spectrum Rules for Analysis of Mechanical Shock"; IMAC XXV, Orlando, FL; Society of Experimental Mechanics; Bethel, CT, www.sem.org; Feb 2007; p 367 2. Gaberson, H.A,, "Pseudo Velocity Shock Spectrum Rules and Concepts", Proceeding of the MFPT Society Annual Meeting; April 2007 3. Gaberson, H.A,, "Pseudo Velocity Shock Spectrum", Training lectures offered yearly at the Shock and Vibration Symposium. Past slide sets can be downloaded from the SAVIAC website, www.saviac.org 2008 4. Bateman, V. I., and Gaberson, H. A., "Shock Testing and Data Analysis", course notes A one week short course offered several times a year by the Shock and Vibration Information and Analysis Center, www.saviac.org 2009 5. Piersol, Allan G., "Recommendations for the Acquisition and Analysis of Pyroshock Data", Sound and Vibration, April 1992, pp 18-21 6. Himelblau, Harry, Piersol, Allan G., Wise, James H., Grundvig, Max R., "Handbook for Dynamic Data Acquisition and Analysis", IEST Recommended Practice 012.1, undated, but listed as current on the IEST website, www.IEST.org 7. Matlab, Version 6, Release 12. The Mathworks. Natick, MA www.mathworks.com 8. Carleton, H.D., "Digital Filters for Routine Data Reduction" U. S. Army Engineer Waterways Experiment Station, Vicksburg, Mississippi; MISCELLANEOUS PAPER N-70-1; AD#705571, March 1970 9.. ASTM D 3332, Reapproved 2004. "Standard Test Methods for Mechanical-Shock Fragility of Products, Using Shock Machines 10. Berners, Dave, "Cut Filters", www.uaudia.com/webzine/2008/august/doctors.html, 2008 11. Gaberson, H.A. "Half Sine Shock Tests to Assure Machinery Survival in Explosive Environments". IMAC XXII, Dearborn, MI; Society of Experimental Mechanics, Jan 29, 2004 12. Smallwood, D. O., and Cap, J. S., "Salvaging Transient Data with Overloads and Zero Offsets", 68th Shock and Vibration Symposium., 1997 pp 209-217 13. Lansmont Corporation; Monterey, CA; www.lansmont.com
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BookID 214574_ChapID 86_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
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[1] C.T. Key, R.W. Six and A.C. Hansen, "A Three-Constituent Multicontinuum Theory for Woven Fabric Composite Materials," Composites Science and Technology, vol. 63, 2003, pp. 1857-1864. [2] R. Fertig, “An accurate and efficient method of constituent-based progressive failure modeling of a woven composite,” Collected Proceedings: Advances in Composite, Cellular, and Natural Materials, Seattle, WA: TMS, 2010. [3] O. Okoli and G. Smith, “High Strain Rate Characterization of a Glass/Epoxy Composite,” Journal of Composites Technology and Research, vol. 22, 2000, pp. 3-11. [4] J.S. Mayes and A.C. Hansen, “Composite laminate failure analysis using multicontinuum theory,” Composites Science and Technology, vol. 64, Mar. 2004, pp. 379-394. [5] J. Mayes and A. Hansen, “A comparison of multicontinuum theory based failure simulation with experimental results,” Composites Science and Technology, vol. 64, Mar. 2004, pp. 517-527. [6] E. Nelson, A. Hansen, and J. Mayes, “Failure analysis of composite laminates subjected to hydrostatic stresses: A multicontinuum approach,” Accepted as part of the Second Worldwide Failure Exercise, 2009. [7] J. Mayes and A. Hansen, “Multicontinuum failure analysis of composite structural laminates,” Mechanics of Composite Material Structures, vol. 8, 2001, pp. 249-262. [8] M. Garnich and A. Hansen, “A Multicontinuum Theory for Thermal-Elastic Finite Element Analysis of Composite Materials,” Journal of Composite Materials, vol. 31, Jan. 1997, pp. 71-86. [9] M.R. Garnich and A.C. Hansen, “A Multicontinuum Approach to Structural Analysis of Linear Viscoelastic Composite Materials,” J Appl Mech, vol. 64, Dec. 1997, pp. 795-803. [10] A. Forghani and R. Vaziri, “Computational Modeling of Damage Development in Composite Laminates Subjected to Transverse Dynamic Loading,” Journal of Applied Mechanics, vol. 76, 2009, pp. 051304-1-051304-11.
BookID 214574_ChapID FM_Proof# 1 - 23/04/2011
BookID 214574_ChapID 87_Proof# 1 - 23/04/2011
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T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_87, © The Society for Experimental Mechanics, Inc. 2011
991
BookID 214574_ChapID 87_Proof# 1 - 23/04/2011
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BookID 214574_ChapID 87_Proof# 1 - 23/04/2011
993 4
5
6
Bar impact
x 10
1.5 Original Windowed Exponential Window
4 3
Pyro test
x 10
1
0.5 Acceleration (g)
Acceleration (g)
2 1 0
0
-0.5
-1 -1 -2
-4
Original Windowed Exponential Window
-1.5
-3
0
0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 Time (sec)
0.01
-2
0
0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 Time (sec)
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0.01
Bar impact (1024-point Hanning window) 7000
Frequency (Hz)
6000 5000 4000 3000 2000 1000 0
0
0.001
0.002
0.003
0.004
0.001
0.002
0.003
0.004
Acceleration (J)
4
5
x 10
0.005 0.006 Time (sec)
0.007
0.008
0.009
0.01
0.007
0.008
0.009
0.01
0
-5
0
0.005
0.006
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BookID 214574_ChapID 87_Proof# 1 - 23/04/2011
994 Pyro test (1024-point Hanning window) 7000
Frequency (Hz)
6000 5000 4000 3000 2000 1000 0
0
0.001
0.002
0.003
0.004
0.005 Time (sec)
0.006
0.007
0.008
0.009
0.01
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
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Acceleration (J)
2
x 10
1 0 -1 -2
0
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BookID 214574_ChapID 87_Proof# 1 - 23/04/2011
995 Single-Sided Amplitude Spectrum of D(W)
2
10
1
|$(I)|
10
0
10
-1
10
Bar impact Pyro test -2
10
0
1000
2000
3000
4000 5000 6000 Frequency (Hz)
7000
8000
9000
10000
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BookID 214574_ChapID 87_Proof# 1 - 23/04/2011
996 Acceleration Data from Pyroshock Test 6000 Axial Lateral 1 Lateral 2
4000
Acceleration (J)
2000
0
-2000
-4000
-6000 -5
0
5
10
15
20
Time (sec)
-3
x 10 )LJXUH$[LDODQGODWHUDOVHQVRUUHVSRQVHWRS\URVKRFNLQSXWRQWKHVWUXFWXUH
Pyro test (1024-point Hanning window) 7000
Frequency (Hz)
6000 5000 4000 3000 2000 1000
Acceleration (J)
0 -5
0
5
10
15
Time (sec)
4000
20 -3
x 10
2000 0 -2000 -5
0
5
10
15
20 -3
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BookID 214574_ChapID 87_Proof# 1 - 23/04/2011
997 Pyro test (1024-point Hanning window) 7000
Frequency (Hz)
6000 5000 4000 3000 2000 1000
Acceleration (J)
0 -5
0
5
10
15
Time (sec)
2000
20 -3
x 10
0 -2000 -4000 -5
0
5
10
15
20 -3
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BookID 214574_ChapID FM_Proof# 1 - 23/04/2011
BookID 214574_ChapID 89_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Identifying Goals For Ares 1-X Modal Testing
Ryan E. Tuttle, Member of the Technical Staff Joshua S. Hwung, Associate Member of the Technical Staff Jeffrey A. Lollock, Director Structural Dynamics Department The Aerospace Corporation P.O. Box 92957 Los Angeles, CA 90009-2957 Nomenclature
[M] [mi ] [ MPF ] [Pi ]
Projection matrix for i subsystem
{qi }
Interface-loaded generalized (modal) coordinate vector for i subsystem
{qsys}
Coupled system generalized (modal) coordinate vector
[ ] [ ] sys
Normalized system modal matrix
i
Normalized interface-loaded modal matrix for i subsystem
System mass matrix for interface-loaded coordinates th
Interface-loaded mass matrix for i subsystem Modal participation factors matrix th
th
th
[ˆ ]
Normalized test-configured modal matrix for i subsystem
{sys}
Coupled system modal vector
Element-by-element multiplication
i
th
ABSTRACT. A series of modal tests was planned for validation of the Ares 1-X Flight Test Vehicle (FTV) analytical model for control purposes. To support the controls analyses, knowledge of the first three pairs of vehicle bending modes is required. Planned modal testing consists of three tests: two subsystem tests and the FTV attached to its Mobile Launch Platform (MLP). Since the FTV as a whole would not be tested in its flight configuration, and the FTV / MLP testing would not be performed until late in the program schedule, it was necessary to identify appropriate goals for the subsystem tests that support the overarching goals for the FTV. This paper describes the analytical process employed to trace structural modes of the FTV to the tested subsystems. This traceability analysis – based on subsystem coupling and energy distribution – identifies subsystem modes important in describing the FTV modes of interest and evaluates the ability of different test configurations to provide data relevant to the system goals. Introduction Prior to launch, the stability and control margins of the National Aeronautics and Space Administration (NASA) Ares 1-X Flight Test Vehicle (FTV) must be shown analytically for all phases of flight. The Ares 1-X dynamic model is required to predict the responses of the first three free-free bending mode pairs of the FTV within the following uncertainty bounds: • 10-20% in frequency • ±100 inches for node locations • 20-50% in amplitude
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_89, © The Society for Experimental Mechanics, Inc. 2011
1011
BookID 214574_ChapID 89_Proof# 1 - 23/04/2011
1012 The Ares 1-X modal test team was tasked with verifying through test the Ares 1-X dynamic properties such that satisfaction of the uncertainty requirements by the analytical models could be judged. This document describes the traceabililty analysis performed to define appropriate goals (i.e. those that would support the system objectives) for subsystem modal testing. Ares 1-X Subsystems Figure 1 shows the Ares 1-X vehicle and identifies the subsystems defined for modal test pruposes. Two subsystems were tested as part of the Ares 1-X modal testing – Super Stack 1 (SS1) and Super Stack 5 (SS5), labeled as “Stack 1” and “Stack 5” in the drawing, respectively. A third test was performed – that of the FTV on the Mobile Launch Platform (MLP) – but this document will be focused on the two subsystem tests. Clearly SS1 and SS5 do not make up the entirety of the FTV. The remaining two subsystems complementary to SS1 and SS5 will be referred to as the First Stage (FS) and the Upper Stage Simulator (USS). The FS and USS comprise, th respectively, all hardware aft of the 5 segment, and the series of subsystems from US-1 Assembly to US-7 Assembly. As limited resources were made available, SS1 and SS5 were made priorities based on engineering judgment and past experience with the Shuttle program. The FS has flown as part of the Shuttle Solid Rocket Booster (SRB), and per NASA direction, was assumed to be a valid model. The USS was judged to have little influence on the dynamic behavior of the first three system bending mode pairs Figure 1: Ares 1-X according to plots of the mode shapes. Thus, in the early planning stages SS1 and Subsystems SS5 were identified as the most critical untested components, and thus the most critical tests for the development of a validated Ares 1-X system model. Subsystem Coupling and Modal Participation Factors Many methods exist for coupling analytical models. Within the loads community different parties and/or contractors are often responsible for different subsystem models. In many situations it is of interest to know how each of the subsystems is contributing to the larger system response. The Benfield-Hruda process [1] can be used to identify subsystem participation in system level dynamics. The final step in the Benfield-Hruda coupling process is diagonalizing the coupled system matrices via triple products with the system eigenvectors, or modes. Prior to the final eigensolution, the system is represented by degrees-of-freedom (DOF) corresponding to the interface-loaded modes of each subsystem. Thus, the final set of modal vectors represent the coupled system modes as combinations of the interface-loaded subspace modes (Equation 1).
q1 q {qsys} = [sys ] 2 qn
(1)
The kinetic energy of each mode is proportional to the triple product of that vector with the mass matrix (Equation 2). T
KE { sys} [ M ]{ sys}
(2)
If the mode shapes are mass-normalized, this product will be unity for each mode shape. The energy product of Equation 2 can be broken down by DOF into modal participation factors (MPF) according to Equation 3.
[ MPF ] = [ M ][sys ] [sys ]
(3)
BookID 214574_ChapID 89_Proof# 1 - 23/04/2011
1013 From Equation 3 it can be seen that each column of the resulting matrix is representative of a system mode and each row corresponds to a subsystem modal coordinate. For mass-normalized mode shapes, the elements in each column of the MPF matrix will add up to one. Thus the total kinetic energy for each system mode is decomposed into contribution from subsystem modes; each element of the MPF matrix defines, according to a kinetic energy basis, the contribution of a subsystem mode to a coupled system mode. Results These MPFs can be used to define the relative importance of subsystem modes. In addition to importance, MPFs can define which subsystem modes are sufficient in defining the system modes. The preceding process was used for defining the goals for the Ares 1-X subsystem tests. Fixed interface Craig-Bampton models of the four subsystems – FS, SS1, USS, and SS5 – were created from finite element models received from NASA. The Craig-Bampton models were coupled through a Benfield-Hruda process with each subsystem subsequently loaded with interface mass and stiffness of all upstream (fore) subsystems. The set of interface-loaded coordinates correspond to modes of the following configurations: (1) SS5 – cantilevered at its interface with USS, (2) USS – cantilevered at its interface to SS1, free at the interface to SS5 with SS5 interface loading, (3) SS1 – cantilevered at its interface to FS and free at interface to USS with interface loading of USS and SS5, and (4) FS – free-free boundary with SS1 interface loading of SS5, USS, and SS1. This set of coupled coordinates precedes the final coupling eigensolution of system modes and are the subsystem DOFs of the MPF matrix (rows). The resulting MPFs are collected in Table 1. The system modes (columns) have been down-selected to show only those modes which are defined as target modes by the Ares 1-X guidance, navigation, and control team – the first three bending mode pairs [Figure 2]. The rows have been down-selected to include only subsystem modes with significant participation in the selected system modes. Table 1: Mass Participation
Mass Participation Factors Subsystem Stack5 USS
Stack1
First Stage
Mode 1 2 1 2 3 4 1 2 3 4 5 7 8 9 10
Target System Modes Mode Shape 1st Bending 1st Bending 2nd Bending 1st Bending 2nd Bending + Torsion 1st Bending 2nd Bending
7
8
1st Bending Freq (Hz) 4.940 4.941 3.374 3.374 17.856 17.858 0.854 0.873 7.365 7.602 7.774 1.688 1.688 6.121 6.131
% Kinetic Energy Captured
1.312 0 0 3.79% 0 0 0 38.60% 0 0 0 0 51.50% 5.34% 0 0
1.323 0 0 0 3.84% 0 0 0 37.53% 0 0 0 5.43% 52.42% 0 0
9
10
2nd Bending 3.664 3.35% 3.92% 15.48% 4.88% 0 0 11.32% 5.68% 3.76% 7.59% 1.66% 25.07% 1.64% 13.64% 0
3.678 4.00% 3.48% 4.85% 15.78% 0 0 5.59% 11.28% 3.65% 5.07% 4.27% 1.75% 24.52% 0 13.69%
11
12
3rd Bending 5.077 80.18% 4.98% 0 0 1.53% 0 3.84% 0 0 0 0 1.47% 0 5.76% 1.40%
5.082 4.92% 80.19% 0 0 0 1.52% 0 3.89% 0 0 0 0 1.45% 1.36% 5.81%
99.97% 99.97% 98.70% 98.70% 99.59% 99.57%
The first pair of free-free system bending modes are comprised mainly of deformations of the FS (approximately 56%) and SS1 (approximately 38%). USS has a minor contribution at less than four percent, and SS5 has virtually no participation at all. The second pair of bending modes has contributions from all subsystems. The third pair of system bending modes is made up largely of SS5 contributions (this mode is primarily bending of the Launch Abort System (LAS) – the tip of the rocket), with less than 15 percent of the energy due to deformations of the other three subsystems. For the interface-loaded configurations of the subsystems to be tested, the first five modes of SS1 and the first two modes of SS5 are necessary for validation of the first three system bending mode pairs.
BookID 214574_ChapID 89_Proof# 1 - 23/04/2011
1014 Boundary Condition Considerations The previous section defined interface-loaded modes of subsystems SS1 and SS5 important in defining the dynamics of the system modes of interest (first three bending mode pairs). Due to program constraints the interface-loaded configurations were not able to be tested. Program restrictions required the modal tests to be performed as part of the normal integration sequence. For this reason, both subsystems were to be tested on non-flight support equipment resting on the Vehicle Assembly Building (VAB) floor. The test configuration for SS1 consisted of the SS1 interface to FS resting on a steel I-beam platform and shims / pads, all resting on concrete / steel platforms built in to the VAB floor. No interface conditioning was applied to the SS1 interface to the USS. For the SS5 test, SS5 was bolted to an access module / transportation cart assembly. The cart casters were removed, and the cart feet shimmed level to the VAB floor. Interface loading for SS5 is not an issue as it is the foremost subsystem and required no interface loading in the Benfield-Hruda coupling process. To complete the traceability study, modes of the interface-loaded configurations of the coupled subsystems need to be mapped to the modes of the test configurations. To accomplish this, the test configuration modes are decomposed using the orthogonality properties of the interface-loaded modes. The decomposition of the testconfigured modal vectors can be captured in a projection matrix. Per the orthogonality properties of normal modes, either the mass or stiffness can be used as the weighting matrix in the projection calculation. Equation 4 shows the calculation of a projection matrix using mass-weighting. Although, the projection matrix looks like a cross-orthogonality matrix, this is only the case for small interface loading or more generally, sufficiently similar boundary conditions; recall that SS5 has no interface-loaded representation, rather this process is necessary due to the inclusion of non-flight hardware in the test configuration. For configurations with significant interface loading the non-loaded modes will not be orthogonal to the interface-loaded mass matrix. However, the nonloaded modes should still be unit normalized to the interface loaded mass or stiffness matrix (depending on the projection being calculated). The reason for looking at both mass- and stiffness-weighted projections matrix is because they represent two different types of energy relevant to structural dynamics, kinetic and strain energy. T
[Pi ] = [ i ] [ mi ][ˆ i ]
(4)
The results of the projection calculations for SS1 are recorded in Tables 2 and 3 and those for SS5 in Tables 4 and 5. The projection calculations are the final step in defining target modes important for satisfying overall system goals. The root sum square (RSS) of all elements in each row of a projection matrix should be equal to unity. From this property, it can be seen that the interface-loaded mode shapes can be completely represented by a linear combination of the test-configured mode shapes. For each interface-loaded mode, a number of testconfigured modes can be identified to sufficiently represent the kinetic and strain energies. It is worth noting that the mass-weighted projection for SS1 is able to reach a closer representation of the interface-loaded modes with a small number of lower order modes. This shows that as boundary conditions change, the system strain energy is redistributed more quickly and with more complexity than the kinetic energy. Table 2: SS1 Mass-Weighted Projection Stack1 Cantilever Modes Mode
1 Shape
Stack1 Interface Loaded Modes
1 2
1st Bending
3 4 5
2nd Bending + Torsion
2
1st Bending Freq
4.64
0.85
-1.00
0.87
6
Bending Torsion + Shell 17.90
23.47
7
8
2nd Bending 24.20
-0.10
7.60
-0.94
RSS All Modes
24.78 1.00
0.99
7.37
7.77
4.73
5
-0.80
0.78
0.99 -0.52
-0.51
-0.20
0.15
0.99
-0.14
0.28
0.99
0.24
0.13
0.98
BookID 214574_ChapID 89_Proof# 1 - 23/04/2011
1015 Table 3: SS1 Stiffness-Weighted Projection
Stack1 Cantilever Modes Mode
1
Stack1 Interface Loaded Modes
Freq
4.64
0.85
-0.87
1st Bending
2 3
2nd Bending + Tosion
4
5
6
7
8
12
Bending Bending 1st Bending Torsion 2nd Bending + Shell + Shell
Shape
1
2
5
4.73
0.87
0.86
7.37
-0.27
7.60
17.90
-0.67
-0.35
7.77
23.47
-0.60
31.57
0.17
-0.33
0.94
-0.14
-0.32
-0.17
0.94
-0.15
-0.35
0.11 0.24
24.20 24.78
RSS All Modes
0.23
0.13
-0.28
0.56
0.39
0.21
0.83 0.73
0.20
0.84
Table 4: SS5 Mass-Weighted Projection Stack5 on Cart Modes (Test Configuration) Mode
1
Stack5 Interface Loaded
1
2 1st Bending
Shape
RSS All Modes
Freq
4.76
4.79
4.94
0.64
0.77
1.00
4.94
-0.76
0.64
1.00
1st Bending 2
Table 5: SS5 Stiffness-Weighted Projection Stack5 on Cart Modes (Test Configuration) Mode
1
Stack5 Interface Loaded
1
2 1st Bending
Shape
RSS All Modes
Freq
4.76
4.79
4.94
0.63
0.76
0.98
4.94
-0.75
0.63
0.98
1st Bending 2
The first six modes of the SS5 test configuration were chosen as target modes. Although only the first two modes of the test configuration for SS5 were shown to be significant with respect to the Ares 1-X system goals, this assessment is based on an analytical representation of the test support hardware. The conclusions of the traceability analysis are sensitive to the test boundary, and the inclusion of additional modes as target modes is a sensible precaution. For the SS1 test, the first eight modes of the test configuration were chosen as target modes. Although there is additional strain energy unaccounted for by this set, this strain energy is distributed over a large number of higher order modes. These target modes are the result of balancing the desire to capture the strain energies and the high costs associated with identifying large numbers of modes during test. Conclusions Large structures often cannot be tested at the system level. In this case and with program goals defined at the system level, it is necessary to show traceability of system modes to the tested subsystems. A process to show
BookID 214574_ChapID 89_Proof# 1 - 23/04/2011
1016 this traceability has been outlined and applied to the Ares 1-X program. The results of this analysis helped to define appropriate goals for the subsystem tests in support of the overall program goals. References 1. Benfield, W. A., and Hruda, R. F., “Vibration Analysis of Structures By Component Mode Substitution”, AIAA Journal, Vol. 9, No. 7, July 1971, pp. 1255-1261. Figure 2: Ares 1-X Free-Free Bending Modes
(a) First bending mode pair (typical)
(b) Second bending mode pair (typical)
(c) Third bending mode pair (typical)
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Ares I-X Launch Vehicle Modal Test Measurements and Data Quality Assessments
Justin D. Templeton, Ralph D. Buehrle and James L. Gaspar NASA Langley Research Center, Mail Stop 424, Hampton, VA 23681 Russel A. Parks and Daniel R. Lazor NASA Marshall Space Flight Center, Huntsville, AL 35812 ABSTRACT The Ares I-X modal test program consisted of three modal tests conducted at the Vehicle Assembly Building at NASA’s Kennedy Space Center. The first test was performed on the 71-foot 53,000-pound top segment of the Ares I-X launch vehicle known as Super Stack 5 and the second test was performed on the 66-foot 146,000pound middle segment known as Super Stack 1. For these tests, two 250 lb-peak electro-dynamic shakers were used to excite bending and shell modes with the test articles resting on the floor. The third modal test was performed on the 327-foot 1,800,000-pound Ares I-X launch vehicle mounted to the Mobile Launcher Platform. The excitation for this test consisted of four 1000+ lb-peak hydraulic shakers arranged to excite the vehicle’s cantilevered bending modes. Because the frequencies of interest for these modal tests ranged from 0.02 to 30 Hz, high sensitivity capacitive accelerometers were used. Excitation techniques included impact, burst random, pure random, and force controlled sine sweep. This paper provides the test details for the companion papers covering the Ares I-X finite element model calibration process. Topics to be discussed include test setups, procedures, measurements, data quality assessments, and consistency of modal parameter estimates. INTRODUCTION The Ares I-X launch vehicle is the first flight test vehicle for the Ares I launch vehicle, which is intended to be NASA’s replacement for the Space Shuttle. Ares I-X consists of a 4 segment solid rocket motor with mass simulated hardware attached. The 4 segment solid rocket motor is the type that is typically used to assemble and launch the Space Shuttle. Because the Ares I vehicle will use a 5 segment solid rocket motor, the lowest set of th mass simulated hardware was a dummy 5 segment. Above this hardware, the rest of the vehicle consists of interstage and upper stage hardware that will not be recovered during the flight test. The upper stage consists of a series of segments with internal platforms to provide access up to the top of the launch vehicle. At the top of the launch vehicle is the mass simulated hardware for the Crew Module and Launch Abort System (CM/LAS). The entire launch vehicle was assembled in the Vehicle Assembly Building of NASA Kennedy Space Center atop a Mobile Launcher Platform, as shown in Figure 1. Ares I-X is scheduled to be rolled out to the launch pad and launched in the fall of 2009. A series of three modal tests were performed on Ares I-X hardware from May through August 2009. These modal tests were intended to calibrate a finite element model so that the first three free-free bending mode pairs of the Ares I-X launch vehicle could be accurately predicted. The free-free modes would then be compared to the modal parameters used in flight control system evaluations to ensure the robustness of the control system design. Based on the free-free modes of interest, the target test modes were identified using pre-test analysis [1]. The target test modes for the flight test vehicle on the Mobile Launcher Platform were the first four bending mode pairs. In order to reduce the uncertainty in the finite element model before the flight test vehicle modal test, two component level modal tests were also proposed. The first of these modal tests was performed on the topmost th hardware of the launch vehicle known as Super Stack 5, and the second was performed on the 5 segment and interstage hardware at the middle of the launch vehicle known as Super Stack 1. This paper will focus primarily on the modal test of the fully assembled Ares I-X launch vehicle, although all of the modal test setups will be discussed. TEST SETUPS The modal test of the fully assembled Ares I-X launch vehicle mounted on the Mobile Launcher Platform was known as the Flight Test Vehicle modal test. The instrumentation used for this modal test consisted of 90 PCB T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_90, © The Society for Experimental Mechanics, Inc. 2011
1017
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1018 series 3701 capacitive accelerometers with 1 V/g sensitivity that were placed at 40 locations on the launch vehicle and Mobile Launcher Platform. Of these 40 acceleration measurement locations, 19 were arranged in a triaxial configuration, 12 were biaxial, and 9 were uniaxial, as shown in Figure 2 below. The instrumentation locations were selected based on pre-test analysis [1] so that the first four bending mode pairs could be measured and distinguished from torsion modes and system modes with Mobile Launcher Platform participation. In addition to the acceleration measurements, one strain gage bridge was measured at each of the four hold down posts where the launch vehicle was bolted to the Mobile Launcher Platform. Additional strain gage bridge measurements at the hold down posts were also acquired during the modal test on a separate data acquisition system.
Figure 1. Fully assembled Ares I-X launch vehicle [2]
Figure 2. Flight test vehicle modal test instrumentation locations
The primary method of excitation for the flight test vehicle modal test used four hydraulic shaker systems with a dynamic force capability of 1560 lb-peak and dynamic stroke of 2 in-peak. Due to safety concerns, the actual dynamic force capability available during the test was 560 lb-peak due to a lowered operating pressure. These
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1019 systems consisted of components manufactured by the Team Corporation. The hydraulic shaker actuators were attached to a mounting frame that allowed the actuator to slide linearly on two hydraulic cylinders with adjustable stiffness as shown in Figure 3. Each shaker was mounted to two existing holes in work platforms through the front I-beam of the mounting frame, and 600 lb of additional weight was added to the rear of the mounting frame to improve the force input to the flight test vehicle. Each hydraulic actuator was connected to the vehicle through the use of a ½” threaded rod connected to a PCB series 223 dynamic load cell mounted to an 8” square aluminum plate, which was attached to the vehicle using Tridox F88 dental cement. An additional static load cell was mounted on the actuator side of the threaded rod to monitor static load inputs. Although the actuators were designed to allow linear movement on the mounting frame, they were operated during the modal test with an additional solenoid active to increase the rigidity of the actuator relative to the work platform. The hydraulic power supplies and solenoids were controlled remotely at a location near the data acquisition system. Team model 2240 valve drivers were used at this location to control the shakers based on Linear Variable Differential Transformer position feedback from each shaker and an input signal from the data acquisition system. The th th shaker systems were located on two work platforms near the 4 and 5 segments of the flight test vehicle’s first stage. Pre-test analysis showed that these locations were able to excite all modes of interest and that the highest nd th chosen platform was an optimal location for exciting the 2 through 4 bending modes. On each work platform, the shakers were oriented at 45° to the preferred direction of travel of the vehicle and 90° relative to each other. This was necessary in order to avoid flight test instrumentation cables and equipment attached to the vehicle while still maintaining an orthogonal shaker setup. The data acquisition system for the modal tests consisted of VXI Technology model VT1432B 24-bit digitizer cards and a Hewlett Packard model 1434A source card in a single Agilent Technologies model E8403A 13-slot mainframe. These data acquisition and source cards were controlled through a Firewire connection to a data acquisition computer running m+p International SmartOffice Analyzer software. The software allowed time history throughput data to be recorded directly to disk while simultaneously computing, displaying and storing autopower, coherence, and frequency response functions during testing.
Figure 3. Hydraulic shaker setup during flight test vehicle modal test
Figure 4. Electrodynamic shaker setup during Super Stack 1 modal test
The Super Stack 5 modal test setup consisted of the top portion of the flight test vehicle bolted to a Super Segment Assembly Stand and heavy weight Upper Stage Simulator transportation cart that rested on a concrete floor, as shown in Figure 5. Test instrumentation consisted of 70 accelerometers placed at 30 locations on the test article in order to measure the first three bending mode pairs of the stack. Of these 30 locations, 10 were arranged in a triaxial configuration and 20 were biaxial. This test used two MB Dynamics model 250 electrodynamic shakers for excitation of the vehicle, shown in Figure 4. Each shaker had a 2” stroke and was capable of providing 250 lb-peak force with supplemental forced air cooling. The shakers for this test were mounted to wooden pallets that were lifted to a height of approximately 16 feet by JLG telehandlers. The shaker locations were identified by the pre-test analysis as optimal locations within elevation constraints imposed by the project.
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1020 The force input spectra to the structure remained flat with no significant dips, and additional triaxial accelerometers mounted to the shakers showed that the telehandler modal frequencies were not at the target modal frequencies. Impact testing at the shaker input locations also showed that the frequency response functions were consistent with those acquired using the shakers. These checks were performed both as a pretest step on another structure and during the actual test to verify that the chosen shaker setup was acceptable.
Figure 5. Test setup for Super Stack 5 modal test [2]
Figure 6. Test setup for Super Stack 1 modal test th
The Super Stack 1 modal test setup consisted of the 5 segment and interstage components in a buildup stand shown in Figure 6. Project constraints resulted in the Super Stack 1 segment being tested while resting on 12 steel shims on an I-beam support structure on top of reinforced concrete pylons. Instrumentation was located at 45 locations, with 7 triaxial, 29 biaxial, and 9 uniaxial locations, totaling 88 accelerometers. The same 250 lbpeak electro-dynamic shakers from the Super Stack 5 test were used for this test, and they were located on a work platform near the middle of the stack. The shaker locations were influenced by project constraints, and pretest analysis confirmed that the locations were satisfactory for exciting the modes of interest. The shakers were lag bolted to wooden fixtures to simplify alignment procedures, and these fixtures were bolted to the work platform in 2 locations near the front of the shaker, as shown in Figure 4. TEST EXECUTION The test execution for all three tests was similar, so only the flight test vehicle test will be described to demonstrate the process. More information on the test execution for the Super Stack 5 and Super Stack 1 modal tests will be included in upcoming NASA technical memorandums (projected release fall 2009). The flight test vehicle modal test data was acquired over the course of 3 test days. The vehicle was clear of personnel during testing on these days. A summary of the high quality datasets that were acquired during this time is given in Table 1. The first datasets to be acquired were the two ambient noise tests. This test data was actually acquired during the test setup days, while people were still working inside and outside the vehicle. During one dataset the high
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1021 bay doors were left open and wind loads were exciting the vehicle bending modes. This data was sampled at 16 Hz with a block size of 512 seconds (8.5 minutes) to achieve the desired resolution of 0.0019 Hz. Autopower data was processed with a Hanning window and a 50% overlap, and 23 averages were acquired for a total test time of 1 hour and 42 minutes. Table 1. Flight Test Vehicle Modal Test Data Summary Type
Quantity
Force Levels
Frequency Range
Resolution
Averages
Ambient Noise
2
n/a
0 - 12.5 Hz
0.0019 Hz
23
Random
3
50, 130, 200 lb-rms
0 - 6.0/12.5 Hz
0.0019 Hz
23
Sine Sweep
9
50, 100, 200 lb-peak
Mode Dependent
Mode Dependent
1
Free Decay
7
n/a
Time Data
0.031 seconds
n/a
Tap Test
11
50 - 800 lb-peak
0 - 200/400 Hz
0.125 Hz
8
Acquisition of the 3 random datasets used similar parameters to the ambient noise tests, but frequency response functions and multiple coherence functions were processed from the data in addition to the autopower. The frequency response functions were computed during acquisition using the H1 estimator. During all but the 200 lbrms random dataset, 4 shakers were operational and a bandwidth of 12.5 Hz was used for the force input. In order to focus more energy into the target modes, a bandwidth of 6.0 Hz was used for the 200 lb-rms random test, but the failure of an analog filter on one of the shaker source channels caused only 3 shakers to be operational. Selection of the resolution for the ambient noise and random datasets was a function of both the desired data quality and the limited test time available. The desired resolution was based on a “rule of thumb” suggested by Elliott and Richardson [3] that “5 or more samples above the 6-dB points, or the noise floor, of a resonance peak is considered acceptable for good curve fitting.” Based upon this rule, the predicted natural frequencies, and an st assumed damping of 0.5%, a block size of 34 minutes was determined to be required for a good curve fit of the 1 bending modes. This can be seen in Figure 7, where the measured modal frequencies and damping values are plotted on a chart that was generated numerically (the predicted frequencies for these modes were very close to the measured frequencies). Unfortunately, the 34 minute block would have resulted in unacceptable test times, nd st so the 8.5 minute block was selected based upon the 2 bending modes. The strategy for the 1 bending modes was to reprocess the data from the time histories so that the desired resolution could be achieved with fewer averages. Previous testing of Super Stacks 1 and 5 had indicated that 20 averages provided acceptable test results, so the plan was to acquire 12 x 8.5 minute blocks, and process them to 23 averages with 50% overlap. If the initial curve fit of the first random dataset was unacceptable, the time records could be increased in later random tests.
Figure 7. Chart of desired block size versus natural frequency for various damping values
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1022 nd
rd
The sine sweep tests were performed with a single shaker and focused on exciting the 2 and 3 bending modes at various force levels as a linearity study. Because the shakers were mounted at 45° to the preferred direction of nd each mode, a single shaker could excite the modes in both axes. For the 2 bending modes, a lower platform rd shaker was used for excitation, and an upper platform shaker was used for the 3 bending modes. This was rd nd because the node locations of the 3 bending modes were very close to the lower shaker platform. The 2 bending modes had similar mode shape magnitudes at either platform, so the lower shaker was chosen for easy observation from the data acquisition system. The sweep rate was chosen based on natural frequency and damping ratios identified during a random test according to an equation listed in Modal Testing by Ewins [4]. ܵ௫ ൏ ͵ͳͲሺ݂ ሻሺߞ ሻଶ ݏ݁ݒܽݐܿȀ݉݅݊
(1)
nd
The resulting sweep rates were 0.003 octaves per minute for the 2 bending mode pair, and 0.01 octaves per rd minute for the 3 bending mode pair. The force was controlled to a constant level during the sine sweeps by the m+p SmartOffice software, but the lowest sweep rate that our version of the software would allow was 0.01 octaves per minute. This was one of the consequences of lower than expected damping values. In order to work around this situation, the plan was to sweep at the 0.01 octaves per minute rate with the knowledge that only nd about 80% of the steady state response would be reached during the sweeps of the 2 bending modes [5]. nd
During the first sine sweep attempts on the 2 bending modes, an issue occurred where the force level went unstable after passing through the resonance and nearly doubled in force. The cause of this issue has not been identified. In order to work around this problem, the sine sweeps were stopped as soon as the force level went unstable. The force instability for sweeps of these modes occurred after passing the 6 dB points, so the data was rd still considered to be acceptable. This instability was not a problem during the sweeps on the 3 bending modes, rd so the 3 bending mode sweeps excited modes in both axes during a single sweep. st
In order to investigate the damping linearity of the 1 bending modes, several free decay tests were performed on the flight test vehicle. From the top work platforms next to the vehicle, several members of the test team pushed on the vehicle in synchronization with the first bending mode period, approximately every 5 seconds. The vehicle eventually responded with approximately 8” deflections at the top of the vehicle, which was the maximum deflection that could be safely reached while maintaining acceptable platform clearances. When the maximum deflection was reached, the vehicle motion was allowed to decay naturally. The resulting time history data allowed the damping to be estimated using a logarithmic decrement approach. One more free decay test was nd also performed in an attempt to target the 2 bending mode in the Y-direction. The raw acceleration responses from this test did not clearly indicate individual peaks, but band-pass filtering did allow this data to be used to estimate damping as well. The final set of tests that were performed on the vehicle were tap tests with a 3 lb instrumented hammer. The tap tests were intended to investigate local modes near three instrumentation locations for the Guidance, Navigation and Control system. These tap tests had a wider frequency bandwidth of excitation than the shaker tests. RESULTS AND DISCUSSION Only the flight test vehicle results will are provided in this section. A companion paper [6] will provide a short overview of the measured mode shapes from all of the Ares I-X modal tests and their use in model calibration. More information on the test results for the Super Stack 5 and Super Stack 1 modal tests will be included in upcoming NASA Technical Memorandums (projected release fall 2009). Sample frequency response data from the 130 lb-rms random dataset is shown in Figure 8 below. The plot nd th shows that the peaks corresponding to the 2 through 4 bending mode pairs are clearly defined with minor nd st coherence drops at the peaks of the 2 bending modes. The 1 bending mode pair, however, had large coherence drops where the peaks were split in half, as shown in Figure 9. Methods were sought to improve the coherence near these peaks, such as the use of load cell signal conditioners that had a lower frequency response and acquisition of unconditioned Integrated Electronics Piezo Electric (IEPE) signals from the load cells. Neither of these methods resulted in improved low frequency coherence. The cause of these drops in coherence is unknown but thought to be related to the locations of the shakers and the ambient movement of the structure. An alternative explanation is related to the fact that the hermetic seals of the load cells may have been breached in a previous modification, possibly changing the time constant of the circuit. The peaks in the autopower from the
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1023 st
ambient noise tests were clearly defined and did not show this split. Due to this difficulty in exciting the 1 bending mode, frequency and damping estimates were acquired using free-decay response measurements.
Figure 8. Frequency response functions and multiple coherence functions from flight test vehicle random test
Figure 9. Detailed view of split peaks in frequency response functions
Another interesting phenomenon from the random datasets was the peaks that showed up in the load cell autopower data as shown in Figure 10. These peaks did not correspond to any of the peak response frequencies identified in the frequency response functions. The load cell spectra were otherwise flat across the bandwidth of interest. Good reciprocity was exhibited for all cases in all of the random test datasets. A sample comparison between the upper platform shakers for the 130 lb-rms dataset is shown in Figure 11.
Figure 10. Load cell autopower measurements from flight test vehicle random test
Figure 11. Reciprocity in flight test vehicle random test data
A comparison of the same frequency response function measurements from the three random tests shows that the target bending modes were linear for the tested excitation levels. A system level mode that included movement of the Mobile Launcher Platform did indicate some nonlinearity, but this was not a problem because it nd th was not a target mode for the test. The peak frequencies of the 2 through 4 bending modes from these random tests are shown in Table 2. The table indicates that frequency shifts were all below 1% from 50 to 200 lbnd rms. The 2 bending mode in the Z-direction had the greatest frequency shift at 0.82%, while the other bending modes did not shift by more than 0.34%.
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Figure 12. Linearity of Y-direction frequency response functions from flight test vehicle random tests
Figure 13. Linearity of Z-direction frequency response functions from flight test vehicle random tests
Table 2. Peak Frequency Changes for Random Excitation Mode
Random 50 lb-rms Peak Frequency (Hz)
Random 130 lb-rms Peak Frequency (Hz)
Random 200 lb-rms Peak Frequency (Hz)
50 vs. 200 lb-rms Percent Difference (%)
2nd bending Y-direction
1.0605
1.0586
1.0586
-0.18
bending Z-direction
1.1953
1.1953
1.1855
-0.82
3 bending Y-direction
rd
3.4649
3.4590
3.4531
-0.34
rd
3.6856
3.6816
3.6738
-0.32
th
4 bending Y-direction
4.6133
4.6113
4.6074
-0.13
4th bending Z-direction
4.7910
4.7832
4.7832
-0.16
2
nd
3 bending Z-direction
nd
rd
The sine sweep test data was also used to investigate modal frequency shifts for the 2 and 3 modes at higher force levels than the random test. The test data was processed by taking a single Discrete Fourier Transform of the entire time history for each channel and computing the frequency response functions by the simple ratio of the acceleration spectra over the force spectrum [7]. The resulting peak frequencies are listed in Table 3, which show frequency shifts less than 1.2% from 50 to 200 lb-peak. Again, the greatest frequency shift was observed for the nd 2 bending mode in the Z-direction, while the other bending modes did not shift by more than 0.18%. Table 3. Peak Frequency Changes for Sine Sweep Excitation Mode
Sweep Up 50 lb-pk Peak Frequency (Hz)
Sweep Up 100 lb-pk Peak Frequency (Hz)
Sweep Up 200 lb-pk Peak Frequency (Hz)
50 vs. 200 lb-pk Percent Difference (%)
2nd bending Y-direction
1.0527
1.0527
1.0508
-0.18
bending Z-direction
1.1836
1.1797
1.1699
-1.2
3 bending Y-direction
rd
3.4600
3.4590
3.4590
-0.03
rd
3.6777
3.6797
3.6729
-0.13
2
nd
3 bending Z-direction rd
st
A sample time history from the 3 free decay test of the 1 Y-direction bending mode is shown in figure 14. The time history was low pass filtered with a 1.0 Hz Bessel filter in order to clean up the peaks, but this had little effect on the overall trends in the time history data. A plot of the natural logarithm of the peaks from the time history versus peak number indicated that the damping was linear over the tested amplitude range because the slope of the line is proportional to the damping ratio [8]. The resulting damping ratios from the free decay tests are listed
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1025 in Table 4. Additionally, the average damped natural frequency of the decay was easily estimated from the time history data. This frequency was used along with the peak acceleration amplitudes to approximate the displacement of the top of the flight test vehicle during the test.
Figure 14. Y-direction free decay test data from the top of the flight test vehicle
Figure 15. Linearity of damping from Y-direction free decay test data
Table 4. Free Decay Test Results from the Flight Test Vehicle Modal Test
Mode
st
1 bending Y-direction
st
1 bending Z-direction
2
nd
bending Y-direction
Decay #
Maximum Unfiltered Amplitude During Push (in)
Maximum Unfiltered Amplitude at Start of Decay (in)
Maximum Fit Amplitude (in)
Free Decay Frequency (Hz)
Damping (%)
1
5.40
4.90
4.75
0.1776
0.81
2
7.80
7.80
6.78
0.1764
0.88
3
8.28
8.28
5.98
0.1764
0.88
1
6.85
6.85
5.55
0.2149
0.45
2
8.29
8.29
6.45
0.2142
0.43
3
7.82
7.82
6.97
0.2145
0.44
1
0.0908
0.0793
0.0392
1.056
0.23
Final modal parameters from the modal test were estimated using different methods by several analysts. Most analysts used the H1 estimates for random excitation with industry standard parameter estimation routines. One st more rigorous approach was used to try and improve the 1 bending estimates using data from the 200 lb-rms random test. The frequency response functions for this curve fit were processed in a different manner than the processing method used during the data acquisition. The time histories from the accelerometers and Integrated Electronics Piezo Electric (IEPE) load cell signals with biases removed were processed by taking cyclic averages, then applying a Hanning window and processing the blocks with a 75% overlap. The Hv estimator was used to compute the frequency response functions. The curve fit used the PolyMAX algorithm [9], and only the Y and Zaxis measurements were included during the pole estimation process. The results from this method and several others are included in Table 5, which summarizes the final target modal parameter estimates for the Ares I-X flight st test vehicle on the Mobile Launcher Platform. Because of the issues with the 1 bending modes in the random datasets, the frequency and damping estimates for these modes were taken from the free decay data. Free nd decay data was also used in addition to the random data to estimate the damping of the 2 bending modes. Overall, the data compiled from several different analysts, parameter estimation routines, and datasets is very consistent. A comparison of these average measured frequencies to the predicted frequencies [1] is listed in Table 6. The comparison shows that the majority of the measured frequencies were within 5% of predictions, with
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1026 rd
one exception at 6.4% for the 3 bending mode in the Y-direction. More information about the use of the measured data in the post-test analysis is included in a companion paper [6]. Table 5. Summary of Modal Parameter Estimates from Ares I-X Flight Test Vehicle Modal Test Mode
Average Freq (Hz)
STD Freq (Hz)
Average Damp (%)
STD Damp(%)
Min Damp(%)
Max Damp (%)
Data Source
1st Bending Y direction
0.178
0.002
0.848
0.038
0.81
0.88
only free-decay data
1st Bending Z direction
0.213
0.002
0.433
0.017
0.41
0.45
only free-decay data
2nd Bending Y direction
1.06
0.001
0.290
0.034
0.23
0.31
free-decay and random
2nd Bending Z direction
1.19
0.004
0.372
0.081
0.25
0.46
free-decay and random
3rd Bending Y direction
3.46
0.003
0.483
0.036
0.43
0.51
random data
3rd Bending Z direction
3.65
0.002
0.388
0.015
0.37
0.40
random data
4th Bending Y direction
4.61
0.001
0.175
0.006
0.17
0.18
random data
4th Bending Z direction
4.78
0.003
0.243
0.010
0.23
0.25
random data
Table 6. Comparison of Predicted and Measured Frequencies Mode
Predicted Freq (Hz) [1]
Measured Freq (Hz)
Percent Difference (%)
1st Bending Y direction
0.176
0.178
1.1
1st Bending Z direction
0.216
0.213
-1.4
2nd Bending Y direction
1.02
1.06
3.7
2nd Bending Z direction
1.17
1.19
2.0
3rd Bending Y direction
3.25
3.46
6.4
3rd Bending Z direction
3.50
3.65
4.2
4th Bending Y direction
4.78
4.61
-3.6
4th Bending Z direction
4.84
4.78
-1.2
CONCLUSIONS The target modal parameters for the Ares I-X flight test vehicle were obtained using both free decay and random st datasets. The 1 bending mode pair was difficult to curve fit from the random data due to inadequate response measurements at lower frequencies that split the peaks of the frequency response functions, so the natural nd th frequencies and damping ratios were determined from free decay data. The 2 through 4 bending modes were nd rd well defined in the random data. Additionally, sine sweep tests were performed on the 2 and 3 bending modes to investigate frequency nonlinearities. Results from multiple levels of random and sine sweep testing indicated linear frequency behavior of the modes, with maximum frequency shifts of 1.2% for quadruple the force levels. Overall, the modal test successfully identified the modal parameters for all of the targeted bending modes of the Ares I-X flight test vehicle, and the measured modal frequencies were in good agreement with pre-test predictions.
[1]
REFERENCES Buehrle, R.D., et. al., “Ares I-X Launch Vehicle Modal Test Overview,” Proceedings of IMAC XXVIII, February 2010.
[2]
Kennedy Space Center Media Gallery, http://mediaarchive.ksc.nasa.gov/search.cfm?cat=166, October 2009.
[3]
Elliott, A.S., and Richardson, M. H., “Virtual Experimental Modal Analysis (VEMA),” Proceedings of IMAC XVI, February 1998.
[4]
Ewins, D.J., Modal Testing: Theory, Practice and Application, 2 231 - 234, 2000.
nd
ed., Research Studies Press Ltd., pp.
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1027 [5]
Lollock, J.A., “The Effect of Swept Sinusoidal Excitation on the Response of a Single-Degree-of-Freedom rd Oscillator,” Proceedings of the 43 AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, April 2002.
[6]
Horta, L. H., et. al., “Finite Element Model Calibration Approach for Ares I-X,” Proceedings of IMAC XXVIII, February 2010.
[7]
Orlando, S., Peeters, B., and Coppotelli, G., “Improved FRF Estimators for MIMO Sine Sweep Data,” Proceedings of the ISMA 2008 International Conference on Noise and Vibration Engineering, Leuven, Belgium, September 2008.
[8]
Meirovitch, Fundamentals of Vibrations, McGraw-Hill, pp. 94 - 98, 2001.
[9]
Peeters, B, et. al., “The PolyMAX Frequency Domain Method: A New Standard for Modal Parameter Estimation?” Journal of Shock and Vibration, Vol. 11, pp. 395 – 409, 2004.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Modal Test Data Adjustment For Interface Compliance
Ryan E. Tuttle, Member of the Technical Staff Jeffrey A. Lollock, Director Structural Dynamics Department The Aerospace Corporation P.O. Box 92957 Los Angeles, CA 90009-2957 Nomenclature
d l μt 2 {q} { x} [K ] [M] [K ] [M ] [M bb ] [12 ] [1 ] [2 ] [ fb ] [rb ] [t ] {} []
Total motion at reference sensor Distance from interface to reference sensor Tip displacement ratio Angular rotation of interface plane Circular frequency for system 2 (updated) mode Generalized (modal) coordinate vector Physical degree of freedom vector System stiffness matrix System mass matrix Differential stiffness for model update Differential mass for model update Differential boundary mass Diagonal matrix of system 2 (updated) eigenvalues System 1 modal matrix System 2 (updated) modal matrix Test modes adjusted to fixed-base configuration Rigid body modes of test article relative to interface Test-measured modes Generalized modal vector Generalized modal matrix
[ i] [˜] [ ]
Generalized matrix with appended rigid body degrees of freedom
{ ˙˙ }
Second time derivative
Matrix for ith system Updated generalized matrix
ABSTRACT. Modal tests are ideally performed in configurations that represent boundary conditions that are both appropriate (facilitate application of analytical methods) and easily modeled (fixed, free, or interfaced to hardware of known stiffness). Due to the constraints of the Ares 1-X program, test fixtures that would guarantee an idealized boundary were unavailable for the desired subsystem test configurations (cantilevered). The decision was made to test subsystems in a configuration as near cantilevered as could be achieved within program constraints. It was decided to test Super Stack 1 (SS1) and Super Stack 5 (SS5) – the two tested subsystems – resting on the test facility floor. In addition, the configuration for SS5 would include non-flight transportation / access hardware
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_91, © The Society for Experimental Mechanics, Inc. 2011
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1030 between it and the floor. Since the facility floor and non-flight hardware were of unknown stiffness, analytical methods were developed to address the effects of this compliance on the measured data. A method was developed to identify interface motion from test measurements, judge when interface motion would affect test results significantly, and adjust the test results to represent those of a fixed-base configuration. Introduction Mode survey testing is used to gather data for validation of analytical models. During such testing careful consideration and measurement of boundary conditions must be undertaken as boundary condition variations can affect the dynamics of a system. Every effort is usually made to sufficiently approximate idealized boundary conditions for ease of modeling. Additionally, the type of boundary condition should be chosen to facilitate the intended analyses of the validated models. For the Ares 1-X modal test program, the desired boundary conditions are fixed at the base of both subsystems planned for test – SS1 and SS5 – as these were used in the traceability study [1]. Fixed boundary conditions for these subsystems eliminate numerous shell-type modes that only serve to unnecessarily complicate the tests and were not needed to meet the goals of this test program based on the traceability study performed. However, due to program constraints, development of fixturing capable of providing fixed boundary conditions could not be pursued. The tests were required to be performed in line with integration processes. Given these constraints, the decision was made to test the subsystems resting on the Vehicle Assembly Building (VAB) floor (with additional support equipment as became necessary) at the National Aeronautic and Space Administrations (NASA) Kennedy Space Center (KSC), relying on the structures weight to provide adequate preload for required reaction forces under test excitation. This configuration introduces the unknown stiffnesses of the floor and support hardware, unfortunately, which may need to be accounted for in subsequent analyses. One possibility of dealing with the interface stiffness is including this unknown stiffness explicitly in the analytical model and removing it after adjustment to the test data. Rather than include the boundary stiffness as a variable in model adjustment, the influence of interface compliance on the test data can be removed and the adjusted data used subsequently in model adjustment of a fixed interface model. This paper outlines such an analytical process and documents its application to the Ares 1-X subsystems. Adjustment Methodology The boundary motion adjustment method is based on a standard mass or stiffness update procedure. This procedure is used commonly throughout the industry to adjust reduced models to account for configuration changes. Consider two systems: system “1”, the nominal system and system “2”, the updated system. The undamped homogeneous equations of motion for each system are,
[ M1 ]{ x˙˙} + [K1]{ x} = {0} [ M 2 ]{ x˙˙} + [K 2 ]{ x} = {0}
(1a) (1b)
We define the delta matrices as the differential mass or stiffness of the updated system relative to the nominal system.
[K ] = [K 2 ] [K1] [M ] = [ M 2 ] [ M1]
(2a) (2b)
At this point, the mode shapes for the nominal system are used to transform the updated system equations.
{ x} = [1 ]{q1}
(3a)
T
[1 ] [ M1][1 ] = [I ] T [1 ] [K1][1 ] = [12 ] T
(3b) (3c) T
[1 ] [ M 2 ][1 ]{q˙˙1} + [1] [K 2 ][1]{q1} = {0}
(4)
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1031 Substituting Equation 2 into 4, T
[1 ] ([ M1 ] + [M ])[1]{q˙˙1} + [1 ]([K1 ] + [K ])[1]{q1} = {0}
(5)
([I] + [ ] [M ][ ]){q˙˙ } + ([ ] + [ ] [K ][ ]){q } = {0}
(6)
[ M˜ ]{q˙˙ } + [K˜ ]{q } = {0}
(7)
T
1
1
1
1
1
2 1
T
1
1
1
As can be seen by the delta terms in Equation 6, the nominal system modes do not diagonalize the updated system matrices. This is accomplished by performing a final eigensolution using the matrices of Equation 7 to determine the modes of the updated system.
[K˜ ]{} [ M˜ ]{} = {0} 2 2
(8)
Computing the solution at the reduced set of coordinates for the nominal system modes gives the updated system modes as a linear combination of nominal system modes.
{q1} = []{q2}
(9)
[2 ] = [1][ ]
(10)
The relationship expressed by Equation 9 defines the necessary condition for this method to be applied: the differential mass or stiffness must result in the updated system modes being linear combinations of the nominal system modes. To put it in other terms, the modal vectors of the nominal and updated systems must span the same space. For Ares 1-X, this method is intended to transform the modal data from the test configuration to that representing a fixed boundary configuration. This can be accomplished in either of two ways: (1) adding stiffness to the interface degrees of freedom (DOFs) to approach a cantilevered system or (2) adding inertia to the boundary DOFs to approach a seismic mass-type configuration. Prior to testing for Ares 1-X commenced, the planned adjustment method was run through a comprehensive example. Theoretical Example A simple beam model was used for the example problem. Each station along the length of the beam contained six degrees of freedom. The beam was free at one end and attached to ground at the other with springs in six DOFs. These springs are meant to represent the unknown stiffness of the test boundary. Of particular interest are the rotational spring stiffnesses about either axis perpendicular to the axial direction of the beam. These are the “rocking” DOFs and are the most influential to the bending behavior of the beam. Initial attempts at correcting boundary motion using this method required a prohibitively large number of modes to be kept in the updated eigensolution (Equation 8). As the number of modes retained was reduced, the results of the update process deviated further from the true fixed-interface configuration. Adding a significant number of modes to the target mode set for Ares 1-X tests was not feasible, so an analytical solution was pursued. Appending additional analytical flexible body vectors to the measured set was seen as problematic as they are products of the analytical model that is assumed to be in error. Generating vectors associated with rigid body motion of the test article requires only knowledge of the structures geometry (assumed to be accurate). However, assigning a frequency to these vectors is still required, but that can be done using only the stiffness of the test boundary (estimation of which will be discussed later) and not that of the test article. Furthermore, for the case of a determinant interface, these vectors are more closely associated with the boundary motion that is contaminating the test data, as they are directly associated with the interface DOFs.
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1032 With the addition of the rigid body vector to the test vectors, the generalized matrices of Equations 3b and c take the form of those in Equations 11a and b.
[ M ] = [t
T
rb ] [ M ][ t
I M = [ ] T[ M] [ rb ] [ ][ t ]
[K ] = [t
[t ] [ M ][rb ] T [rb ] [ M ][rb ] T
T
rb ] [K ][t
12 ] [ K = [ ] T [rb ] [K ][ t ]
rb ] (11a)
rb ]
[t ] [K ][rb ] T [rb ] [K ][rb ] T
(11b)
The update method for Ares 1-X is best illustrated by assuming the physical DOFs are partitioned between those corresponding to the interface and those that are part of the test article. Equation 12 shows the eigenproblem corresponding to the mass update problem. The delta matrix is made large enough such that the updated frequencies and modal vectors are converged. A simple diagonal matrix can be used with elements increased iteratively until the resulting shapes and frequencies converge. The frequencies resulting from the solution to Equation 12 are the fixed-interface modal frequencies with additional zero values corresponding to the rigid body (or suspension) vectors. The eigenvectors can be transformed back to physical space using the test modes and analytical rigid body vectors (Equation 13).
0
[K ]{} 22[ M ] + []0
0 []{} = {0} M bb
(12)
[ ] = [][]
(13)
fb
An analogous form for the stiffness update option can be formed in a similar manner.
Figure 1: Tip Displacement Ratio Metric
In order to assess the affects of interface motion on the test measurements, the degree of interface motion must be quantified. As a comparison tool, a tip displacement ratio metric was devised. The total motion of the test article is assumed to be the sum of a flexible portion due to deformation of the structure and a rigid portion due to interface motion. The tip displacement ratio metric is defined as the ratio of the rigid portion of this motion to the total motion (Figure 1, Equation 14). It is noted that this metric is valid only for the pair of first bending mode shapes. This is appropriate as these are the modes most sensitive to the interface stiffness.
μt =
l d
d
l
(14)
This update process was performed using a range of interface stiffnesses resulting in corresponding amounts of interface motion. The method converges very near the
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1033 fixed-interface modes for smaller amounts of interface motion. As the interface motion increases, however, large errors in converged frequency relative to the fixedinterface values are present (Figure 2). It is concluded that there may exist some amounts of interface motion that cannot be corrected to a sufficient error bound using this update process. This sensitivity also can be used to define an acceptable amount of interface motion where no correction is needed based on an error budget. This acceptable value can then be compared to what is measured in test to determine if adjustments to the test results are necessary. These sensitivities are not absolute and require calculation for each new structure or configuration encountered.
Figure 2: Adjustment Sensitivity To Interface Motion
To better simulate a real world application of the method, errors were introduced into the analytical stiffnesses of the ground and test article. The update process was then applied to models with varying amounts of error. Stiffness and modal errors are tracked separately and the resulting differences between converged updated frequencies and the truth (cantilevered) model are recorded in a three-dimensional plot (Figures 3a and b). In both plots (mass or stiffness update) there is a valley of small frequency error (approximately two percent) around the intersection of no modeling errors. The valleys show that the proposed update method can converge to acceptable values (given some acceptable error) even in the presence of modeling errors. The steep gradient to the right in each plot shows that grossly overestimating the interface stiffness results in large errors for the converged values. This should be avoided by using the sensitivities defined in Figure 1. Figure 3: Update Method Results In the Presence of Noise
(a) Mass update
(b) Stiffness update
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1034 The process described in the Figure 4: Interface Stiffness Sensitivity preceding requires an estimate of the stiffness of the test article and the test boundary. The estimate for the test article is generally derived from a pre-test finite element model (FEM). Often such an estimate for the test boundary is not available. However, this stiffness can be estimated directly from the test data if the proper sensitivities are calculated beforehand. Figure 4 shows the relationship between interface stiffness and the tip displacement ratio (for the primary bending modes) for this example problem. The upper portion of this curve represents a very soft boundary as the test article motion is due almost entirely to rigid body motion relative to the interface. The far right is a very stiff boundary as the tip displacement ratio is very small. In the middle region deformation of both the boundary and test are exhibited. If the tip displacement ratio is calculated from the test measurements, this sensitivity curve can be used to estimate interface stiffness by reading across from the measured tip displacement ratio to the curve and then down to the interface stiffness value. Again, these sensitivities are problem dependent, and it is the responsibility of the analyst to define appropriate interface parameters. Adjustment of SS5 Data The mode survey test of SS5 was performed in a configuration that included non-flight support hardware below its interface to the Upper Stage Simulator (USS). This interface was chosen as that at which to perform the update procedure discussed in the preceding text. In this manner, all influence of non-flight hardware can be removed from the test data, resulting in parameters equivalent to those that would be estimated from a fixed-base test of SS5. Calculation of the tip displacement ratio metric from the SS5 test data resulted in values of 11 to 12 percent rigid body motion. This was significantly higher than the analytical values calculated using the FEMs of SS5 and the non-flight support hardware. Cross-orthogonality matrices were calculated for the test configuration and the cantilevered (fixed interface) configuration (Tables 1 and 2). Comparing the tables between the two configurations, it can be seen that the comparison between test and model is better when the non-flight support hardware is removed; the shape matches improve, and the modal frequency differences are reduced. This is particularly evident for the first pair of modes which are the most sensitive to interface flexibility. The second pair of corrected modes actually corresponds closest to the third pair in the test configuration. The fact that the test-tomodel correlation is worse for the test configuration is evidence that there are errors present in the model of the support hardware. This highlights the advantage of the adjustment method. By adjusting the modal test resutls, the modeling of the non-flight hardware can be ignored, and all effort for model adjustment can be focused on the flight hardware. The third pair of bending modes for the cantilevered configuration could not be estimated as there was an insufficient number of modes measured to correct this pair.
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1035 Table 1: SS5 Test-To-Model Cross Orthogonality (Test Configuration)
Table 2: SS5 Test-To-Model Cross Orthogonality (Cantilevered Configuration)
Adjustment of SS1 Data The configuration for the SS1 mode survey test consisted of the SS1 flight hardware resting on shims and pads to an I-beam frame structure, with the I-beam frame in turn resting on concrete piers built into the Vehicle Assembly Building (VAB) floor. The configuration resulted in a tip displacement ratio of approximately 85 percent. This degree of interface motion is beyond the region of application for the adjustment method and thus was not applied. Conclusions The Ares 1-X modal test program encountered significant technical challenges as a result of program constraints. One such challenge was incorporating modal testing into the normal integration progress without the use of dedicated test fixtures. An analytical method was proposed to adjust modal test data to negate effects of interface motion. This method was shown to work on an example problem, and subsequently applied with success to the SS5 modal test results. Reference 1. Tuttle, R. E., Lollock, J. A., and Hwung, J. S., “Identifying Goals For Ares 1-X Modal Testing”, International Modal Analysis Conference XXVIII.
BookID 214574_ChapID FM_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Finite Element Model Calibration Approach for Ares I-X Lucas G. Horta 1 , Mercedes C. Reaves1, Ralph D. Buehrle 2 , Justin D. Templeton2, Daniel R. Lazor 3 , James L. Gaspar 4 , Russel A. Parks3, and Paul A. Bartolotta 5 ABSTRACT Ares I-X is a pathfinder vehicle concept under development by NASA to demonstrate a new class of launch vehicles. Although this vehicle is essentially a shell of what the Ares I vehicle will be, efforts are underway to model and calibrate the analytical models before its maiden flight. Work reported in this document will summarize the model calibration approach used including uncertainty quantification of vehicle responses and the use of nonconventional boundary conditions during component testing. Since finite element modeling is the primary modeling tool, the calibration process uses these models, often developed by different groups, to assess model deficiencies and to update parameters to reconcile test with predictions. Data for two major component tests and the flight vehicle are presented along with the calibration results. For calibration, sensitivity analysis is conducted using Analysis of Variance (ANOVA). To reduce the computational burden associated with ANOVA calculations, response surface models are used in lieu of computationally intensive finite element solutions. From the sensitivity studies, parameter importance is assessed as a function of frequency. In addition, the work presents an approach to evaluate the probability that a parameter set exists to reconcile test with analysis. Comparisons of pre-test predictions of frequency response uncertainty bounds with measured data, results from the variancebased sensitivity analysis, and results from component test models with calibrated boundary stiffness models are all presented. 1 INTRODUCTION The process of model calibration involves reconciling differences between test and analysis. This process requires both heuristics and quantitative methods to assess model deficiencies, to evaluate parameter importance, and to compute required changes. Although model update (calibration) has been a very prolific area of research in the US and abroad, no single technique is universally accepted and at times, when viewed from a deterministic viewpoint, Avitabile in [1] referred to it as a problem with endless possibilities. For years, the commercially available tools to address the update problem relied on sensitivity information to judge the relative importance of parameters and to assist in making model changes. Friswell in [2] discusses many of the conventional sensitivity based approaches, some implemented in many commercial tools. These tools, in the hands of experienced engineers, provide heuristic approaches for model calibration that work very well in reconciling differences between test and analysis, but often provide unrealistic parameter changes and give little insight into the probabilistic nature of the problem. This aspect of the problem that has prompted extensive research with noteworthy contributions from Hasselman et al., [3]; Herendeen et al., [4]; Alvin, [5]; Farhat and Hemez, [6], aimed at addressing uncertainty. Hasselman discussed propagation of parameter uncertainty in frequency response calculations and presented various approaches to handle variability of response values near dynamic resonant conditions. Herendeen et al. [4] discussed a mathematical procedure using multi-disciplinary optimization to conduct analysis/test calibration studies of frequency response data. Alvin [5] extended a procedure developed by Farhat and Hemez, [6] to improve convergence and to incorporate uncertainty information into the estimation process, taking full advantage of the model structure and sensitivity. To their credit, very high dimensional problems have been calibrated successfully using these techniques. However, in the end the question about realism of updated parameters is still unanswered. To properly address this question one would need to exploit the work that Montgomery [7] has done in terms of design of experiments and Analysis of Variance (ANOVA) as a means to judge parameter adequacy. In the work of Uebelhart, [8] tools from modern designs of experiments are relied heavily for uncertainty quantification and parameter selection. Regardless of the parameter selection approach, engineering judgment will always play a key role, and these tools are available to guide the analyst. In the computational fluid dynamics (CFD) area, work by Oberkampf [9] and Roach [10] is leading the verification and validation of mathematical models effort. In this area, techniques for systematic assessments of model 1
NASALangleyResearchCenter,MS230,Hampton,VA23681 NASALangleyResearchCenter,MS424,Hampton,VA23681 3 NASAMarshallSpaceFlightCenter,Huntsville,AL35812 4 NASALangleyResearchCenter,MS434,Hampton,VA23681 5 NASAGlennResearchCenter,MS49ͲB,Cleveland,OH44135 2
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_92, © The Society for Experimental Mechanics, Inc. 2011
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1038 adequacy and uncertainty quantification have been established along with standards, and a common language. Unfortunately, these efforts and the methods associated with them are just now slowly migrating to other disciplines; see, for example, Thacker [11]. The principles for model verification and validation set forth by the CFD community are applicable to many other disciplines but the metrics for assessment need to be modified. Using current terminology, the work presented here is primarily a model calibration effort. Fundamental to the success of the model calibration effort is a clear understanding of the ability of a particular model to predict the observe behavior even when the observed behavior is uncertain. The approach proposed in this paper is focused primarily on model calibration using parameter uncertainty propagation and quantification, as opposed to a search for a reconciling solution. The process set forth follows a two-step approach. First, ANOVA is used for parameter sensitivity, which is followed by uncertainty propagation to evaluate uncertainty bounds and to gage the ability of the model to explain the observed behavior. Once this process is completed, optimization is used to find a reconciling solution. This approach was demonstrated by Horta [12]. Key to uncertainty propagation is the calculation of both single and multi-parameter variances. Because of the computational burden associated with variance-based sensitivity estimates, the approach in this report uses response surface models to estimate the frequency response functions (FRF) from which the variance is computed. Response surface modeling is based on the Extended Radial Basis Functions (ERBF) as described by Mullur [13-14]. To quantify the agreement between test and analysis for dynamic problems, it is common to use FRF data. Normally hundreds of sensors are placed on a structure to identify frequencies and mode shapes. If FRFs are used as calibration metrics, the number of metrics equals the product of the number of sensors times the number of inputs, which is often a large number. To reduce the number of metrics used for comparison, the Principal Values (PV) (also known as Principal Components (PC)) of the FRF are used instead. Incidentally, to expand this metric across multiple models, PV uncertainty bounds for the maximum of all maxima and the minimum of all minima are easily computed along with the probability of observing such values as a function of frequency, see Horta [15]. These bounds are then used to determine the probability that a solution exists that reconciles test results with analysis. To obtain a parameter set that reconciles the model with test, nonlinear optimization as described by Lewis [16] is used with a quadratic objective function to minimize the error between test and analysis. Hasselman [17] also used PC in a procedure he calls a PC-based statistical energy analysis, for generic uncertainty quantification and comparison of frequency responses. Our approach is very similar with the addition of PV maximum and minimum bounds. Future work needs to incorporate a probabilistic performance metric. 2 DESCRIPTION OF ARES I-X VEHICLE The Ares I-X vehicle shown in figure 1, consists of the First Stage (FS), Upper Stage Simulator (USS), and the Crew Module and Launch Abort System (CM/LAS). This demonstration vehicle uses heritage shuttle first stage booster technology with newly designed components to mimic the mass loading of the Ares I vehicle. Because heritage hardware is used, and due to scheduling issues, only two components of the vehicle are tested prior to the flight vehicle modal test. Stack 5 (SS5), shown in figure 1, is the first major component tested. To minimize interference with the vehicle assembly schedule, SS5 is tested on a non-flight interface adapter while awaiting integration into the flight vehicle. This non-flight boundary interface adapter, facilitates access and transportation, but is not part of vehicle. Nonetheless, a finite element model (FEM) of the adapter is also part of the model evaluation. Stack 1 (SS1) is the second major component tested and is shown in Fig. 1. It is comprised of the Interstage 2, th Interstage 1, Frustum, Forward Skirt Extension, Forward Skirt, and 5 Segment Simulator. This component is also tested while resting on a metal frame at the Vehicle Assembly Building (VAB) at the Kennedy Space Center. Because of the unconventional boundary conditions for both component tests, data analysis presented some unusual challenges. Finally, a flight vehicle modal test is completed with the vehicle resting on hold-down posts on the Mobile Launcher Platform (MLP), shown in figure 2. This configuration is the closest to the flight configuration, and therefore, it is the most relevant configuration for guidance and navigation model verification. 3 ARES I-X FINITE ELEMENT MODEL DESCRIPTION Figure 2 shows the Ares I-X flight vehicle FEM. The FS, USS, and CM/LAS finite element models were delivered by independent product teams (IPT). The three FEMs are connected with rigid elements (RBE2) at each interface.
1039 Between each of the three FEMs there are 24 connection points. These models reference a global coordinate system and have a unique numbering system to simplify the model integration effort. The Ares I-X modeling approach is consistent with current modeling practices for this type of structure. The vehicle skin is modeled using shell elements (CQUAD4 and CTRIA3), and section flanges and support bracing are modeled using beam elements. Lump masses such as nose tip, engines, splice plates, and umbilical are modeled using concentrated masses and are attached to the vehicle using rigid bar elements (RBE2). Constraint elements (RBE3) are used to attach platforms, secondary structures, and concentrated masses. Joints on the upper stage and first stage segments as well as ground and non-flight hardware boundary interface for the SS5 and SS1 test configurations are modeled with CBUSH spring elements. 4 PROBLEM FORMULATION Model calibration begins with an initial assessment of the model adequacy for its intended use. For example, if the model is being developed to predict loads, detailed analysis of critically loaded regions must be emphasized. On the other hand, if the model is developed to support control design, critical structural modes must be measured and calibrated against analysis. The intended use determines the type of test and calibration process to follow. Metrics for the vehicle are established by the Guidance and Navigation Group in terms of allowable excursions from the nominal Ares I-X free-free model. Specifically, discrepancies in the frequency from test and st analysis must be less than 10% for the 1 bending pair and less than 20% for higher frequency modes; node st locations for the 1 three bending pairs must be within +/- 100 inches of the nominal, and modal displacement differences must be less than 20% for the first bending pair and 50% for higher frequency modes. Although these metrics are for the free-free configuration, they are also used to evaluate the vehicle model on the MLP. The ability to observe the modes of interest of any modal test depends heavily on proper pre-test analysis, adequate sensor count, and sensor/shaker placement. Readers are referred to Buehrle [18] for details on the pre-test analysis and test configuration. Finally, the calibration process followed during test is shown in figure 3. Specific elements of the process are described next. 4.1 Parameter Selection This step is perhaps the most difficult one because it requires first-hand knowledge of the assumptions and approximations used during model development. Also, because heritage hardware components are involved, those components and their parameters have not been considered for calibration; specifically, the first stage booster and the MLP. Collecting the necessary information for this step requires interviews with model developers and complete familiarization with all model elements.. After selecting an initial parameter set, the next step is the selection of parameter uncertainty models. These uncertainty models describe any a priori knowledge of the parameter variations in the form of distribution functions. In this problem, since there is no uncertainty information available, uniform distributions functions are used throughout the paper. As a consequence, all parameter updates resulting from this process are equally acceptable. Furthermore, parameter bounds are selected to assure that updates are acceptable from an engineering viewpoint. The uncertainty propagation and quantification step requires executing the FEM multiple times with parameter variations as prescribed by the uncertainty model. This phase reveals outcomes that are highly probable, determines bounds for the response of interest, and is aimed at answering the question: what is the probability that the model can predict the measured data? Probability assessments are all based on discrete probabilities computed using a prescribed number of FEM runs. For example, using solutions obtained from n different parameter sets, the discrete probability of observing a particular output is simply 1/n. Our goal in selecting a parameter set is to make the probability of capturing the measured response high. In other words, to find a parameter set that makes the analysis bounds as encompassing as possible. Results from the parameter uncertainty study are used directly to create response surface models for use with a variance-based approach to determine parameter importance. Once the important parameters are identified, our ability to calibrate the model is assessed in terms of probability bounds. If the probability of finding a reconciling solution is greater than zero (i.e., test is within the uncertainty bounds), parameter updates are sought using the nonlinear optimization. Otherwise the parameter selection and bounds must be re-visited and modified. 4.2 Analysis of Variance Parameter sensitivity in most engineering fields is often associated with derivative calculations at specific parameter values. However, for analysis of systems with uncertainties, sensitivity studies are often conducted using ANOVA. In classical ANOVA studies, data is collected from multiple experiments while varying all parameters (factors) and also while varying one parameter at a time. These results are then used to quantify the output response variance due to variations of a particular parameter as compared to the total output variance
1040 when varying all the parameters simultaneously. The ratio of these two variance contributions is a direct measure of the parameter importance. Sobol in [19] and others [20-22] have studied the problem of global sensitivity analysis using variance measures. Sobol developed the "so called" Sobol indices to provide a measure of parameter sensitivity. In his formulation the variance is computed using the following expressions;
P#
1 N
N
¦ y(Q ) i
i 1
D P2 # Dx P 2 #
1 N
N
2
¦y
1 N
(Q i )
(1)
i 1
N
¦ y (Q ) y( x , z ) i
i
i
i 1
where P is the mean value of the response of interest y, D is the total variance computed using N samples, and Dx is the single parameter variance due to parameter x. Note that, Qi refers to the ith sample of the parameter vector and xi is a parameter within Qi about which the variance is being evaluated, whereas zi are all other parameters that comprise Qi not including xi. To properly evaluate Eq. (1), one needs at least 2N function evaluations; one where all variables are randomly sampled, and a second set where all but xi are re-sampled. With this information the 1st Sobol index is computed as Sx Dx / D . Of course, for problems with m parameters, there are m possible first order factors. Equation (1) is easily extended to study two or more parameter (factor) interactions, as described in [19], simply by adjusting the number of parameter that gets re-sampled. Only first factors are studied in this paper. Depending on the type of structure being analyzed, FEM and their solutions can be computationally intensive. Because the parameter selection process relies on statistical analysis of the response data, it is important that no FEM solution is wasted. A way to capture information from every computed solution, as parameters are changed, is to use response surface models, i.e., surrogates. For this purpose, the Extended Radial Basis Function ERBF as developed by Mullur [13-14] is used. With this surrogate, it is now possible to conduct ANOVA in cases where computing thousands of FEM solutions directly is prohibitively expensive. A final note on ANOVA using Sobol's approach regards convergence of the variance estimates. Although Ref. [21] discusses the asymptotic behavior of the variance estimates when using Eq. (1), for cases where a surrogate model is used instead of the FEM, the variance estimates are only as good as our surrogate model. Nonetheless, this approach provides an excellent way to rank variable importance when only a limited number of FEM solutions are possible. 4.3 Goodness Metrics Metrics for dynamic problems are more appropriately provided in terms of mode shapes, frequencies, and FRF. One commonly used metric is the orthogonality criterion. This metric compares mode shapes extracted from measured FRF with FEM mode shapes weighted using the analytical mass matrix. Because this metric has been historically used, it is one of two metrics computed and reported. A second metric compares differences in the FRF from test and analysis directly. For optimization and uncertainty quantification studies this second metric is preferred and used for model calibration. When studying the effect of parameter changes, Sobol is used to evaluate variances in the FRF at each frequency due to parameter variations. Instead of comparing each input/output pair, data is compared in terms of the Principal Values (PV) of the FRF. These PVs are computed as the singular value decomposition of the FRF matrix at each frequency. For cases with multiple singular values, only the maximum and minimum are compared. 5 DISCUSSION OF RESULTS Results are presented for the component test first to make it easier for the reader to follow. Results for Stack 5 are presented first followed by Stack 1, and then the flight vehicle.
1041 5.1 Stack 5 Model Calibration The first major component tested is the Command Module and Launch Abort System (SS5) as shown in figure 4. To facilitate access to the interior of the structure while on the ground, a service access structure (bottom yellow section) is mated and tested with the SS5 hardware, as shown in the photograph. Although this section is not part of the flight hardware, it is modeled, tested, and reported with the results. Figure 5 shows pre-test prediction st of the 1 three pairs of bending modes with their corresponding frequencies. Results from an independent traceability study by Tuttle [23] that considered subsystem coupling of SS5 with the flight vehicle concluded that although several modes of SS5 contribute to the total strain energy, the 1st bending pair is the most important set and therefore should be the focus for this component test. Analysis of Variance for SS5- is one of two tools used for selection of critical parameters. To begin this analysis, a second order ERBF response surface model with 8 parameters, as defined in Table 1, is initially created from 400 FEM eigenvalue solutions to provide modal frequencies and mode shapes as a function of boundary stiffness. From these 400 runs, the FRF is synthesized for displacements outputs and is used to evaluate FRF variations as a result of parameter variations. Initially, FEM runs are executed with parameter values uniformly distributed between the bounds shown in Table 1. By construction, the ERBF surrogate model matches each of the 400 solutions exactly. To study parameter contributions to the FRF variance, the ERBF surrogate is used instead of the FEM to generate hundreds (often thousands) of predictions and to compute the Sobol indices according to Eq. (1). Figure 6 shows the results from the Sobol calculation as a function of frequency with N set to 1000 in equation 1. Colors are used to depict contribution of the individual parameter to the total variance where the total contribution from all 8 parameters is less than or equal to 1. Frequency values with no colors are simply areas where the variance is not computed. As an example on how to use this information, suppose that changes to regions lower than 14 Hz are needed, then changes to parameters KV1 and KV3 (vertical stiffness of quadrant I and III) will produce the largest variation in the maximum principal value of the FRF. Note that quadrants, labeled in figure 6 using roman numerals I-IV, correspond to parameters labeled 1-4. For reference, at the top of figure 6 is a plot of the maximum PV as a function of frequency (with the nominal model) to indicate resonant frequencies near 4 Hz, 12 Hz, and 26 Hz. This sensitivity information is critical to understand how to properly set parameters before any optimization is performed. More importantly, it provides users with a per frequency map of parameter importance. Although individual parameters can dominate the variance in certain frequency ranges, variance alone does not provide information on the magnitude of the changes. To study magnitude variations, the approach using PV bounds is presented next. SS5 Principal Value Uncertainty Analysis- The objective of this step is to assess variations in the FRF and to determine FRF bounds due to parameter variations. Using 400 parameters sets the maximum and minimum principal value across all models are computed and plotted as a function of frequency. These results are used not only as bounds for the measured FRF but also to estimate probabilities. Intuitively, if the response from 400 FEM models falls within the bounds then the probability of measuring a value outside the bounds must be less than 1/400, if the parameter uncertainty model is adequate. Figure 7a shows in solid-red the analysis bounds due to variations in the boundary stiffness computed prior to test, and the blue-dashed line shows the test data PV. Although the test data is from 58 accelerometers (with two shakers), the acceleration FRFs are scaled using frequency to compare results in terms of displacements. Incidentally, all analytical modes are assumed to have damping levels of 0.5%. Two observations are in order regarding the results shown in figure 7a: 1) variations in the PV values due to boundary stiffness changes are so small that the bounds looks just like the PV (see figure 8a), and 2) since our test data is outside the uncertainty bounds, the model representation is not adequate. The problem with this model is that the structure is on the ground without physical constraints, which makes the boundary stiffness unknown (epistemic uncertainty). It is clear that our initial estimate is incorrect and requires a second look. Because of project time constraints, model calibration for SS5 concentrated exclusively on the boundary stiffness as the main source of discrepancies. After a post-test look at the model, it is determined that the boundary stiffness is two orders of magnitude lower than initially estimated. In addition, a discrepancy was found involving inertia properties used to simulate the engine mass. After adjusting the boundary stiffness uncertainty model as shown in Table 2 and correcting for the simulated engine inertias, results with the updated bounds are shown in figure 7b along with the test data PV. With these corrections, the measured response is now within the analysis st uncertainty bounds (for the 1 two pair of modes) and parameter optimization can now proceed to determine stiffness parameters to reconcile test with analysis. Figure 8a shows the PV for the pre-test nominal model (solid-
1042 red) and test (dashed-blue); whereas figure 8b shows results with the optimized boundary stiffness. Although results now are more in line with the measured FRF data, improvements are also seen in the orthogonality results. Figure 9a shows orthogonality results comparing the pre-test FEM mode shapes (ordinate) with mode shapes identified from test (abscissa); whereas, figure 9b shows results with the optimized boundary parameters. When examining these results, recall that orthogonality values range from 0 to 1 with black squares corresponding to values of 1; i.e. black squares indicate exact match. It is clear that updates to the boundary stiffness helped correct for errors in the 1st pair of modes principal direction as well as reducing their frequency error. With this correction, the 1st mode frequencies are within 2.9%. Since the traceability study in [23] indicated that only the first bending mode pair is critical, no further updates are pursued. Because models with uncertainty are being compared to test, test uncertainty is also critical. Unfortunately, in order to properly address this question one would need additional data, which for economic reasons have not been collected. Of the small number of data sets collected, PV values for test data is studied to determine if there is significant variability across different tests. With the limited number of tests, this variability, although not shown in the paper, is not significant enough to warrant additional tests. 5.2 Stack 1 model calibration Stack1 is the second major component tested and is shown in figure 10. In this configuration, pre-test predictions of target modes are shown in figure 11 with the boundary modeled using springs to ground. As with SS5, results from an independent traceability study identified modes 1, 2, 6, 7 and 8 as our target modes for this component test. SS1 Analysis of Variance- begins again by creating a second order ERBF response surface model with 8 parameters from 300 FEM eigenvalue solutions computed while varying the boundary stiffness. In this case the boundary is also divided into quadrants and stiffness parameters for each quadrant are varied independently. As before, the FRF response is synthesized for displacements outputs. To evaluate variations in the FRF as a result of parameter variations, FEM runs are executed with parameter values uniformly distributed between the bounds shown in Table 3. Variance results for Stack 1 are shown in figure 12 for each of the 8 boundary stiffness parameters. Again, only results for the variance of the maximum PV are shown; the nominal model PV is shown at the top of figure 12. In contrast to results for Stack 5, for frequencies less than 5 Hz, the lateral stiffness in all quadrants contribute significantly to the total variation. SS1 Principal Value Uncertainty Analysis- For 300 parameter sets, the maximum and minimum PV across all FEM models are computed and plotted as a function of frequency. Figure 13a shows in solid-red the FEM PV bounds due to variations in the boundary stiffness computed prior to test and the blue-dashed line shows the test data PV. For this case, test data from 64 accelerometers (with 2 shakers) are used to recover the FRF and then converted to displacement FRF for comparison. It should be obvious that the measurements are outside the pretest model uncertainty bounds, and consequently the model is not adequate. As with SS5, the boundary stiffness is found to be two orders of magnitude softer than the initial estimate. After adjusting the uncertainty model for the boundary, the updated parameter ranges are defined in Table 4. With this updated set, the computed uncertainty bounds are now shown in figure 13b; solid-red is analysis and dashed blue is test. Damping levels for the model are again set to 0.5 % for all modes. Since the boundary stiffness is unknown, it is selected in such a way as to ensure that the test data is captured within the bounds for the target modes. Figure 13b shows this assumption to be true; and, therefore, one can proceed to compute updated parameters using optimization. For comparison, figure 14a shows the pre-test maximum and minimum PV with the nominal model in solid-red versus test in dashed-blue. Similarly, Figure 14b shows the results for SS1 with the optimized boundary parameters, shown in Table 4. Note that PV matching of test with analysis is improved significantly for the 1st mode pair. Finally, orthogonality results for SS1 are shown in figure 15a using the pre-test model and 15b uses the optimized boundary stiffness. Concentrating on the 1st pair, the optimized values corrected a problem with the principal directions and also reduced the frequency errors. Although high frequency modes are difficult to observe with the limited instrumentation suite, qualitative assessments of test modes with orthogonality values greater than 0.8
1043 revealed that many of these modes resemble modes found in the analysis. No further updates are performed due to time constraints and the need to prepare for the flight vehicle test. 5.3 Flight Test Vehicle (FTV) Model Calibration The flight vehicle calibration process follows the same steps as those described with the component tests. Since the Ares I-X vehicle is tested while mounted on four hold-down posts on the MLP, as opposed to free-free, target modes in this configuration are the 1st four bending mode pairs. Figure 16 shows only the first mode of each pair. It is assumed that after calibrating the analytical model of the vehicle on the MLP, the MLP can be removed analytically to predict the free-free configuration. In addition, damping values obtained from this test can be allocated to the free-free modes. FTV Analysis of Variance- Unlike the SS1 and SS5 component tests, the FTV boundary condition is relatively well defined. Consequently, parameters selected for uncertainty analysis and their variations must be supported with data collected from various sources. After a careful study of potential sources of uncertainty, the parameters defined in Table 5 are selected. The seven parameters used in this study included the USS joint stiffness, FTV/MLP interface stiffness, forward skirt Young modulus XL_SK_E, the CM/LAS Young modulus LAS_E, as well as ballast mass densities B_US_1/B_US_7. Ranges used are based on collected information regarding modeling assumptions and uncertainty in the information collected. Figure 17 shows results for the variance analysis with the seven parameters. Note that the variance for the first two modes (about 0.2 Hz) is dominated by ballast densities, whereas modes between 3 to 4 Hz are dominated by the USS joint stiffness. FTV Principal Value Uncertainty Analysis- Due to time constraints, only 200 FEM runs are used to compute the PV bounds. Figure 18 shows results for the FEM analysis uncertainty bounds in solid-red and test in dashed blue. Two aspects of these results should be highlighted; 1) variations in the PV due to parameter variations are relatively small, and 2) the uncertainty bounds captures relatively well the measured data with only a few exceptions. Damping for the analysis is again assumed to be 0.5 % for all flexible modes. Figure 19 shows the PV using the pre-test nominal model (solid-red) as compared to the measured values (dashed-blue). Again the agreement is good. A final check examines orthogonality results of the pre-test nominal model with the identified modes from test. Results for this metric are shown in figure 20. Along the abscissa are test mode frequencies and along the ordinate are analysis frequencies. Modes circled correspond to target modes that need to be verified for controls. Frequency errors for those modes are under 6.5 %. Table 6 shows a summary of the identified frequencies and measured damping ranges (in brackets) using only random excitation data. With this information and the requirements initially provided by the guidance and navigation team, the nominal Ares I-X model is found to be adequate for evaluation of the flight control system without need for additional parameters adjustments. Although there are certain areas where the model can be improved, these areas should not impact guidance performance of the vehicle based on the initial requirements. 6 CONCLUDING REMARKS A procedure to conduct model calibration for the Ares I-X vehicle has been presented The approach uses a variance-based approach for parameter selection, nonlinear optimization to minimize the error between test and analysis, and multiple FEM models to bound the system response and to assess the probability of finding a reconciling solution. To alleviate the computational expense in performing variance-based sensitivity analysis, the approach uses a surrogate model to predict changes in the frequency response functions as parameters are varied. Uncertainty in the parameters and their effect on the frequency response function is studied in terms of Principal Values. Uncertainty bounds of the principal values are established across multiple models to allow one to determine the probability of finding a solution that reconciles test with analysis results. Model adequacy is easily ascertained by comparing the measured responses against PV uncertainty bounds. Because of the use of unconventional boundary conditions during Stack 5 and Stack1 tests, the model calibration effort concentrated on boundary stiffness uncertainty characterization. In both cases, the initial boundary stiffness estimates were two orders of magnitude lower than initially thought. After adjusting the uncertainty models for the
1044 boundary stiffness to properly capture the measured data, optimization is used to find a set of parameters that reconciles test with analysis. This optimized parameter set is found to minimize the error between the test and analysis FRF. Because uncertainty data is not available, it is impossible to make probabilistic statements regarding confidence in the optimized boundary stiffness values. Although, the boundary stiffness parameters do not alter vehicle parameters directly, agreement of test with analysis for the target modes improved significantly after boundary adjustments. Likewise, our confidence in the nominal model is also improved. Finally, results for the Ares I-X vehicle are presented and compared to predictions using the nominal model with uncertainty bounds computed prior to test. These results show that the nominal model is remarkably close to the pre-test predictions and complies with model verification metrics provided by the guidance and navigation group. For this reason, changes to the nominal model are not recommended. 7
ACKNOWLEDGEMENTS
Special thanks to the FEM development group for their continued support: Winifred Feldhaus (LaRC,FTV), Genevieve Dixon (LaRC,CM/LAS), Troy Mann (LaRC,CM/LAS), Richard Pappa (LaRC,CM\LAS), Nickolas A. Vitullo (ATK, weight), Andrew Panetta (LaRC,weight/cg), William K. Thompson (GRC,upper stage) , Jeremy Redden (ATK,first stage) , Carl Pray (ATA,uncertainty analysis, Jones Trevor (GRC,upper stage) , Jennifer Lange (GRC,upper stage). Thanks to our independent verification team of Jeff Lollock, Ryan Tuttle, and Joshua Hwung from Aerospace Corporation. Finally thanks to the KSC team; Russ Brucker, Trip Healey, Frank Walker, Stephanie Heffernan, Teresa Kinney, Todd Reeves, Kara Schmitt, Mark Tillett, and Karl Meyer. 8 REFERENCES th 1. Avitabile, P.: “Model Updating- Endless Possibilities.” Proceedings of the 18 International Modal Analysis Conference, San Antonio, TX, Feb. 2000, pp. 562-570. 2. Friswell, M.I., and Mottershead, J.E.: Finite Element Model Updating in Structural Dynamics, Kluwer Academic Press, Dordrecht, 1995. 3. Hasselman, T.K., Chrostowski, J.D., and Ross, T.J.: “Propagation of Modeling Uncertainty Through Structural th Dynamic Models,” Proceedings of the 35 AIAA/ ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 1994, Paper No. AIAA 94-1316, SC. 4. Herendeen, D.L., Woo, L., Hasselman, T.K., and Zimmerman, D.C.: “Analysis-Test Correlation and Model Updating of Dynamic Systems Using MDO Software Tools,” Proceedings of the 7th AIAA/USAF/NASA/SSMO Symposium on Multidisciplinary Analysis and Optimization, 1998, Paper No. 98-4730, St. Louis, MO. 5. Alvin, K.F.: “Finite Element Model Update via Bayesian Estimation and Minimization of Weighted Residuals,” AIAA Journal, Vol. 35, No. 5, 1997. 6. Farhat, C. and Hemez, F.M.: “Updating Finite Element Dynamic Models Using an Element-by-Element Sensitivity Methodology,” AIAA Journal, Vol. 31, No. 9, 1993, pp. 1702-1711. th 7. Montgomery D.C., Design and Analysis of Experiments, John Wiley & Sons, New York, 5 ed. 2001 8. Uebelhart, S.A., Miller, D.W., and Blaurock, C.: “Uncertainty Characterization in Integrated Modeling.” th Proceedings of the 46 AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, & Materials Conference, April 2005, Austin TX., AIAA 2005-2142. 9. Oberkampf, W.L., Trucano, T.G., and Hirsch, C.: “Verification, Validation, and Predictive Capability in Computational Engineering and Physics.” SAND 2003-3769, Feb. 2003. 10. Roach, P.P., Verification and Validation in Computational Science and Engineering, Hermosa Publishers, Albuquerque, NM, 1998. 11. Thacker, B.H.: “The Role of Non-determinism in Computational Model Verification and Validation.” Proceedings of the 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, & Materials Conference, April 2005, Austin TX, AIAA 2005-1902. 12. Horta, L.G., Reaves, M.C., and Lew, J.-S.: “A Procedure for Static and Dynamic Model Update of Finite Element Models: Application to an Inflated/Rigidized Torus.” Proceedings of the IMAC XXIV: Conference and Exposition on Structural Dynamics, 2006, St. Louis Missouri. 13. Mullur, A. and Messac, A.: “Extended Radial Basis Functions: More Flexible and Effective Metamodeling.” AIAA Journal, Vol., 43, No. 6, June 2005. 14. Mullur, A. and Messac, A.: “Metamodeling Using Extended Radial Basis Functions: A Comparative Approach.” Engineering with Computers (2006) 21: 203-217 DOI 10.1007/s00366-0005-7.
1045 15. Horta, L.G., Kenny, S.P., Crespo, L.G., and Elliott, K.B.: “NASA Langley’s Approach to the Sandia’s Structural Dynamics Challenge Problem.” Computer Methods in Applied Mechanics and Engineering, Vol. 197, Issues 29-32, May 2008, also available at www.sciencedirect.com. 16. Lewis. R.W., Shepherd A., and Torczon, V.: “Implementing Generating Set Search Methods for Linearly Constrained Minimization.” SIAM Journal on Scientific Computing, Vol. 29, No. 6, pp 2507-2530. 17. Hasselman T., Yap K., Yan H. and Parret, A.: “Statistical Energy Analysis by Principal Components for MidFrequency Analysis.” Proceedings of the 43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, April 2002, Denver, CO. AIAA 2002-1395. 18. Buehrle, R.D., Templeton, J.D., Reaves, M.C., Horta, L.G., Gaspar, J.L, Bartolotta, P.A., Parks, R.A., and Lazor, D.R,: “Ares I-X Launch Vehicle Modal Test Overview.” Proceedings of IMAC XXVIII, Jacksonville, FL., February 1-4, 2010. 19. Sobol, I.M., Tarantola, S., Gatelli, D., Kucherenko, S.S, and Mauntz, W.: "Estimating Approximation Error When Fixing Unessential Factors in Global Sensitivity Analysis," Reliability Engineering and System Safety 92 (2007) 957-960, © 2006 Elsevier LTD. 20. Mullershon, H., and Liebsher, M.: “Statistics and Non-Linear Sensitivity Analysis with LS-OPT and DSPEX.” th 10 International LS-DYNA Users Conference, pp. 4-1,4-13. 21. Homma, T, and Saltelli, A.: “Importance Measures in Global Sensitivity Analysis of Nonlinear Models.” Reliability Engineering and System Safety 52 (1996) 1-17, © 1996 Elsevier LTD. 22. Sudret, B.: "Global Sensitivity Analysis Using Polynomial Chaos Expansion," Reliability Engineering and System Safety 93 (2008) 964-979, © 2007 Elsevier LTD. 23. Tuttle, R., Lollock, J.A, and Hwung, J.S.: “Identifying Goals for Ares I-X Modal Testing,” Proceedings of IMAC XXVIII, Jacksonville, FL., February 1-4, 2010. Table 1 Stack 5 Pre-test parameter definition No. 1 2 3 4 5 6 7 8
Lower Bound
Nominal
Upper Bound
4.84E+07 4.84E+07 4.84E+07 4.84E+07 4.84E+07 4.84E+07 4.84E+07 4.84E+07
6.06E+07 6.06E+07 6.06E+07 6.06E+07 6.06E+07 6.06E+07 6.06E+07 6.06E+07
6.06E+07 6.06E+07 6.06E+07 6.06E+07 6.06E+07 6.06E+07 6.06E+07 6.06E+07
Parameter ID Kv1 Kl1 Kv2 Kl2 Kv3 Kl3 Kv4 Kl4
(lbs//in) (lbs//in) (lbs//in) (lbs//in) (lbs//in) (lbs//in) (lbs//in) (lbs//in)
Table 2 Stack 5 Post-test parameter definition and optimized solution No. 1 2 3 4 5 6 7 8
Parameter ID Kv1 Kl1 Kv2 Kl2 Kv3 Kl3 Kv4 Kl4
(lbs//in) (lbs//in) (lbs//in) (lbs//in) (lbs//in) (lbs//in) (lbs//in) (lbs//in)
Lower Bound
Nominal
Upper Bound
Optimized Solution
5.45E+05 5.45E+05 5.45E+05 5.45E+05 5.45E+05 5.45E+05 5.45E+05 5.45E+05
6.06E+05 6.06E+05 6.06E+05 6.06E+05 6.06E+05 6.06E+05 6.06E+05 6.06E+05
3.33E+06 3.33E+06 3.33E+06 3.33E+06 3.33E+06 3.33E+06 3.33E+06 3.33E+06
5.45E+05 6.01E+05 3.33E+06 1.09E+06 5.47E+05 5.94E+05 3.33E+06 3.33E+06
1046 Table 3 Stack 1 pre-test parameter definition No.
Parameter ID
1 2 3 4 5 6 7 8
KQ2RTh (lbs/in) KVQ2 (lbs/in) KQ3RTh (lbs/in) KVQ3 (lbs/in) KQ4RTh (lbs/in) KVQ4 (lbs/in) KQ1RTh (lbs/in) KVQ1 (lbs/in)
Lower Bound
Nominal
Upper Bound
1.00E+06 1.00E+06 1.00E+06 1.00E+06 1.00E+06 1.00E+06 1.00E+06 1.00E+06
1.00E+07 1.00E+07 1.00E+07 1.00E+07 1.00E+07 1.00E+07 1.00E+07 1.00E+07
1.00E+08 1.00E+08 1.00E+08 1.00E+08 1.00E+08 1.00E+08 1.00E+08 1.00E+08
Table 4 Stack 1 Post-test parameter definition and optimized solution No. 1 2 3 4 5 6 7 8
Parameter ID KQ2RTh (lbs/in) KVQ2 (lbs/in) KQ3RTh (lbs/in) KVQ3 (lbs/in) KQ4RTh (lbs/in) KVQ4 (lbs/in) KQ1RTh (lbs/in) KVQ1 (lbs/in)
Lower Bound 2.50E+05 2.50E+05 2.50E+05 2.50E+05 2.50E+05 2.50E+05 2.50E+05 2.50E+05
Nominal 3.75E+05 3.75E+05 3.75E+05 3.75E+05 3.75E+05 3.75E+05 3.75E+05 3.75E+05
Upper Bound 1.00E+06 1.00E+06 1.00E+06 1.00E+06 1.00E+06 1.00E+06 1.00E+06 1.00E+06
Optimized Solution 1.00E+06 8.03E+05 1.00E+06 2.50E+05 1.00E+06 6.32E+05 1.00E+06 4.27E+05
Table 5 Flight vehicle pre-test parameter ranges No.
Parameter ID
Lower Bound
Nominal
Upper Bound
1 2 3 4 5 6 7
USS_J (lbs/in) FTV/MLP_V (lbs/in) FTV/MLP_L (lbs/in) B_US_1 (lbs-s2/in4) B_US_7 (lbs-s2/in4) XL_Fwd_Skirt_E (lbs/in2) LAS_E (lbs/in2)
4.84E+06 2.43E+08 2.43E+08 6.44E-04 5.63E-04 2.70E+07 8.91E+06
1.00E+07 2.70E+08 2.70E+08 7.32E-04 7.32E-04 3.00E+07 9.90E+06
2.63E+07 2.97E+08 2.97E+08 8.27E-04 8.20E-04 3.30E+07 1.09E+07
1047 Table 6 Summary of flight vehicle identified frequencies and damping Mode
Freq. (Hz)
Damp. (%)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.18 0.21 1.06 1.19 1.87 2.07 2.91 3.46 3.64 3.68 4.61 4.78 6.17 6.41 6.66
[0.48, 0.83] [0.36, 0.44] [0.28, 0.35] [0.39, 0.43] [2.22, 2.95] [1.28, 1.43] [0.96, 1.40] [0.43, 0.52] [0.37, 0.41] [0.38, 0.41] [0.17, 0.18] [0.23, 0.25] [1.35, 1.38] [0.66, 0.69] [1.29, 1.40]
Y-X 1st bend Z-X 1st bend Y-X 2nd bend Z-X 2nd bend Z-X bend &MLP Y-X bend & MLP 3rd bend % MLP 3rd Y-X bend 3rd Z-X bend Torsion 4th bend (45deg) 4th bend X-Y system system system
Figure 1 Ares I-X vehicle schematic and subcomponents tested
1048
Figure 2 Finite element model schematic of the flight vehicle
Figure 3 Model calibration process
1049
SS5 Test Hardware
FEM Model
Figure 4 Ares I-X Command Module and Launch Abort System (CM-LAS) 4.63-4.66 Hz
13.4-14.4 Hz
26.2-26.3 Hz
z y
x
Figure 5 Pre-test predictions of first three pairs of bending modes
z
y x
Quadrants
Figure 6 Parameter contributions to variance of the 1st principal value due to longitudinal and vertical boundary stiffness variations.
1050
-4
-4
10
Analysis Max Analysis Min Test Max Test Min
-5
10 Principal values
-5
10 Principal Values
10
Analysis Max Test Max Analysis Min Test Min
-6
10
-7
10
-6
10
-7
10
-8
10
-8
10
-9
10
0
1
10
-9
10
2
10 Frequency (Hz)
10
0
1
10
a) pre-test model bounds
2
10 Frequency
10
b) Updated bounds
Figure 7 Comparison of SS5 principal value bounds for a) pre-test model and b) after updates to parameter ranges
Principal Values
10
10
10
10
10
-4
10 Analysis Max Test Max Analysis Min Test Min
-5
10 Principal values
10
-6
-7
-8
10
10
10
-9
10
0
1
10
2
10 Frequency (Hz)
10
-4
Analysis Max Analysis Min Test Max Test Min
-5
-6
-7
-8
-9
10
a) pre-test nominal model
0
1
10 Frequency
10
2
b) optimized
0.9
52.02 48.08 47.88 46.39 44.74 42.65 41.90 41.86 41.28 40.61 38.45 38.44 36.60 35.20 26.25 26.06 14.72 12.19 4.68 4.60
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 1
2
3
4
5
a) pre-test model
6 7 8 9 Test Mode No.
10
11
12
13
14
27.17
26.71
26.59
26.14
24.77
24.06
23.58
23.30
23.13
9.90
22.64
8.84
4.32
4.18
27.17
26.71
26.59
26.14
24.77
24.06
23.58
23.30
MASS Ortho Frequency (Hz)
23.13
9.90
22.64
8.84
4.32
MASS Ortho Frequency (Hz)
Analysis Freq. (Hz)
Analysis Freq. (Hz)
4.18
Figure 8 Comparison of SS5 principal values for a) pre-test model b) model with optimized boundary parameters
45.49 44.49 43.90 41.65 41.29 39.72 38.37 38.28 37.99 36.78 34.91 33.34 30.66 26.44 25.12 25.04 9.24 8.73 4.38 4.30
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 1
2
3
4
5
6 7 8 9 Test Mode No.
10
11
12
13
14
b) optimized
Figure 9 SS5 orthogonality results with a) pre-test model and b) with optimized boundary parameters
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Figure 10 Ares I-X SS1 test article
4.12Hz, 4.19 Hz
22.8Hz, 23.29
24.09Hz
Figure 11 SS1 pre-test predictions
Quadrants
Figure 12 SS1 parameter contributions to variance of the 1st principal value due to boundary stiffness variations
1052
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10
10
Analysis Max Analysis Min Test Max Test min
Principal values
10 Principal values
-5
10
-6
10
-7
10
10
10
10
-8
10
0
1
10
10
2
10 Frequency
10
-4
Analysis Max Analysis Min Test Max Test min
-5
-6
-7
-8
-9
10
a) Pre-test model bounds
0
1
10 Frequency
10
2
b) updated bounds
Figure 13 Comparison of SS1 PV bounds for a) pre-test model and b) updated boundary stiffnes bounds
-4
10
10
Analysis Max Test Max Analysis Min Test Min
Principal Values
10 Principal values
-5
10
-6
10
-7
10
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10
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10
-4
Analysis Max Analysis Min Test Max Test Min
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-6
-7
-8
10
a) Pre-test model
1
0
10 Frequency
10
2
b) optimized boundary
47.57 47.32 44.66 44.05 43.42 42.64 39.95 37.98 30.20 29.51 27.46 26.54 23.26 22.71 22.04 17.32 16.73 16.18 4.14 4.07
0.9 0.8 0.7
0.6
0.5
0.4
0.3
0.2 0.1 1
2
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7 8 9 10 11 12 13 14 15 16 17 18 Test Mode No.
a) Pre-test model
25.56
24.78
23.70
22.35
21.74
21.50
21.03
19.97
16.98
16.79
16.35
15.95
15.18
12.86
8.75
11.34
2.03
1.64
25.56
24.78
23.70
22.35
21.74
21.50
21.03
19.97
16.98
16.79
Frequency (Hz)
16.35
MASS Ortho
Frequency (Hz) 15.95
15.18
12.86
8.75
11.34
2.03
MASS Ortho
Analysis Freq. (Hz)
Analysis Freq. (Hz)
1.64
Figure 14 Comparison of SS1 PV with a) pre-test model and b) optimized boundary stiffness model
44.00 43.69 43.38 42.22 38.05 38.05 32.36 31.25 30.19 27.10 25.59 19.63 19.42 19.08 17.51 17.08 14.18 11.09 2.05 1.67
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 1
2
3
4
5
6
7 8 9 10 11 12 13 14 15 16 17 18 Test Mode No.
b) updated
Figure 15 SS1 orthogonality results with pre-test model and with updated boundary stiffness model
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0.176 Hz
1.017 Hz
3.248 Hz
4.78 Hz
Figure 16 Flight vehicle target modes
st
Figure 17 Parameter contributions to variance of the 1 PV for the flight vehicle
-2
10
Analysis Max Analysis min Test Max Test min
-4
Principal values
10
-6
10
-8
10
-10
10
-1
10
0
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1
10
Figure 18 Comparison of flight vehicle PV bounds for pre-test model and test (assumed damping for model is 0.5 %)
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-4
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-6
10
-8
10
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10
-1
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0
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1
10
Figure 19 Comparison of flight vehicle PV for pre-test model and test
Figure 20 Flight test vehicle orthogonality results with pre-test and updated model
BookID 214574_ChapID 93_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Model Reduction and Substructuring Techniques for the Vibro-Acoustic Simulation of Automotive Piping and Exhaust Systems Jan Herrmann, Michael Junge and Lothar Gaul Institute of Applied and Experimental Mechanics, University of Stuttgart Pfaffenwaldring 9, 70569 Stuttgart, Germany
ABSTRACT The influence of the acoustic field on the structural dynamics is a common issue in automotive applications. An example is the pressure-induced structure-borne sound of piping and exhaust systems. Efficient model order reduction and substructuring techniques accelerate the finite element analysis and enable the vibro-acoustic optimization of such complex systems with acoustic fluid-structure interaction. This research reviews the application of the Craig-Bampton and Rubin method to fluid-structure coupled systems and presents two automotive applications. First, a fluid-filled brake-pipe system is assembled by substructures or superelements according to the Craig-Bampton method. Fluid and structural partitions are fully coupled in order to capture the interaction between the pipe shell and the heavy fluid inside the pipe. Second, a rear muffler with an air-borne excitation is analyzed. Here, the Rubin and the Craig-Bampton method are used to separately compute the uncoupled component modes of both the acoustic and structural domain. These modes are then used to compute a reduced model which incorporates full acoustic-structure coupling. For both applications, transfer functions are computed and compared to the results of dynamic measurements.
Nomenclature M , K, D, C s, a f (t), q(t) p, u q I, F ρ0 Φ
CB ˆ Φ ˆ Ψ Hp→u
1
Mass matrix, stiffness matrix, damping matrix, acoustic-structure coupling matrix Index for solid partition, index for acoustic partition Structural forces, nodal acoustic fluxes Nodal acoustic excess pressures, nodal displacements Modal coordinates Index for interface DOF, index for free (inner) DOF Fluid mass density Eigenspace of the constrained problem – fixed interface modes Superscript for reduced component model matrices Craig-Bampton method Free-interface modes used for Rubin method Attachment modes used for Rubin method Transfer function between pressure at inlet and structural deflection
Introduction
Elastic piping and exhaust systems are often characterized by (hydro-)acoustic sources. An example for a hydroacoustic excitation in hydraulic pipes such as fuel and brake-pipes is the operation of pumps and valves which leads to oscillating pressure pulsations within the pipe. As a result, pressure waves propagate along the pipe and excite the pipe shell due to the heavy fluid-structure coupling [1, 2, 3]. Finally, the pressure-induced structure-borne sound is transmitted to attachment structures which leads to undesired noise and vibration levels [4]. A similar excitation scenario is found in automotive exhaust systems where the exhaust gas acts as a strong acoustic source which also leads to pressure-induced structure-borne sound and undesired sound radiation [5].
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_93, © The Society for Experimental Mechanics, Inc. 2011
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To predict the vibro-acoustic behavior of such mechanical systems, three-dimensional models including full coupling of the two-field problem are needed. It is particularly important to include bending modes of the structure, which are predominantly responsible for sound and vibration harshness. The finite element method [6] is considered as the appropriate discretization method to investigate the dynamics of the interior vibro-acoustic problem including the coupling between the inner fluid and the pipe shell. The boundary element method might be used to determine the sound radiation in the exterior field [7]. The main problem of fully discretized models are large computation times and extensive computer memory. Model order reduction and substructuring techniques such as the well-known component mode synthesis overcome this limitation. This research shows how the Craig-Bampton [8] and the Rubin method [9] are applied to efficiently compute the hydroand vibro-acoustic response of two automotive applications. First, a fluid-filled brake-pipe system is analyzed which is characterized by a heavy fluid-structure coupling between the water inside the pipe and the flexible pipe shell. The CraigBampton method is applied which leads to a considerable model order reduction and to moderate computation times. Moreover, the influence of a cross-section change on the hydraulic transfer function is analyzed and the results are validated by a hydraulic test bench operating in the kHz-range. Secondly, an automotive exhaust system is analyzed which is an example of light fluid-structure coupling between the exhaust gas and the structure of the expansion chamber. Here, the Rubin and the Craig-Bampton are used to separately determine the uncoupled component modes of both the acoustic and structural domain. These modes are then used to compute a reduced model including full acoustic-structure coupling. Measurements are performed and the vibro-acoustic response is compared to the results of the numerical simulation including the proposed model reduction technique.
2
Finite Element Based Substructuring Techniques
This chapter briefly summarizes the application of the Craig-Bampton and the Rubin method on fluid-structure coupled systems. Both methods are used to reduce the order of the corresponding finite element model of each component and to assemble the overall mechanical system. The linear wave equation for a fluid at rest is used for low Mach numbers as encountered in most piping and exhaust systems [1]. The acoustic field is coupled to the structural dynamic equations at the fluid-structure interface [6], where two coupling conditions hold, namely the Euler equation and the reaction force axiom. The corresponding finite element formulation leads to coupled discretized equations in terms of nodal structural displacements u and nodal acoustic excess pressures p Ms 0 ¨ u Ds 0 u˙ K s −C u f (t) = . (1) + + ¨ p 0 Da p˙ 0 Ka p q(t) ρ0 C T M a In the above equation, index “s” denotes the structural partition, whereas index “a” characterizes the acoustic fluid. The discretized equations include mass matrices M s,a , stiffness matrices K s,a , viscous damping matrices Ds,a , as well as the coupling matrix C. Forces and fluxes are given as f (t) and q(t). Hereby, the classical unsymmetric formulation with displacements and pressures as field variables is used. It is worthy of note that alternative representations use the acoustic velocity potential as field variable [10] which leads to a symmetric formulation of Eq. (1).
2.1
Craig-Bampton Method
The adaptation of the Craig-Bampton method to mechanical systems with acoustic fluid-structure coupling has been developed in [11]. For clarity, the critical steps are briefly summarized in this section. To apply the Craig-Bampton method to the acoustic-structure coupled problem, displacement and pressure DOFs are both separated into interface DOFs with T T T T T index “I” and inner (free) DOFs denoted by index “F”, i.e. u = [uT I uF ] and p = [pI pF ] , respectively. The matrices are partitioned accordingly. As explained in [8], the Craig-Bampton reduction basis consists of fixed interface modes and constraint modes. The fixed interface modes are obtained by solving the eigenvalue problem for the system with constrained interface DOFs M s,FF 0 ˆ j,F u K s,FF −C FF 2 − ωj (2) ˆ j,F = 0. p 0 K a,FF ρ0 C T M a,FF FF The reduction bases Φ are enriched by the fixed interface modes up to the m-th eigenvector or a certain frequency threshold (which is usually at least two times the maximum frequency of interest), such that ˆ m,F ] and Φa = p ˆ 1,F , ... , p ˆ m,F . A modification of an iterative subspace solver is applied to compute Φs = [ˆ u1,F , ... , u the eigenvectors [12, 1]. The eigenspaces are mass-normalized in order to avoid ill-conditioning or numerical break-down.
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The constraint modes follow from a static of Guyan condensation [13]. The resulting reduction bases Γs and Γa are built according to Craig-Bampton method by discarding any coupling terms, i.e. I 0 uI uI = = Γs q s (3) uF ud −K −1 s,FF K s,FI Φs,FF for the structural domain and
pI pF
=
I
−K −1 a,FF K a,FI
0 Φa,FF
pI pd
= Γa q a
(4)
for the fluid domain, respectively. Hereby, index “d” denotes dominant modal coordinates as compared to neglected or truncated modal coordinates. Hence, q s and q a are the generalized coordinates of the reduced system. The overall reduction basis is assembled by u Γs 0 qs = (5) p 0 Γa qa
Γ∈Rn×nr
and reduces the coupled system in Eq. (1) to nr n DOFs M s 0 ¨s q K s + T ¨a q 0 ρ0 C Ma
−C K a
qs qa
=
fs fa
(6)
where M s C
T M a = ΓT a M a Γa , K s = Γs K s Γs,
= ΓT s M s Γs , =
ΓT a CΓs ,
fs =
ΓT s f
fa =
K a = ΓT a K a Γa ,
(7)
ΓT a q.
Eq. (6) describes the superelement for the component mode synthesis. It is important to note that both structural and fluid interface DOFs are kept as physical DOFs which simplifies the component coupling procedure and which allows the integration of an impedance boundary condition in the reduced equations. The Craig-Bampton method is particularly efficient for piping systems, where the number of interface DOFs between the components is small compared to the inner DOFs and where repeating superelements occur. An additional reduction of the remaining interface DOFs leads to a further computational speedup and may be applied as explained in [3, 14, 15]. The coupling between the reduced component models is defined by single point contraints on the component interfaces. This holds for both the solid and the fluid domain, and implicit coupling conditions are established. They are used for the generation of an explicit transformation matrix Q to transform the component coordinates in coordinates of the assembled piping system [16]. This coupling procedure allows the subsequent summation of nsub substructure contributions. An alternative way to couple the superelements using Lagrangian multipliers is shown in [17]. The global dynamic system of equations with the global reduced coordinates q = [q s q a ]T is given as ¨ + D g q˙ + K g q = f g , M gq
(8)
whereas the system matrices M g and K g are assembled as
Mg =
n sub i=1
Kg =
QT s,i M s,i Qs,i T ρ0 Qa,i C T i Qs,i
n sub i=1
QT s,i K s,i Qs,i 0
0 , QT a,i M a,i Qa,i −QT s,i C i Qa,i . QT a,i K a,i Qa,i
(9)
(10)
The global damping matrix is given as Dg =
n sub i=1
T αs,i QT s,i M s,i Qs,i + βs,i Qs,i K s,i Qs,i 0
0 , βa,i QT a,i K a,i Qa,i
(11)
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assuming a Rayleigh damping model for each substructure with the corresponding Rayleigh damping parameters αs and βs for the structural domain and βa for the fluid partition. For fluid-filled pipes, a considerable improvement of the fluid damping model is achieved using an advanced modeling approach including wall friction effects which is the dominant damping mechanism in thin pipes. The advanced fluid damping model is based on a complex wave number and incorporates the frequency dependent wall friction between the acoustic fluid and the pipe shell. A complete derivation of the improved fluid damping model and its integration in the finite element analysis is beyond the scope of this paper. The interested reader is refered to [18, 19]. In the frequency domain, Eq. (8) is given as
ˆ = fˆg , −M g ω 2 + iωDg + K g q
(12)
such that a harmonic analysis is performed using the inverse of the dynamic stiffness matrix as transfer function. The results are expanded to full space in order to obtain the transfer function of interest.
2.2
Rubin Method
In contrast to the Craig-Bampton method, the Rubin method is a free-interface method, i.e. neither the interface DOFs ˆ which is assembled nor the free DOFs are additionally constrained for the computation of the component modes in Γ, by free-interface normal modes and attachment modes [9]. In what follows, the basic principle of the Rubin method is explained on behalf of the structural domain. The free-interface normal modes are computed by solving the eigenvalue problem of the unconstrained system
2 ˆ sj = 0 . −ωj M s + K s Φ
(13)
Analogue to the Craig-Bampton method, only a small number of free-interface normal modes are retained. Attachment ˆ s , augment the component modes matrix accounting for the modal truncation error. The i-th attachment mode modes, Ψ is defined by the static solution vector due to a single unit force applied to the i-th interface DOF K s,II ˆ Ψsi = K s,FI
K s,IF K s,FF
−1 [0
...
1 ...
0]
T
.
(14)
ˆ is then given by In other words, attachment modes are columns of the associated flexibility matrix. The reduction basis Γ ˆs = Φ ˆs Ψ ˆs . Γ (15) Attachment modes cannot be computed directly, if the structure pocesses rigid body degrees of freedom. Then, instead of the standard attachment modes, inertia-relief attachment modes represent one alternative [20]. The coupling between the reduced component models is analogue to the Craig-Bampton method. Since for the CraigBampton method all interface DOFs are retained, the displacement coupling conditions are typically fulfilled more accurately than with the Rubin method. Yet, if experimentally determined modal damping ratios are to be incorporated in the simulation model, the Rubin method is favorable. The free-floating boundary conditions of the free-interface normal modes can be realized in an experimental setup. Thus, the obtained modal damping ratios may be mapped directly.
2.3
Combination of Craig-Bampton and Rubin Method for Light Fluid-Structure Coupling
It is observed, that for structures with high impedance mismatch between the structure and the contained fluid, the coupled eigenfrequencies and eigenvectors are altered only marginally compared to the uncoupled ones. Therefore, in such a case, the reduced-order model is constructed by using the uncoupled eigenvectors, since they span approximately the same subspace as the coupled eigenvectors This is equivalent to reducing each domain separately. Please note, that the reduced-order model is still a fully coupled system. This approach has the advantage that for each domain the favorable reduction method might be applied. If for example, the Rubin method is applied for the reduction of the structure and the Craig-Bampton method is used for the fluid the reduction basis reads ˆs Ψ ˆs 0 Φ 0 ˆ Γ= . (16) 0 0 Φa Ψa ˆ are obtained by the solution of the uncoupled problem. Please note, that all component modes vectors in Γ
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3
Applications
So far, the application of substructuring techniques to problems with acoustic-structure coupling has been elaborated. Now, two industrial applications are presented, where the different model reduction techniques are applied.
3.1
Hydroacoustic Analysis of Automotive Piping Systems
The first example is a fluid-filled piping system which is characterized by a heavy fluid-structure coupling between water inside the pipe and the flexible shell. The Craig-Bampton method as explained in Section 2.1 is used as model reduction and substructuring technique to solve the fully coupled system equations. The focus of this section is the hydroacoustic analysis of thin piping systems as found in brake and fuel pipes. 3.1.1
Straight Pipe With a Cross Section Change
A water-filled piping system with a cross section change is assembled using three substructures as shown in Fig. 1 (steel pipe: E=206 GPa, ρs =7900 kg/m3 , water: c = 1460 m/s, ρ0 =1000 kg/m3 ). The first superelement is a straight pipe section (length 262 mm) with an outer radius of 3 mm and a wall thickness of 0.7 mm. The second substructure is an orifice with a length of l = 4 mm which is characterized by a cross section change as depicted in Fig. 1. The inner diameter of the orifice is d = 1.2 mm. The third component is another straight pipe section (length 305 mm) with the same uniform cross section as the first substructure and a closed end. To achieve a compatible mesh between the component interfaces, the substructure with the small inner pipe radius includes a thin element layer at both ends to enable the transition between the two different pipe diameters as illustrated in Fig. 1(a). It is also important to model the casing around the orifice which is needed according to the design. The piping system is clamped on the left end and an acoustic pressure excitation is applied as boundary condition. The reduced system has 550 DOFs as opposed to 8762 DOFs of the full FE model. The additional interface reduction as introduced in [3] further reduces the model order to 389 DOFs. p1
1
2
3 p2 pipe PC
solid
measurement system + Matlab-interface charge ampl. di da l d
vacuumpump
p0
fluid FSI pump (a) substructuring
p1 orifice
p2
pipe pressure source power ampl. function generator
(b) experimental setup for validation
Figure 1: Left: assembled piping system using three substructures including boundary conditions and cut through the discretized Substructure 2 with the cross section change. Right: experimental setup of the hydraulic test bench. The experimental setup of the hydraulic test bench is illustrated in Fig. 1(b) and is also described in [21, 3]. The setup consists of a hydroacoustic pressure source and a fluid-filled steel pipe with the dimensions mentioned before. The pipe is characterized by a cross section change due to an orifice as shown in the technical drawing of Fig. 1(b). The pressure source consists of two piezo stack-actuators which are arranged perpendicularly to the direction of wave propagation and on opposite sides of the hydraulic pipe. The actuators are driven by a power amplifier and a function generator and oscillate with opposite phase in order to excite pressure pulsations in the fluid column. The supply pipe with the additional pump is required to fill and compress the fluid to ensure a stable fluid column without any air bubbles, which is a challenging task especially for small orifice diameters. The dynamic pressure pulsations are measured with piezoelectric pressure sensors. A sweep excitation is chosen in order to excite a wide frequency range. The repeated sweep signals have a cycle duration of 200 ms. The hydraulic transfer function between the pressure p1 and the pressure p2 is estimated using the averaged auto-power and cross-power spectra of 15 swept pressure signals.
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The hydraulic transfer function Hp1→p2 is measured using the hydraulic test bench and the result is compared to the finite element simulation using the substructuring technique as described before. The result of the hydraulic or hydroacoustic transfer function for the pipe with an orifice diameter of d = 1.2 mm and an orifice length of l = 4 mm is shown in Fig. 2. The coherence of the measurement is plotted in the same figure. The correlation between experiment and simulation is very good, both for the hydraulic resonances and the predicted damping. A kinematic viscosity of ν = 1 · 10−6 m2 /s is assumed for the advanced fluid damping model [18]. Two coupled modes are visible where a structural resonance of the pipe shell interacts with the acoustic fluid. The first coupled peak around 1.5 kHz is characterized by a longitudinal mode of the pipe structure. The second coupled mode with the typical antiresonance behavior around 5 kHz strongly depends on the pipe section with the orifice. It is worthy to note that a cross section change as investigated in this section leads to a shift of the hydraulic resonance frequencies to lower frequencies when compared with a uniform pipe. The reason for this phenomenon is the increasing hydraulic inertia for decreasing orifice diameters. The frequency shift is particularly strong for odd-order modes.
experiment Craig−Bampton
|Hp1→p2 | [dB]
40 30 20 10 0
−10 −20
γ2
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 1 0.5 0
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 frequency [Hz]
Figure 2: Top: hydraulic transfer function – comparison between experiment and simulation for a fluid-filled pipe with an orifice. Bottom: coherence of the measurement.
3.1.2
Elbow Pipe With Attachment Part
The Craig-Bampton method is now applied to a more complex piping system consisting of a curved brake pipe (E=206 GPa, ρs = 7900 kg/m3 , ν = 0.3, lengths 0.7 + 0.3 m) with an outer radius of 3 mm and a wall thickness of 0.7 mm, two steel joints (the so-called clips) and a plate as target structure (E=180 GPa, ρs = 7900 kg/m3 , ν = 0.3, lengths 0.3 x 0.3 x 0.001 m). The pipe is filled with water (c = 1460 m/s, ρ0 = 1000 kg/m3 ). The fluid wave strongly interacts with the flexible pipe shell, which leads to a fluid wave speed of 1408 m/s as opposed to the free sound speed of c = 1460 m/s. This phenomenon is described by Korteweg’s equation [22]. The investigated pipe configuration is of practical importance since pressure pulsations in the fluid and strong fluid-structure coupling result in a structural excitation of the brake pipe. The pressure-induced structure-borne sound is transferred to the target structure by the clips as explained in [4]. Fig. 3 shows the pipe configuration assembled by 9 substructures (five straight pipe sections, one elbow, two clips and the plate). The application of the Craig-Bampton method reduces the model order from 55818 DOFs to 2566 DOFs. The additional interface reduction leads to 1118 DOFs. To validate the overall simulation method, the measured and computed hydroacoustic (or hydraulic) transfer function Hp1 →p2 is shown in Fig. 3. The measurement is conducted with the same setup as explained before. Again, the correlation between experiment and simulation is very good, both for the obtained hydraulic resonances and the predicted damping in the fluid path. Strong acoustic-structure coupling is observed for a frequency around 1800 Hz, where a hydraulic and a structural resonance coincide. Another fluid-structure coupled mode is visible around 3600 Hz. Note that the observed coupled modes are very sensitive with respect to the structural configuration and the pipe mounting position.
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p2 full: 55818 DOFs CB: 2566 DOFs CB+interface-red.: 1118 DOFs
|Hp1 →p2 | [dB]
30 20 10 0
−10
−30 0 1
1000
2000
3000
4000
2000
3000
4000
γ2
p1
experiment Craig−Bampton
−20
0 0
1000
frequency [Hz] (a) assembled piping system
(b) result
Figure 3: Left: finite element model of the assembled brake-pipe system and boundary conditions (vibration mode at 1030 Hz). Right: measured and computed hydraulic transfer function |Hp1 →p2 | (top) and coherence of the measurement (bottom).
3.2
Vibro-Acoustic Analysis of an Exhaust System
In this section, pressure-induced vibrations of a production series rear muffler as depicted in Fig. 4 are investigated. Please note, that for simplicity the inner structural parts of the rear muffler are removed. The periodically blown out exhaust gas leads to pressure pulsation within the exhaust system. These pulsations excite structural vibrations, which then additionally contribute to the sound radiation of the system. It is reported that this so-called surface radiated noise might dominate the noise radiated at the orifice [23, 24, 25, 5]. In this work, the focus is set on the pressure-induced vibrations and not on the sound radiation.
(a)
(b)
Figure 4: Picture of a production series rear muffler and corresponding CAE model. The system is depicted upside-down. The number on the right mark the location of the investigated nodes. In order to quantify the pressure-induced vibrations of the rear muffler, the transfer function, Hp→u , between the acoustic pressure at the inlet and the structural deflection at 4 locations on the surface is determined (cf. Fig. 4(b)). The acoustic pressure at the inlet is measured by making use of the two-microphone-method [26]. For the simulation a finite element model is set up with 179808 structural DOFs and 143602 fluid DOFs. For an efficient simulation a reduced model is computed as described in Section 2.3. The Rubin method is applied for the structural domain. The modal damping values obtained from an experimental modal analysis are incorporated in the damping matrix. The Craig-Bampton method is employed for the fluid domain. For each domain, 40 free-interface and fixed-interface normal modes are retained, respectively. The interface DOFs on the inlet and outlets sum up to 444 structural DOFs and 211 fluid DOFs yielding the same number of constraint modes and attachment modes, respectively. It is worth noting, that the interior acoustic fluid intersect with the surrounding fluid at the orifices. Previous investigation showed, that a radiation impedance condition approximates sufficiently accurate the occurring interaction at this cross-section [27]. The impedance
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condition yields complex entries in the damping matrix D a [28]. For the frequency sweep computations between 300 Hz and 600 Hz (with a step step size of 1 Hz) the reduced-order models clearly outperform the full-order solution. A speedup of approximately factor 100 is obtained.
10
measured simulated
-6
3
-6
|Hpinlet →usn | [m /N]
10
measured simulated
3
|Hpinlet →usn | [m /N]
The plots in Fig. 5 show a comparison between the experimental (solid, black line) and simulative results (dashed, blue line). Each subplot represents the magnitude of the transfer function Hp→u for one node on the surface of the rear muffler as depicted in Fig 4(b). A strong excitability via the the acoustic path is observed for all points – Hp→u spans more than four orders of magnitude within the depicted frequency range between 300 Hz and 600 Hz. A comparison of the eigenfrequencies with the results of an experimental modal analysis reveals that the reasonance frequencies are reached at eigenfrequencies of the structure. This explains the fact that the surface radiated noise shows a strongly tonal characteristic. All four subplots show a good agreement between experiments and simulations. It is worth noting, that the simulation is capable to predict the transfer function both qualitatively and quantitatively. The proposed method is thus suitable to efficiently predict pressure-induced vibrations in an early development stage and is capable to prevent cost-intensive modifications and time-consuming experiments.
10
10
-8
-10 300
10
10 350
400
450
500
550
-8
-10
600
300
350
400
frequency [Hz]
600
10
measured simulated
-6
3
|Hpinlet →usn | [m /N]
3
|Hpinlet →usn | [m /N]
550
(b) Node 3290
measured simulated
-6
500
frequency [Hz]
(a) Node 3287
10
450
10
10
-8
-10 300
10
10 350
400
450
frequency [Hz] (c) Node 3756
500
550
600
-8
-10 300
350
400
450
500
550
600
frequency [Hz] (d) Node 3765
Figure 5: Pressure-induced structural vibrations. Comparison of experimental and simulative results.
4
Conclusion
The Craig-Bampton and the Rubin method are successfully applied to fluid-structure coupled systems in order to achieve moderate computation times and moderate computer memory. The hydro- and vibro-acoustic response of two automotive applications is analyzed in this research showing the applicability of the described component mode synthesis. For heavy fluid-structure coupling as in the case of fuel and brake pipes, the fluid and structural partition need to be fully coupled
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to compute the corresponding component modes and to capture the strong interaction between the fluid and the flexible pipe shell. For light acoustic fluid-structure coupling as in the case of an exhaust system, uncoupled component modes of both the acoustic and structural domain can be used to compute a reduced model. The results of the numerical simulation are compared to dynamic measurements. Good agreement is achieved with respect to hydro- and vibro-acoustic transfer functions of complex fluid-structure coupled systems.
5
Acknowledgement
The authors gratefully acknowledge the funding of this project by the German Research Society DFG in the transfer unit TFB 51 and by the Friedrich-und-Elisabeth-Boysen-Stiftung. Moreover, the authors wish to thank Dr. Matthias Maess at Robert Bosch GmbH for his ongoing advice.
References [1] M. Maess. Methods for Efficient Acoustic-Structure Simulation of Piping Systems. Ph.D. thesis, Institute of Applied and Experimental Mechanics, University of Stuttgart, 2006. [2] M. Maess, J. Herrmann, and L. Gaul. Finite element analysis of guided waves in fluid-filled corrugated pipes. Journal of the Acoustical Society of America, 121:1313–1323, 2007. [3] J. Herrmann, M. Maess, and L. Gaul. Substructuring including interface reduction for the efficient vibro-acoustic simulation of fluid-filled piping systems. Mechanical Systems and Signal Processing, 24:153–163, 2010. [4] J. Herrmann, T. Haag, S. Engelke, and L. Gaul. Experimental and numerical investigation of the dynamics in spatial fluid-filled piping systems. In Proc. of Acoustics, Paris, 2008. [5] M. Junge, F. Schube, and L. Gaul. Sound Radiation of an Expansion Chamber due to Pressure Induced Structural Vibrations. In Proc. of DAGA, Stuttgart, 2007. [6] O. Zienkiewicz and R. Taylor. The Finite Element Method. Butterworth-Heinemann, Oxford, 2000. [7] D. Brunner, M. Junge, and L. Gaul. A comparison of fe-be coupling schemes for large scale problems with fluid-structure interaction. International Journal for Numerical Methods in Engineering, 77:664–688, 2009. [8] R. R. Craig and M. C. C. Bampton. Coupling of substructures for dynamic analysis. AIAA Journal, 6:1313–1319, 1968. [9] S. Rubin. Improved component-mode representation for structural dynamic analysis. AIAA Journal, 13:995–1006, 1975. [10] G. C. Everstine. A symmetric potential formulation for fluid-structure interaction. Journal of Sound and Vibration, 79:157–160, 1981. [11] M. Maess and L. Gaul. Substructuring and model reduction of pipe components interacting with acoustic fluids. Mechanical Systems and Signal Processing, 20:45–64, 2006. [12] E. Balm`es A. Bobillot. Iterative technique for eigenvalue solutions of damped structures coupled with fluids. AIAA 43. Structures, Structural Dynamics, and Materials Conference, 168:1–9, 2002. [13] R. J. Guyan. Reduction of stiffness and mass matrices. AIAA Journal, 3:380, 1965. [14] J. Herrmann, M. Maess, and L. Gaul. Efficient substructuring techniques for the investigation of fluid-filled piping system. In Proc. of IMAC XXVII, Orlando, 2009. [15] M. Junge, D. Brunner, J. Becker, and L. Gaul. Interface reduction for the Craig-Bampton and Rubin Method applied to FE-BE coupling with a large fluid-structure interfaces. International Journal for Numerical Methods in Engineering, 77(12):1731 – 1752, 2008. [16] E. Brechlin and L. Gaul. Two methodological improvements for component mode synthesis. In 25. International Seminar on Modal Analysis, Leuven, 2000. [17] D. Rixen. A dual Craig-Bampton method for dynamic substructuring. Journal of Computational and Applied Mathematics, 168:383–391, 2004. [18] J. Herrmann, M. Spitznagel, and L. Gaul. Fast FE-analysis and measurement of the hydraulic transfer function of pipes with non-uniform cross section. In Proc. of NAG/DAGA, Netherlands, 2009. [19] H. Theissen. Die Ber¨ ucksichtigung instation¨ arer Rohrstr¨ omung bei der Simulation hydraulischer Anlagen. Ph.D. thesis, RWTH Aachen, 1983. [20] R. R. Craig Jr. Coupling of substructures for dynamic analyses: An overview. In Proceedings of the AIAA Dynamics Specialists Conference, Atlanta, GA, 3–6 April 2000. Paper No. 2000-1573.
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1064 [21] J. Herrmann, T. Haag, L. Gaul., K. Bendel, and H. G. Horst. Leitungssystemen. In Proc. of DAGA, Dresden, 2008.
Experimentelle Untersuchung der Hydroakustik in Kfz-
¨ [22] D.J. Korteweg. On the velocity of sound propagation in elastic pipes (in German: Uber die Fortpflanzungsgeschwindigkeit des Schalles in elastischen R¨ ohren). Ann. Phys., 5:525–542, 1878. [23] J.-F. Brand, P. Garcia, and D. Wiemeler. Surface radiated noise of exhaust systems - calculation and optimization with CAE. In Proceedings of the SAE 2004 World Congress & Exhibition, volume 01, 2004. [24] J.-F. Brand and D. Wiemeler. Surface radiated noise of exhaust systems – structural transmission loss test rig, part 1. In Proceedings of the CFA/DAGA, Strasbourg, 22– 25 March 2004. [25] J.-F. Brand and D. Wiemeler. Surface radiated noise of exhaust systems – structural transmission loss test rig, part 2. In Proceedings of the ISMA, 20–22 September 2004. [26] A. F. Seybert and D. F. Ross. Experimental determination of acoustic properties using a two-microphone random-excitation technique. Journal of the Acoustical Society of America, 61(5):1362 – 1370, 1977. [27] H. Levine and J. Schwinger. On the radiation of sound from an unflanged circular pipe. Physical Review, 73(4):383 – 406, 1948. [28] L. Gaul, D. Brunner, and M. Junge. Coupling a fast boundary element method wih a finite element formulation for fluidstructure interaction. In S. Marburg and B. Nolte, editors, Computational Acoustics of Noise Propagation in Fluids. Springer, Berlin Heidelberg, 2008.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Examples of Hybrid Dynamic Models Combining Experimental and Finite Element Substructures
Randy L. Mayes, Michael Ross and Patrick S. Hunter * Sandia National Laboratories P.O. Box 5800 - MS0557 Albuquerque, NM, 87185 [email protected] [email protected] [email protected]
Nomenclature DOF CMS FBS FE FRF TS % I ) +& +(;3 +)( 5 V ; Ȧ Ȧ[[ ȍU T ĭ Ȍ )( (;3 PHDV
Degree of Freedom Component Mode Synthesis Frequency Based Substructuring Finite Element Frequency Response Function Transmission Simulator Constraint matrix Physical force vector Modal Force vector Frequency Response Function matrix from combined system Frequency Response Function matrix from experimental substructure Frequency Response Function matrix from finite element substructure Transformation between all modal DOF and independent modal DOF Vector of generalized degrees of freedom Vector of physical degrees of freedom Response Frequency in radians/sec Natural Frequency in radians/sec of substructure Natural Frequency in radians/sec of final combined system modal DOF kept mass normalized mode shapes from any substructure combined system mode shapes in terms of T subscript for FE substructure subscript for experimental substructure subscript for DOF at measured locations on the transmission simulator
*Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the U.S. Department of Energy under Contract DE-AC04-94AL85000.
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_94, © The Society for Experimental Mechanics, Inc. 2011
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1066 ABSTRACT Substructuring methods have been used for many years to reduce the size of FE dynamic models and maintain satisfactory response for a limited bandwidth of interest. However, experimental substructures have been used only in a very limited number of cases, generally where there was only a single connection point idealized to six connection degrees of freedom. This is because it is difficult to experimentally characterize the moments and rotations at multiple connection points. Mayes and Allen developed a method to practically characterize continuity and equilibrium at multiple connection points through the instrumentation of a flexible fixture. The instrumented fixture is transformed into a force and response sensor that is expressed in terms of the modal connection forces and modal connection responses of the fixture. The sensor is called the transmission simulator. Two previous results of the method for coupling R&D pieces of hardware are given. The primary emphasis of this paper is on a set of representative system hardware, for which two designs of a transmission simulator are compared with the view to discover design characteristics that optimize the sensor effectiveness. Frequency response functions (FRF) from the assembled system hardware are measured and designated as the truth against which to compare. The accuracy of the experimental substructure is evaluated by attaching it to a finite element substructure and predicting full system response FRFs which are compared against the truth FRFs. The result is that the transmission simulator design is relatively robust to the stiffness chosen, although the stiff transmission simulator appears to be a slightly better choice in the structure evaluated in this work. Substructure modes tend to maintain the shapes of the bare transmission simulator if it is stiff. A one piece transmission simulator design with no joints is easier to model accurately, which is of value in this methodology. INTRODUCTION AND MOTIVATION Substructuring methods generally combine reduced order dynamic models of substructures to generate a full system dynamic model. In common usage, substructures have been generated only by reducing finite element models from a large number of degrees of freedom (DOF) to a smaller number of DOF. In the reduction, the frequency response at higher frequencies is degraded. However if enough degrees of freedom are retained in the substructure, it can be satisfactory for frequency response through a desired bandwidth. In some cases there is motivation to develop experimental substructure models. If hardware is available which represents the substructure, theoretically, it may require fewer resources and less time to experimentally characterize the substructure than to develop a finite element model. The authors work with systems that are assemblies of multiple substructures from various agencies. Certain substructures are important for our agency to model because we want to use the finite element (FE) model for design improvement or to apply environmental loads to the system. Other assemblies are important to the dynamics, but our agency has no design authority for the substructure. Some of these substructures are extremely complicated, and could theoretically be developed from an experimental modal test in much less time and for much less money than to develop a FE model and validate it against a modal test. However experimental substructures are not in common use because of several difficulties with the classical approaches. These difficulties are in x x x
Measuring connection rotational DOF, Applying connection point moments, Performing measurements at multiple connection points.
In one successful application, Carne and Dohrmann[9] have developed experimental substructures with single point attachment by utilizing a rigid fixture on which enough translational response measurements are made and to which enough translational forces are applied to fully characterize the six DOF at the idealized connection point to provide the continuity and equilibrium information. Then the inertia of the fixture was subtracted. However, when there are multiple connection points, generally experimentalists have neglected moments and rotations at the connection points. In some cases rotations have been estimated from differencing multiple translation measurements. In general, the moments have not been applied in the hope that they are not important. Mayes and Allen also attempted to use the rigid fixture method to attach an experimental beam substructure to
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a FE beam substructure[4,5]. However, in their problem, the fixture was elastically deforming even in the lowest axial modes of the combined structure. They developed two techniques to address the flexible fixture that improved the results. The results from the best method for a full system axial FRF are shown in Figure 1. Mayes et al [6] realized that the flexible fixture approach could be applied to a multiple connection problem. This is because their approach reduces the continuity and equilibrium to the equivalent modal DOF and modal forces of the fixture. The instrumented fixture is now called the transmission simulator, since it is a sensor that captures measurement of both the modal forces and the modal responses of the sensor. Mayes, et al. [6] demonstrated the capability for a multiple connection system as shown in the FRFs of Figure 2. The system was a cylinder with a flange, modeled as a FE substructure, coupled to a plate with a beam, modeled from experimental measurements, shown in Figure 3. The response is an axial driving point response on the end of the beam. X Direction
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Figure 3 - Combined FE Cylinder and Experimental Plate and Beam Substructures Encouraged by the laboratory success on a real, but simple piece of hardware, a realistic set of hardware was addressed which is the subject of the rest of this paper. A FE substructure of a shell was developed which was coupled to an experimental model of a payload. In this work two different transmission simulators were designed and implemented in the modal test of the payload substructure. They are hereinafter denoted as the soft and stiff transmission simulators (TS). Comparisons will be made of the results for the two simulators, as well as the relative difficulty of the two associated modal tests.
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To begin, the application hardware and a description of the truth model will be provided. Then the TS concept will be explained. Then the soft and stiff TSs will be described and results addressed. SYSTEM HARDWARE AND TRUTH MODEL The rest of this paper is based on representative system hardware that is depicted notionally in Figure 4. The system consists of a shell substructure connected through a bolted flange to a complex internal payload substructure. The shell substructure is modeled with FE, and the internal payload is the experimental substructure. Experimental FRF measurements on the full system were acquired with axial and lateral hammer impacts at the base of the shell and axial and lateral responses measured at both the base of the shell and the top of the shell. These FRFs are the measure of truth to which the full system predictions from the coupled FE and experimental substructures will be compared.
Figure 4 - Cutaway of Full System (Force Inputs - Red, Acceleration Responses - Blue) SUBSTRUCTURE CONCEPT The TS concept is based on developing a sensor which has structural dynamics that are easily modeled analytically. In Figure 5 both substructures are represented. The experimental substructure on the left has the payload bolted to the TS that is instrumented. This substructure includes the joint between the payload and the shell. On the right is the FE substructure of the shell. This substructure includes an analytical model of the TS. Part of the analytical TS actually occupies the same space as the flange inside the shell to which the payload is bolted. The coupling of the two substructures is accomplished by forcing the two TSs to have the same motion through a constraint equation. The constraint represents the motion through linear sums of the mode shapes of the analytical model of the TS at only the measured DOF. The degree to which the analytical mode shapes span the space of the actual motion determines the accuracy of the method. The equations are written in terms of the modal DOF of the TS instead of the measured physical DOF. This allows a large reduction in the number of connection DOF. This reduction enforces a softer constraint that eliminates ill-conditioning that is associated with experimental data errors in the experimental substructure. The modal connection DOF of the TS are determined from the measurement DOF on the TS. The measurement DOF do not have to be at connection locations. The connection concept may be visualized by constraining both transmission simulators to have the
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1070 same motion using the kept modes of the TS. Then the reduced mass and stiffness matrices of the analytical TS are subtracted twice to remove its effect from the combined substructures. Both the component mode synthesis (CMS) and frequency based substructuring (FBS) theory sections using the transmission simulator are in Appendix 1 and 2. For either case, the experimental substructure is developed from a free modal test extracting modal frequencies, damping and scaled mode shapes. The rigid body modes are also required from mass properties testing, analysis or some combination.
Figure 5 - Experimental Payload Substructure (Left) and FE Substructure (Right) This method provides the following advantages: x Only translation DOF need to be measured, x The rotational DOFs are inherently captured in the modal DOF of the TS, x Excitation of the TS inherently applies moments to the connection DOF, x Measurements at the true connection DOF are not required (this is not trivial, as many times there is no way to get either an accelerometer or force sensor mounted at the connection point), x Fewer experimental mode shapes are required than with free-free modes of the experimental substructure without the simulator because mass of the TS exercises the attachment interface, x Residual flexibilities are not required (which are very difficult to measure), x The method is applicable for any number of connections with no increase in complexity (the approach works for even a continuous connection), x The stiffness and damping of the joint is naturally included in the experimental model, x Matrix inversions (for FRF substructuring methods) become practically well conditioned. Having enumerated the incredible advantages over classical experimental methods, consider also the limitations which should be investigated in future work: x The frequency bandwidth of accuracy is limited by the number of kept modes of the TS, x The frequency bandwidth of accuracy is limited by the number of modes extracted from the substructure test,
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The accuracy is limited by the condition number of the analytical mode shape matrix at the measurement DOF (as this matrix becomes difficult to invert, the accuracy degrades), The accuracy is limited by the extent that the modes of the TS span the space of the coupled substructure's true connection motion, The accuracy is limited by the fidelity of the extracted modal parameters (properly scaled shapes, frequency and damping), The method is bound by the assumption of linearity, Subtraction of the TS can lead to reduced order mass or stiffness matrices with negative eigenvalues creating unpredictable results (sometimes these effects are negligible but not always).
Although the uncertainties associated with this method need much further investigation, currently we believe that the following items are very important to a good experimental substructure: x An accurate FE model of the TS, x An accurate modal extraction for the experimental substructure, x Experimental TS design for which the kept modes of the analytical substructure span the space of the coupled system motion, x Sensor placement on the TS that minimizes the condition number of the kept mode shape matrix at sensor DOF. TRANSMISSION SIMULATOR PURPOSE AND DESIRED CHARACTERISTICS The TS serves several important purposes listed here. 1. It loads the interface with mass which reduces the number of modes required to represent the substructure. 2. It inherently captures the joint stiffness/damping in the experiment. 3. Its mode shapes are used to span the space of the connection motion of full system. 4. Its modal DOF inherently contain all the rotational (as well as translational) connection DOF. 5. Its mode shapes provide a reduction from the number of physical connection DOF down to the number of kept mode shapes of the TS. 6. Its mode shapes also provide the transformation to a smaller number of modal connection forces. 7. Mitigation of experimental errors results from the reduction from physical connection DOF and forces to modal DOF and forces because the continuity and equilibrium constraints are softened. 8. The modal connection DOF of the TS do not have to be obtained at connection locations but can be measured at convenient locations on the TS as long as the kept mode shape matrix is easily invertible. Inherent in the TS method is a modal analysis of the experiment either to provide the raw modal parameters for the component mode synthesis (CMS) method or to provide synthesized FRFs for the frequency based substructuring (FBS) method. A poor extraction of the modal parameters from the experiment can drastically degrade the experimental substructure. There are several important design aspects for the TS that are known and some that are probably yet to be recognized. In this work we specifically wish to investigate the advantages or disadvantages of stiff vs. soft TSs. First, let us list the desirable features already known about TSs. 1. The transducers must be placed so that the condition number of the kept mode shape matrix is low, i.e. the mode shapes are independent. 2. The TS must provide driving points to enable the extraction of all the modes. (This means that driving point response sensors must be available at each forcing location for scaling the mode shapes, and that all the modes must be excited. In addition the ability to extract closely spaced modes may be an issue, as shown later). 3. The TS must have enough mass to bring a number of modes down into the testable bandwidth, in order to span the space of the combined system motion. The TS must also exercise the joint physics in a manner similar to the way the full system does in order to accurately capture the important dynamics in the substructures. (There is a tradeoff between bringing more modes down into the testable bandwidth and having to add more sensors to measure the modes as well as forcing at more locations to excite the modes). 4. The TS must have the same connection interface geometry as the real hardware.
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1072 5. In many cases the TS must also be designed to suspend the test article for the free-free modal test. 6. The TS must be simple to model analytically, since the analytical model accuracy is important in order to subtract the effects of the TS. SOFT TRANSMISSION SIMULATOR DESCRIPTION Now the soft TS will be described. It was designed as a single piece of machined aluminum for attachment on a flange with more than twenty bolts. As seen in Figure 6, it is a simple ring with an interface to the experimental substructure that matches the real hardware. Four tabs are added which provide extra mass to bring modes of the experiment down into the testable frequency band. This extra mass is hopefully enough to eliminate the necessity of measuring flexibility terms to supplement the free modes as necessary in other works [11]. The tabs also provide easy places to instrument for response and force input as well as a place to hook bungee cords to support the experimental substructure. The experimental payload is bolted to all the bolt holes of the TS, and is basically a cylinder full of potted parts. Forty-two accelerometers were placed on the bare TS. The placement was designed by choosing from candidate locations of a FE model of the TS. The condition number as defined by the ratio of the maximum and minimum singular values was minimized for 33 modes and was 3.1 (it was 126 for 42 modes). Blocks were fabricated for the ends of the tabs to provide driving point locations for mode shape scaling. FRF data were collected using an impact hammer. The tabs were designed to have modes below 2000 Hz.
Figure 6 - Soft Transmission Simulator with Instrumentation The finite element model of the soft TS consists of tetrahedral 10-node elements (Tet10s), point masses (for accelerometers), and rigid bars connecting the point masses to the main structure. There are 29,264 elements and 144,141 DOF. A modal test of the bare soft TS was performed and compared with the FE modal analysis, which is depicted in Figure 7.
Table 1. The agreement of all modal frequencies extracted was within one percent. A comparison of one of the FRFs between the FE model and the soft fixture experiment is shown in
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1073 Table 1: Soft TS Modal Comparison between Test and Model % Test Model Difference Weight (lb) 3.676 3.68 -0.1 % 1st Frequency (Hz) 178 178.8 -0.4 % 2nd Frequency (Hz) 263 264.2 -0.5 % 3rd Frequency (Hz) 375 373.9 0.3 % 4th Frequency (Hz) 437 438.0 -0.2 % 5th Frequency (Hz) 539 542.1 -0.6 % 6th Frequency (Hz) 548 550.2 -0.4 %
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Figure 7 - Comparison of TS Simulator FRFs (FE-Blue, Experiment-Red) STIFF TRANSMISSION SIMULATOR The stiff TS shown in Figure 8 took advantage of a previously designed piece of hardware, which was a much stiffer ring than the soft TS. Four tabs were also included on the TS which were much stiffer and more massive than the previous tabs. These tabs were attached by bolts and glue. Because of the joints connecting the tabs, the comparison of the tested hardware was not as good as for the soft TS, see Table 2. The finite element model of the stiff TS also mainly consists of Tet10s with 73,464 elements and 361,323 DOFs. The worst difference in modal frequencies that were compared was less then four percent on the sixth mode. Investigations of physical differences between the hardware and the model found minor errors that were corrected, but the 3.7 percent difference for the sixth mode never improved. In the end, it will be seen that this did not seem to have a large negative impact on the results. However, we plan to make all future TS designs as one piece of hardware for more accurate modeling. A large enough error in the analytical model will ultimately degrade the subtraction of the TS from the experimental model, since it is real hardware. This is one quantity for which errors should be able to be minimized by a simple one piece design.
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Figure 8 - Stiff Transmission Simulator with Instrumentation Table 2: Stiff TS Modal Comparison between Test and Model Test Model % Difference Weight (lb) 8.78 8.84 -0.7 % 1st Frequency (Hz) 285.9 289.3 -1.2 % 2nd Frequency (Hz) 330.9 339.7 -2.7 % 3rd Frequency (Hz) 348.0 360.2 -3.5 % 4th Frequency (Hz) 539.0 543.3 -0.8 % 5th Frequency (Hz) 898.3 929.7 -3.5 % 6th Frequency (Hz) 902.8 935.8 -3.7 % FE SUBSTRUCTURE DESCRIPTION The FE substructure primarily consists of 277,698 elements (the majority are HEX-elements with 20-nodes Hex20) and 3,747,681 DOF. The shell substructure model was compared to a modal test of the shell substructure to help validate the material properties. The comparison with the modal test is shown in
Table 3. The model compares fairly well in regards to the weight. However, the model is generally stiffer than the true substructure; especially, in the first three modes. The lower modes are ovaling modes.
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1075 Table 3: FE Substructure Modal Comparison Between Test and Model Test Model % Difference 1st Frequency (Hz) 198.0 207.1 -4.6 % 2nd Frequency (Hz) 208.0 216.9 -4.3 % 3rd Frequency (Hz) 284.2 294.4 -3.6 % 4th Frequency (Hz) 292.1 296.8 -1.6 % 5th Frequency (Hz) 509.8 519.0 -1.8 % 6th Frequency (Hz) 520.0 522.6 -0.5 % ST 1 Bending Mode (Hz) 968.6 956.0 1.3 %
EXPERIMENTAL MODAL EXTRACTION DIFFICULTIES When the soft TS was tested with the experimental payload, the data were taken with impact excitation. Very important elastic modes were associated with the tabs bending in the x direction of Figure 6. The attachment of the TS to the payload caused the four vertical tab modes to be closely spaced in frequency. However, the structure was nonlinear enough that the frequencies were not consistent from the different input locations, therefore, a multi-reference curve fit could not separate the modes accurately. The best modal fit that could be obtained fit one mode for the impact reference located on that tab. This was repeated for all four tabs. From a single reference, all four closely spaced modes could not be extracted. Therefore some pollution of the mode shape would occur from the adjacent modes that were not extracted. Reconstruction of cross FRFs were not as good as desired. A multi-reference test with shakers was attempted to improve the extraction of the four tab modes. However, the mass of the force gages drastically affected the tab modes (20% reduction in frequency) and the shaker effort was abandoned. Only six elastic modes were extracted below 700 Hz. When the stiff TS was tested, no significant difficulties were encountered. Because it was so stiff and massive, it drove the experiment in mode shapes that looked similar to those of the bare TS, only at higher frequencies, due to the attachment to the payload. Although there was some slight nonlinearity that could be observed from different impact levels, the modal analysis was performed without great difficulty. A maximum of two modes were closely spaced, as opposed to the four in the soft TS test. Eleven elastic modes were extracted below 2000 Hz. LESSONS LEARNED FOR EXPERIMENTAL EXTRACTION When designing "soft" transmission simulators, one lesson learned was that the tabs should be different lengths so that their modes are not all at the same frequency when bolted to the experimental test article. However, this was not necessary for the stiff transmission simulator, as its modes were more separated and appeared similar to the bare TS simulator modes, only at higher frequencies. Impact testing still appears to be a better option than shaker testing, unless the TS is so massive that the mass of the force gage does not affect it. Impact testing also has the advantage that there is good force signal to noise ratio at the modal frequencies. Shaker testing tends to have dropouts in the autospectrum of the force at the modal frequencies if the input is not controlled in some way. COMPARISON OF RESULTS FOR THE SOFT AND STIFF TRANSMISSION SIMULATOR Results for the coupling of the FE and experimental substructures using the soft and stiff TS are given in this section, but some comment on the finite element substructure is warranted. It should be noted that the FE
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1076 model frequencies of the shell substructure were up to 5% higher than those of a modal test (see Table 3), so it is a bit too stiff. The FE substructure was represented with five hundred modes with either the soft or stiff analytical TS attached. This included modes up to 5,348 Hz for the version with the soft TS and 5,860 Hz for the version with the stiff TS. To connect the TS to the FE substructure, nodes along shared surfaces were equivalenced to each other. This forced the TS and shell to have the same motion at the interface where they are joined. The joint stiffness is characterized in the experimental model since the TS attachment to the experimental payload exactly reproduces the bolted joint. In the following figures (Figure 9 - Figure 12), the red coupled results are compared with the blue measured FRFs on the actual hardware when it was assembled together. On the left is the stiff TS result, while on the right is the soft TS result. Figure 9 and Figure 10 show driving points at the large end of the shell as shown in Figure 4 where the response is at the same location as the input. Figure 11 and Figure 12 show the cross points showing response at the small end of the shell to inputs at the large end of the shell. The driving point results would probably be considered by most subject matter experts to be adequate for establishing specification envelopes. The cross points are much poorer. In general, the driving points are usually considered the most difficult to match since all the modes that are excited are present. In cross points, not all excited modes are evident. The reason the cross points are worse is not currently understood. Overall, the results are noticeably better with the stiff TS than with the soft TS. That being said, there is not a tremendous difference between the two sets of results.
Figure 9 - Lateral Driving Point FRF Comparison - Coupled FE and Experimental Results with Stiff Transmission Simulator (Left) and Soft Transmission Simulator (right)
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Figure 10 - Axial Driving Point FRF Comparisons - Coupled FE and Experimental Results with Stiff TS (left) and Soft Transmission Simulator (right)
Figure 11 - Lateral Cross FRF Comparisons - Coupled Experimental and FE Results with Stiff Transmission Simulator (left) and Soft Transmission Simulator (right)
Figure 12 - Axial Cross FRF Comparisons - Coupled Experimental and FE Results with Stiff Transmission Simulator (left) and Soft Transmission Simulator (right)
LESSONS LEARNED FOR TRANSMISSION SIMULATOR DESIGN x
For soft TS, the lesson learned was that the tabs should be different lengths to avoid many modes being at the same frequency which significantly increases the difficulty and degrades the accuracy of the modal analysis and resulting substructure.
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One piece designs are much easier to model than designs that have multiple pieces with joints, which are difficult to model. The stiff TS attached to the experimental payload hardware had mode shapes that appeared more similar to the bare TS, which also appeared to eliminate the problem of many modes falling at the same frequency (although this is not guaranteed for every system).
CONCLUSIONS The transmission simulator approach overcomes many of the difficulties that caused the classical approaches to fail completely, especially for multiple attachment problems. It is shown to work well for driving point responses, i.e. where the force is applied near the response point. Problems appear for cross points, i.e. where the force is applied far away from the response point. The stiff transmission simulator produced better results than the soft TS in this work. The modal test was also easier to perform, allowing more modes to be extracted from the experimental substructure, which may be at least part of the reason why the stiff TS was better. The fact that the mode shapes for the experimental substructure modal test were similar to those of the bare TS is also an advantage from the aspect of planning the TS design and instrumentation, since the number of accelerometers and their placement is dependent upon the number of modes desired to be kept from the TS for the reduction matrix. Although the stiff transmission simulator produced better results than the soft transmission simulator, there was not a large difference between the two sets of results. We conclude that the method is not tremendously sensitive to the transmission simulator design stiffness and mass, if extremes are avoided. Theoretically, a TS design that was infinitely massive and stiff would fail because signal to noise ratio would be too low in the experiment. If the TS design is much too low in stiffness and mass, it will not exercise the joint well and relatively many mode shapes (and therefore additional accelerometers) will be required to span the space of the final connection motion. In this work only six elastic modes were extracted from the experimental substructure using the soft TS and 11 using the stiff TS. If no responses on the experimental substructure are required for predictions, the result is that not a large number of modes are required to be extracted on the experimental substructure for this class of problem. This is often the case for Sandia applications. The transmission simulator should be designed as one piece to make the FE modeling of the TS simple, since joints tend to complicate generating an accurate model. Future work should address the errors that were evident in cross point predictions. REFERENCES 1 Maia, Nuno M. M., and Silva, Julio M. M., Martinez, Theoretical and Experimental Modal Analysis, Research Studies Press LTD., Baldock, Hertfordshire, England, 1997. 2 Simmons, Leslie A., Smith, Gregory E., Mayes, Randall L., and Epp, David S., “Quantifying Uncertainty rd in an Admittance Model Due to a Test Fixture”, Proceedings of the 23 International Modal Analysis Conference, February 2005. 3 DeKlerk, Dennis, “Dynamic Response Characterization of Complex Systems Through Operational Identification and Dynamic Substructuring”, PhD Thesis, Technische Universiteit Delft, 2009. 4 Allen, Matthew S. and Mayes, Randall L., "Comparison of FRF and Modal Methods for Combining Experimental and Analytical Substructures", Proceedings of the 25th IMAC Conference on Structural Dynamics, Paper #269, February 2007. 5 Mayes, Randy L., and Stasiunas, Eric C., "Combining Lightly Damped Experimental Substructures With Analytical Substructures", Proceedings of the 25th IMAC Conference on Structural Dynamics, Paper #207,
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February 2007. 6 Mayes, Randy L., Hunter, Patrick L., Simmermacher, Todd W. and Allen, Matthew S., "Combining Experimental and Analytical Substructures With Multiple Connections", Proceedings of the 26th IMAC Conference on Structural Dynamics, Paper #164, February 2008. 7 Rixen, Daniel, "How Measurement Inaccuracies Produce Spurious Peaks in Frequency Based Substructuring", Proceedings of the 26th IMAC Conference on Structural Dynamics, Paper #87, February 2008. 8 Ind, Phillip, “The Nonintrusive Modal Testing of Delicate and Critical Structures”, PhD Thesis, Imperial College, London, 2004. 9 Carne, T.G. and Dohrmann, C.R., "Improving Experimental Frequency Response Functions for Admittance Modeling", Proceedings of the 24th IMAC Conference on Structural Dynamics, February 2006. 10 Craig, R., and Bampton, M., "Coupling of Substructures for Dynamic Analysis", AIAA Journal Vol. 6, No. 7, July 1968, pp. 1313-1319. 11 Martinez, David R., Miller, A. Keith and Carne, Thomas G., "Combined Experimental / Analytical Modeling of Shell Payload Structures", Sandia National Laboratories, Sandia Report SAND 84-2598, December 1985. APPENDIX 1 - THEORY FOR COMPONENT MODE SYNTHESIS WITH TRANSMISSION SIMULATOR The authors have been involved in investigating the method using the TS for both the Component Mode Synthesis (CMS) and Frequency Based Substructure (FBS) approaches. Here the assumption is that there are two substructures, one generated from the FE model that will be coupled to one generated from an experiment. The method will be derived assuming that reduced models from both substructures are assumed, although this is not required for the FE substructure. The derivation given here is not new theory. A combination of derivations from Allen[4] and Mayes[6] and DeKlerk[3] will be used. The constraint is the one introduced by Allen and Mayes. Although there are only two substructures being coupled, both substructures include the TS, which is a sensor that allows the continuity and equilibrium constraints to be applied with a reduced number of constraints from the number of physical connection DOF. The two TSs must be subtracted, so they are included as what appears to be a third negative substructure, multiplied by two. Using the )( subscript for the finite element model, the (;3 subscript for the experimental model and the 76 subscript for the TS, Ȧ[[ as the natural frequency in rad/sec, Ȧ as any desired response frequency, T as the uncoupled modal DOF and ĭ as the kept mass normalized mode shapes from each substructure, one has for the uncoupled equations (neglecting damping) ªZ )( « « « ¬
Z
(;3
º T )( ½ º T )( ½ ª, »° ° ° ° « »» ®T (;3 ¾ » ®T (;3 ¾ Z « , »° ° «¬ , »¼ °¯ T76 °¿ Z76 ¼ ¯ T76 ¿
7 ªI )( « « « ¬
I
7 (;3
The constraint is applied only at the measurement DOF as ;)(PHDV ;(;3PHDV ;76PHDV (2) One can rewrite this as a Boolean matrix of zeros, ones and minus ones
; )(PHDV ½ ª , , º ° ° « , , » ® ; (;3PHDV ¾ ¬ ¼° ; ° ¯ 76PHDV ¿
(3)
º I )( ½ »° ° » ® I (;3 ¾ (1) 7 »° ° I76 ¼ ¯ I 76 ¿
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1080 as in DeKlerk[3]. Approximating each set of physical measurement DOF using the modal substitution gives
º T )(PHDV ½ ªI )(PHDV ª , , º « ° ° I(;3PHDV »» ®T (;3PHDV ¾ . « , , » « ¬ ¼« I76PHDV »¼ °¯ T76PHDV °¿ ¬
(4)
The TS method multiplies the Boolean matrix by the pseudo-inverse of the TS mode shape matrix as
º T )(PHDV ½ ªI )(PHDV SLQY I76PHDV º« ° ° I(;3PHDV »» ®T(;3PHDV ¾ . » « SLQY I76PHDV SLQY I76PHDV ¼ «¬ I76PHDV »¼ °¯ T76PHDV °¿
ª SLQY I76PHDV « ¬
(5)
Eqn (5) reduces the number of constraints from twice the number of measured DOF to twice the number of kept TS mode shapes. This softening of the constraint helps to eliminate errors in the constraint matrix due to experimental errors in shape extraction. The number of measurement DOF should be more than the number of kept modes of the TS for an overdetermined least squares fit of the modal DOF constraints to the physical DOF. Combine the first two matrices in eqn (5) and name them the B matrix to represent the constraint equation as
T)(PHDV ½ ° ° % ®T(;3PHDV ¾ . °T ° ¯ 76PHDV ¿
(6)
Now the coupling is accomplished by expressing q through a reduced number of independent DOF qr which satisfy the constraints as
^T`
5^TU `
(7)
which when substituted back into eqn (6) yields
%5^TU ` .
(8)
One can pick 5 to be orthogonal to % to satisfy eqn (8) as (9)
5 QXOO %
T
The coupled equations are given by substituting eqn (7) into (1) and premultiplying by R as ªZ )( Z « 57 « « ¬
Z (;3 Z
Z
76
7 º ªI )( » 7 « » 5^T U ` 5 « « Z »¼ ¬
I
7 (;3
º I )( ½ »° ° » ® I (;3 ¾ . (10) 7 »° ° I76 ¼ ¯ I 76 ¿
Solving the eigenvalue problem for the homogenous version of eqn (13) yields the new eigenfrequencies ȍU and eigenvectors Ȍ which satisfy the uncoupled equations with generalized DOF S as
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>:
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U
Z
@^S`
7 ªI )( « < 7 57 « « ¬
I
7 (;3
º I )( ½ »° ° » ® I (;3 ¾ 7 »° ° I76 ¼ ¯ I 76 ¿
(11)
S can solved when specific forces are applied and ultimate displacements, ;, come from multiple modal substitutions as
^; `
ĭ^T` ĭ5^TU ` ĭ5Ȍ ^S`
(12)
so the coupled mode shapes are given as ĭ 5 Ȍ
(13).
Although the mathematics is just manipulation utilizing standard eigenvalue equations, there are some subtle difficulties that can be encountered. This manipulation assumes that the analytical fixture shapes are just the same as the experimental fixture shapes, which will never be perfectly true. Because of this, the analytical fixture subtraction from both substructures may not be a perfect subtraction of the experimental fixture. Also, the experimental modal parameters will have errors due to experimental response measurement errors as well as modal extraction errors. These errors can produce reduced mass and stiffness matrices with negative eigenvalues, which is not a realizable physical system. In certain cases, if the errors are large enough, this may produce unacceptable response errors. On the positive side, the reduction of the number of constraints using the TS mode shapes tends to mitigate the effects of experimental errors, which is a huge advantage this approach has over classical methods which require continuity of all the physical DOF responses, even though there are experimental errors in those responses. APPENDIX 2 - THEORY FOR FREQUENCY BASED SUBSTRUCTURING WITH TRANSMISSION SIMULATOR Here we use DeKlerk's[3] terminology of Frequency Based Substructuring (FBS) for modeling that is commonly called admittance or impedance modeling in other texts and papers. Nominally, the authors have been working with accelerance FRFs for the frequency based substructure. Idealized FRFs are utilized for the experimental substructure that are synthesized from the elastic modal parameters extracted from a modal test and the rigid body modes provided from mass properties measurements or analytical model mass properties. The classical FBS methods are derived by several authors such as Maia and Silva in their chapter 5.4.2 on FRF coupling[1]. The author will develop the TS based modification for one commonly used partition of these matrices to connect the FE and EXP substructures. The subtraction of the TS will be addressed later. The FRF matrix of the combined system is HC and can be written as +& +(;3 >+(;3+)(@ +)(
.
(F1)
Here we address the FRF from an input at DOF 2 on the FE substructure to DOF 1 on the EXP substructure. The connection DOF are designated with subscript 0. This matrix exists for every frequency line, Ȧ. Other partitions of the FBS matrices can be handled in an analogous fashion. The inversion of the FRF matrix associated with the experiment can be difficult, as documented by many authors (Ind[8], Rixen[7], Simmons and Mayes[2]). If we project this FRF onto the space of the TS analytical mode shapes with fewer modes than measurement DOF, the experimental errors are mitigated. We do this through approximating the response DOF for both the FE and EXP substructures as
^; )( ` ^; (;3 `
ĭ76 ^V )( `
ĭ76 ^V (;3 `
(F2)
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1082 The forces at physical DOF, I, are approximated as modal forces, ), on the TS by
^))( ` ^)(;3 `
7 ĭ76
^ I )( `
7 ĭ76
^ I (;3 `
(F3)
The physical forces can be represented by the modal forces by pre multiplying eqn F3 by the pseudoinverse of
) 776 giving
^ I )( ` ^ I (;3 `
7 SLQY ĭ76 ^))( `
7 SLQY ĭ76 ^)(;3 `
(F5)
We can write the FRF expressions in eqn F1 as
^; (;3 ` and
^; )( `
+(;3 ^ I (;3 ` +)( ^ I )( `
(F6)
Upon substituting eqn F2 and F5 into F6 yields ĭ76 ^V(;3` +(;3 SLQY ĭ767 ^)(;3` ĭ76 ^V)(` +)( SLQY ĭ767 ^))(`
(F7)
Premultiplying both sides by the pseudoinverse of ĭTS gives ^V(;3` SLQY ĭ76 +(;3 SLQY ĭ767 ^)(;3` ^V)(` SLQY ĭ76 +)( SLQY ĭ767 ^))(`
(F8)
Eqn F8 provides the projection of +(;3 and +)( onto the TS modal coordinates, which generally is a reduction assuming there are more measured DOF than kept modes. Using this conversion in eqn F1, a projection of HC in the TS modal coordinates for the connections is written as +& +(;3 SLQY ĭ767 >SLQY ĭ76 +(;3+)( SLQY ĭ767 @ SLQY ĭ76 +)(
(F9)
Recall that the connection DOF using the TS are the measurement DOF. A way to understand the connection is to visualize that the TS on the EXP substructure and on the FE substructure are being forced to have the same motion. If fewer modes are retained in ĭ76 than measurement DOF, the size of the FRF matrix that is inverted is reduced. This reduction softens the continuity and equilibrium constraints, which provides some relief from experimental errors that will be present in HEXP. The analytical mode shapes of the TS also provide a common basis for connecting the motions of the EXP and FE substructures.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
PROPAGATION OF UNCERTAINTY IN TEST/ANALYSIS CORRELATION FOR SUBSTRUCTURED SPACECRAFT Daniel C. Kammer Sonny Nimityongskul Department of Engineering Physics University of Wisconsin Madison, WI 53706 [email protected] (608) 262-5724 Determining a finite element model's (FEM) fidelity-to-data, or the process of test/analysis correlation, is a vital component of the overall verification and validation procedure that is applied to analytical models of spacecraft prior to flight. The goal of this work is to develop a methodology for studying the effects of uncertainty on metrics used for test/analysis correlation of complex spacecraft that are validated on a substructure-by-substructure basis. Uncertainty will be propagated from substructures to system for low frequency applications. The objective of this work is to quantify the level of accuracy required at the substructure level to produce acceptable accuracy at the system level. Organizations, such as NASA and the Air Force make critical decisions on spacecraft performance and survivability based on the results of test/analysis correlation metrics. Currently there is no uncertainty quantification performed or even required for test/analysis correlation in the low-frequency regime. The aerospace community has traditionally relied on modal analysis in conjunction with the finite element method for the efficient linear dynamic analysis of spacecraft in the low frequency range. At low frequencies, a relatively small number of modes with widely spaced frequencies can be used to capture the system behavior. Prior to flight, a test-validated FEM of the spacecraft must be developed to provide accurate loads analysis. The state-of-the-art in FEM validation of space structures is based on modal analysis. The validation process is comprised of several activities, such as determining the fidelity of the model to test data, uncertainty quantification, and determining predictive accuracy. Determining fidelity-to-data is a component of fundamental importance. It is the process of comparing test and analysis predictions, also called test/analysis correlation, and then determining optimum changes in parameters that will update, or tune the model. A final accuracy assessment is performed by comparing updated model predictions with an independent set of test results. In modal based validation, the accuracy of the FEM is determined by comparing test and analysis modal parameters. Frequencies are compared directly, while the corresponding mode shapes are compared using metrics based on orthogonality and cross-orthogonality of the modes with respect to a reduced analytical mass matrix, called a Test-Analysis model (TAM). The use of these metrics, and the required values for test/analysis correlation, are dictated by agencies such as NASA and the United States Air Force. Depending on the agency, the requirements are different. For example, the Air Force requires test/analysis frequency errors less than or equal to 3.0%, cross-generalized mass values greater than 0.95, and coupling terms between modes of less than 0.10 in both cross-orthogonality and orthogonality. Recently, work in the structural dynamics community on analytical model validation has focused on the quantification of model uncertainty within large numerical simulations, and its propagation into predicted results. The concept of model uncertainty is the reality of the design problem. An engineer may design a single structure based on drawings, analysis, and experiments, but then the item produced is one of a statistical population due to variations and uncertainties in geometry, material parameters, construction, etc. This leads to random populations of frequencies and mode shapes. Obviously, there is a corresponding uncertainty and error in the measured test data. In the low frequency regime of T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_95, © The Society for Experimental Mechanics, Inc. 2011
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modal-based test/analysis correlation and model updating, it is common practice to ignore the effects of model and test uncertainty. If one does not examine the agreement between measurements and predictions relative to uncertainty, very erroneous and dangerous decisions can be made regarding the models ability to make accurate predictions within untested regimes. As spacecraft become larger and more complex, ground based vibration tests of the entire structure become impossible due to lack of structural integrity, cost, complexity of the test, etc. The spacecraft must then be validated on a substructure-by-substructure basis. A full-up vibration test of the structure might not be possible until the spacecraft is on-orbit. For this new type of model validation problem, there are several critical questions that must be addressed. What level of accuracy or correlation do the substructures need to exhibit, to have a required level of correlation at the system level? How does uncertainty and error at the substructure level propagate into, and affect model validation at the system level? Finally, how does uncertainty and error in the connections between substructures propagate into the system correlation? New approaches must be developed to answer these questions. This work uses component mode synthesis (CMS) and reduced order modeling of substructures to quantify and propagate uncertainty in the substructures and their connections into the full synthesized system representation. The CMS approach has been used for years to solve large structural dynamics problems in the low frequency range. It is already built into many standard finite element analysis (FEA) codes. The substructure representation that is used in this work is the one developed by Craig and Bampton, which is standard in most FEA codes, such as NASTRAN. The efficacy of this substructure representation for studying the effects of model uncertainty lies in the fact that the interior and interface of the substructure are modeled separately. Therefore, the effects of uncertainty from each source can also be considered independently. Within the interior of the substructure, uncertainty in model parameters can be propagated into uncertainty in the fixed interface modal parameters. This uncertainty can then be propagated into the standard modal based test/analysis correlation metrics. In fact, the orthogonality metric, and the frequency error metric appear directly in the equations of motion for the CB substructure representation. Uncertainty at the interface will be propagated into the constraint modes, either directly, or through their corresponding modal basis, given by the characteristic constraint modes. Several different approaches to uncertainty propagation are investigated, such as linear perturbation methods, Monte Carlo sampling, and linear covariance propagation. This investigation at the substructure level will result in an understanding of the impact of model uncertainty on the standard test/analysis correlation metrics. The model uncertainty at the substructure level is then propagated into uncertainty at the synthesized system level to determine its impact upon system model correlation metrics within the low-frequency regime. Even if the spacecraft will be tested as a system prior to launch, an understanding of the required level of substructure correlation, and how the related uncertainty propagates into the system, will save a great deal of time, effort, and cost during the system level test and analysis.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Experimental Modal Substructuring to Extract Fixed-Base Modes from a Substructure Attached to a Flexible Fixture Matthew S. Allen Assistant Professor [email protected]
Harrison M. Gindlin, Undergraduate Student Department of Engineering Physics University of Wisconsin-Madison 535 Engineering Research Building 1500 Engineering Drive Madison, WI 53706 &
Randall L. Mayes Distinguished Member of Technical Staff Sandia National Laboratories PO Box 5800 Albuquerque, NM 87185 Abstract: It is well known that fixed boundary conditions are often difficult if not impossible to simulate experimentally, but they are important to consider in many applications. Mayes and Bridgers ["Extracting Fixed Base Modal Models from Vibration Tests on Flexible Tables," IMAC XXVII, Orlando, Florda, 2009.] recently presented a method whereby one can determine the fixed base modes of a structure from measurements on the system and the fixture to which it is attached. They used modal constraints in conjunction with an admittance approach to perform the substructuring computation, demonstrating that the first fixed-base mode could be estimated accurately using the coupling procedure, even though the free fixture had a mode at almost the same frequency. This work builds on that by Mayes and Bridgers, but employs modal substructuring instead of frequency-response based methods to estimate the fixed-interface modes. The method is validated by applying it to a experimental measurements from a simple test system meant to mimic a flexible satellite on a shaker table. A finite element model of the subcomponents was also created and the method is applied to its modes in order to separate the effects of measurement errors and modal truncation. Excellent predictions are obtained for many modes of the fixed-base structure, so long as modal truncation is minimized, verifying that this modal substructuring approach can be used to estimate fixed-base modes of a structure without having to measure the connection point displacements and rotations and even with the limitations inherent to real measurements. 1. Introduction Testing and model validation campaigns often include both a low-vibration level modal test, used to extract the modal parameters of the system, and subsequent shaker testing at higher amplitudes to evaluate the durability of the system. Modal parameters extracted from the former are correlated with FEA models which are then used to predict the life of the structure in a specified environment. The latter tests are meant to verify the FEA predictions either by verifying that the test article survives the vibration environment or that the strains measured at critical points are the same as those predicted by the model. There are a number of reasons for desiring to combine those two tests. First, the high amplitude shaker tests better describe the system in the environment of interest, so it would be preferable to extract modal parameters from those tests in case the structure has any nonlinearity that would change its effective stiffness or damping with excitation level. This
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assures that the appropriate model is used for the environment of interest. Second, each test increases the time and cost required to develop the system, so significant savings might be realized if one of the tests can be shortened or eliminated. Finally, this approach may help minimize uncertainty in the boundary conditions that are applied in shaker testing, and it may be preferable to use the fixed-base modal parameters for model updating rather than the free-free modal parameters. Blair discussed some of these issues in the context of model validation for space shuttle payloads in [1]. However, it is well known that fixed boundary conditions can be exceedingly difficult to achieve in practice. Any real boundary condition has some flexibility, and it very often has an important effect on the system’s modal parameters. If the article of interest is made from stiff engineering materials then often no material exists that can provide an adequately rigid boundary condition out to sufficiently high frequencies. A few previous works have explored methods for predicting fixed-base modal parameters from test measurements, as outlined by Mayes and Bridgers [2]. Using classical frequency-based substructuring (FBS) [3] or impedance coupling [4, 5] methods, one can, in principle, estimate the fixed-base properties of a test article if all of the displacements, rotations, forces and moments between the test article and the supporting structure are measured. This is rarely possible and often leads to ill-conditioning in the coupling equations even if the necessary moments and forces can be measured. Mayes and Bridgers presented a method that avoids the need to measure the interface forces and displacements and which seems to minimize this ill conditioning [2]. They used FBS to couple the modal motions of a shaker table to ground using Allen & Mayes’ modal constraint approach [6-8]. An experiment verified that their approach was capable of extracting the first fixed-base mode of a cantilever beam from measurements on a simplified shake table. This work expands upon Mayes and Bridgers’ work in two ways. First, this work employs modal substructuring rather than FBS, so one only needs to manipulate the modal parameters of the structure rather than all of the FRFs. Second, this work considers a large number of modes of both the fixture alone and the fixture+substructure, showing that the approach also works well for higher modes and exploring how high of a test bandwidth is needed to estimate fixed-base modes over a desired frequency range. The proposed substructuring approach is illustrated schematically in Figure 1, where a structure of interest is attached to a flexible fixture (plate). The modes of the substructure of interest are found when it is attached to the flexible fixture, and then modal constraints are used to constrain the motions of the fixture to ground. Modal constraints are constraints applied to modal motions, which are estimated using a modal filter [9], rather than to the motions of physical points or surfaces. Structure of Interest
Flexible Fixture Modal Constraints Grounded, Infinitely Stiff Fixture
Figure 1: Schematic of proposed modal substructuring approach, which estimates the fixed-base modes of the structure of interest from the measured modes of the structure+fixture.
The rest of this paper is organized as follows. Section 2 presents the proposed modal substructuring technique, which estimates fixed base modal parameters from measurements on a flexible fixture. Section 3 presents the hardware used to evaluate the proposed methodology, a stiff rectangular plate with a beam attached perpendicular to the plate, and discusses experiments used to find the modes of both the fixture (plate) and the fixture+structure (plate+beam). In Section 4, the technique is applied with finite element derived modes for both subcomponents, in order to assess the proposed substructuring technique when the measurements are perfect. Section 5 summarizes the conclusions.
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2. Theoretical Development Suppose that one performs a modal test [4] to obtain N modes of vibration of a structure of interest at a certain set of measurement points, when it is attached to a flexible fixture. One then has estimates for the following modal parameters,
Zr
]r
,
ªI f º «I » ¬ s¼
,
(1)
where Zr are natural frequencies of the structure, ] r its damping ratios, If and Is denote the mass-normalized mode shapes at the measurement points on the fixture and substructure of interest, respectively, and r=1…N. The fixture is a dynamic system itself, although it is meant to approximate a rigid boundary condition. For example, the shake tables discussed in [2] can be thought of as stiff, translating fixtures. One can also obtain the modal parameters of the fixture alone (without the structure of interest attached), either through test or analysis, so its modal parameters are denoted
Z fixt r
] fixt r
,
I fixt f
,
,
(2)
The natural frequencies and damping ratios of the fixture are not needed for this approach, only the mass normalized mode shapes of the fixture, Ifixtf. If the fixture were truly rigid and perfectly fixed to ground, then If would be zero for all of the measured modes of the fixture+structure. In practice this will not be the case, but one can approximate the measured fixture motion, yfixt, as follows in terms of Nfixt modes of the fixture,
y fixt | I fixt f q fixt ,
(3)
fixt
where q , denotes the modal coordinates of the fixture. One can then estimate the participation of each of the fixture modes in the measured response of the fixture+structure by multiplying the upper partition of the mode shapes in eq. (1) by the pseudo-inverse of the fixture mode shapes (i.e. using a modal filter [9]),
q fixt | I fixt f
If
,
(4)
where ()+ denotes the Moore-Penrose pseudo-inverse of the matrix. The estimate of the fixture modal amplitudes is only meaningful if one has at least as many measurement points as modes of interest and if the measurement locations on the fixture are chosen such that Ifixtf has full column rank. Our desire is to estimate the modal parameters of the substructure when attached to a base that is truly fixed. This work proposes to do that by applying the constraints,
q fixt
0,
(5)
to the modes of the fixture+structure using the Ritz method [6, 7, 10]. In terms of the modal coordinates of the fixture+subsystem, q, the constraint equations are
ª I fixt f ¬«
I f º» q 0 , ¼
(6)
where the term in brackets is an Nfixt by N matrix of constraint equations. This matrix is denoted [a] in the text by Ginsberg [10], or B in the review by De Klerk, Rixen and Voormeeren [3]. The procedure described in either of those works can be used to enforce these constraints and hence to estimate the modes of the fixture+subsystem with the fixture motion nullified. The “ritzscomb” Matlab routine, which is freely available from the first author, was used to perform the calculations for this work. It is important to note that the constraints above only enforce zero motion at the fixture measurement points if the number of measurement points equals the number of fixture modes. In practice one should use more measurement points on the fixture than there are active modes in order to be able to average out noise or measurement errors on the mode shapes. Hence, the motions of the physical measurement points may not be exactly zero. There may also be residual motion in the fixture that is physical, since one is seeking to constrain an infinite dimensional system with a finite number of constraints. Fortunately, one can readily observe the fixture motions after applying the constraints to see whether the constraints were effective in enforcing a rigid boundary condition. This is illustrated in the following sections and provides a valuable way to check whether enough modes were used in eq. (6).
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3. Experimental Application A simple system, consisting of a 12.35-in. long steel beam attached to a 12 by 10-in. steel plate was constructed to evaluate the proposed approach. The 1-in. by 0.5-in. cross-section beam is the substructure of interest, and the 0.625-in. plate is stiff and so approximates a rigid boundary condition. A schematic of the system is shown in Figure 2, with the left picture looking down the beam onto the plate (the beam extends from the origin in the negative x-direction), and the right figure (green) looking parallel to the surface of the plate showing the beam standing up. The plate was designed such that many of its natural frequencies would be equal to those of the fixed-base beam. This assured that the modes of the plate and beam would interact creating an interesting case study. The measurement grid used in the experiments is also shown, along with the locations used for three uni-axial accelerometers. The red circles show the locations of the accelerometers used to test the plate+beam, the blue show those used when the plate was tested alone to find Ifixtf. The accelerometers placed on the plate were actually placed on the underside of the plate so that the structure could be excited from above at each of the points shown. ½”
101y 156z …
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0.95”
Figure 2: Schematic of plate-beam system used to estimate the fixed-base modes of a steel beam.
The assembly was placed on an inflated rubber tube to simulate free-boundary conditions, which were realized quite effectively since the first vibration mode of the system was about ten times higher than the highest rigid body mode of the system on the inner tube, which conforms to the guidelines in [11]. Figure 3 shows pictures of the setup. The beam is attached to the plate with two hex screws. Some initial measurements were processed with the zeroed-fast Fourier transform nonlinearity detection method described in [12], which revealed that the natural frequencies of the system did decrease significantly with increasing excitation amplitude. To remedy this, the system was disassembled and reassembled with adhesive between the beam and plate and with the bolts very tight. After doing this, the method in [12] no longer revealed significant nonlinearity. Table 1 lists all of the equipment used to perform a roving hammer modal test of the structure. Three hits were applied to each of the 36 points on the plate, 6 on the y-side, 6 on the z-side, 12 on the y-side of the beam, and 12 also on the z-side of the beam giving 72 total measurement points. Initial analysis revealed that the phase of certain FRFs sometimes showed large and unreasonable delays, apparently due to a fault in the data acquisition software, resulting in distortions to the measurements, so those points were re-tested. Once a reasonable set of FRFs had been obtained, the modes were identified using the AMI algorithm [13-15] and the mode shapes and natural frequencies were imported into Matlab to perform the substructuring analysis. A similar set of tests was also performed on the plate without the beam attached to estimate Ifixtf.
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Figure 3: Photographs of the experimental setup. The left picture shows the entire setup with the plate on top of an inflated tube, and the beam mounted on the plate. The right picture shows the two accelerometers located on the tip of the beam.
Equipment Data Acquisition Software Accelerometers Hammer
Specifications LDS Dactron Photon II, Model #5880283 RT Pro Photon 6.33 PCB Piezoelectronics, Model #J351B11 PCB Piezoelectronics – Modally Tuned Model #086C01
Table 1: List of equipment used, manufacturer and model.
Table 2 lists the natural frequencies found for the plate and plate+beam. The experiments were designed to extract all modes below 3kHz and the results show that those and quite a few more were extracted. Comparing the natural frequencies before and after adding the beam, we see that the beam causes the first mode of the plate to split, the familiar vibration absorber effect, and similarly due to the interaction of the 2z and 3y modes of the beam with the 3rd and 4th elastic modes of the plate. This is illustrated in Figures 4 and 5, which shows the 3rd and 4th mode shapes of the plate+beam. The surface plot shows the deflection of the plate and the two line plots show the bending of the beam in both the y- and z- directions. The beam bending has an opposite sense in the 4th mode as in the 3rd mode, demonstrating that the beam is acting as a vibration absorber for the plate. In either mode shape the motion of the beam resembles the 2nd analytical mode shape for a cantilever in the y-direction, so one would expect this pair of modes to merge to a single bending mode of the beam once the constraints at the base of the beam are enforced. The beam’s displacement in the z-direction is small and presumably dominated by noise or errors in the alignment of the hammer blows.
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1090 Nat. Freqs (Hz) of Elastic Modes Plate Alone Mode (Fixture) Mode 1 1 670.5 2 2 893.7 3 3 1344.0 4 4 1620.3 5 5 1850.4 6 6 2538.0 7 7 3107.8 8 8 3143.6 9 9 3591.1 10 4082.0 10 11 11 4814.5 12 12 4837.7 13 13 5220.6 14 14 5561.4 15 15 6765.1 16 16 7093.4 17 17 7147.8 18 18 7179.0 19 7808.4 20 8288.7 21 8414.8
Plate + Beam 130.7 224.0 628.3 693.8 902.4 1254.4 1350.2 1657.8 1770.7 1797.8 1906.1 2311.9 2995.7 3107.7 3233.0 3424.5 3522.8 3845.5
Plate+Beam Modes: Beam 1y Beam 1z Beam 2y + Plate Beam 2y + Plate Plate Beam 2z + Plate Beam 2z + Plate Plate Beam 3y + Plate Beam 3y + Plate Beam 3y + Plate Plate Plate Plate Plate Beam 3z + Plate Beam 4y + Plate Plate
Table 2: Experimentally measured natural frequencies (Hz) of the Plate and Plate+Beam.
Mode 3 at 628.3 Hz
ion Beam Deformation
y disp
2
Beam
1
x disp
0.5
0 í2 í4
0
0
5 10 15 Beam Axial Position (íx)
2
í0.5 í1 10 15
e id "s 10 5
10
) (y
5
" 12
z) e( sid
z disp
1 0 í1 í2 í3
5 10 15 Beam Axial Position (íx) Figure 4: Mode shape of 3rd elastic mode of the plate+beam. The left plot shows the deformation of the plate, while the panes on the right show the deflection of the beam in the y- and z- directions. The beam is fixed to the plate midway between the two square markers, as illustrated schematically on the top. 0 0
0
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Beam Deformation 2
y disp
1 1
-1
0.5 x disp
0
-2
0
0
5 10 15 Beam Axial Position (-x)
0
5 10 15 Beam Axial Position (-x)
2
-0.5
15 5
10 5
10" side (y)
0
0
z disp
1 -1 10
0 -1 -2
12" side (z)
Figure 5: Mode shape of 4th elastic mode of the plate+beam (see description for Figure 4Error! Reference source not found.). These mode shapes show that the 2nd mode of the beam interacts with the 1st mode of the plate, so the dynamics of the plate must be accounted for to estimate the fixed-base modes of the beam.
The experimentally measured plate and plate+beam modes were used in the procedure described in Section 2 to estimate the fixed-base modes of the beam. That procedure requires an estimate of the rigid body modes of the system, and rather than measure those, a finite element (FEA) model (the same one described in Section 4) was used to estimate them. Hence, CMS was performed by combining six FEA rigid body modes with 18 experimentally measured plate+beam modes, and then creating Ifixtf with six FEA rigid body modes for the plate alone and six measured plate modes for a total of 12 constraints in eq. (6). The density of the FEA model was adjusted to reproduce the measured mass of the system. (The FEA model was also validated by comparing it with the experimental results, as described in Section 4, but that is not particularly relevant to the results presented here.) Table 3 shows the modal substructuring (CMS) predictions as well as the analytical natural frequencies of a cantilever beam with the same properties as the actual beam. The first five CMS predicted natural frequencies agree fairly well with the analytical ones, but beyond the fifth the predicted natural frequencies are inaccurate. The usual rule of thumb for CMS predictions is that coupled system predictions are typically valid over a bandwidth, BW, if modes for each substructure are used out to 1.5*BW or 2*BW. Here, modes out to 3000 Hz were used in the CMS predictions, so one would expect the results to be accurate out to 1500 or 2000 Hz. The errors in the CMS predicted natural frequencies are below 10% for the first five modes, which suggests that this rule of thumb may be valid for this CMS procedure. One also observes that the modes that involve zdirection motion have larger percent errors. The z-direction is the stiffer of the two bending directions of the beam.
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CMS Prediction Analytical Percent with 12 Fixed-Base Error in MCFS Beam Freq. CMS Constraints (Hz) Prediction 104.1 195.2 651.4 1230.5 1777.1 1823.0 2369.4 2923.8 3045.3 3352.8 3503.8 3620.2
107.35 214.70 672.8 1345.5 1883.4 3691.4 3766.8 6102.2 7382.9 9115.6 12204.3 12731.7
-3.0% -9.1% -3.2% -8.6% -5.6% -50.6% -37.1% -52.1% -58.8% -63.2% -71.3% -71.6%
Table 3: Natural frequencies of plate+beam and the estimated fixed-base natural frequencies of the beam found using the proposed procedure with 12 modes of the plate and 24 plate+beam modes. The natural frequencies of the Plate+Beam before substructuring are shown in Table 2.
As mentioned in Section 2, it is advisable to observe the motion of the fixture after applying the constraints in eq. (6) to check whether enough constraints were used to force the fixture motion to zero. This was done and the norm of the motion over the plate was found to be between 0.6% and 4.3% for the first six modes estimated by CMS, suggesting that the constraints were quite effective. The shape of the plate after applying the constraints had no recognizable pattern, suggesting that it was due to noise in the measured mode shapes, so they are not shown. The mode shapes over the beam are shown in Figure 6. Each plot overlays the estimated ybending shape, z-bending shape and the analytical shape of a cantilever with the same properties as the experimental. The shapes agree very well. There is a scale difference between the first bending modes and the analytical, but otherwise the shapes are quite similar. The second modes also agree closely, but the shapes near the root of the beam suggest that the rotation there may not be exactly zero. Some of the rotation at the connection point may be physical, since the beam is connected to the plate with a real joint that does have finite stiffness.
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1093 3 Exper y
2.5
Exper z Analytical
2 1.5 1 0.5 0
0
2
4
6 8 x-position (in)
10
12
14
6 8 x-position (in)
10
12
14
3 Exper y
2
Exper z Analytical
1 0 -1 -2 -3
0
2
4
Figure 6: Mode shapes 1 and 2 of the beam in both the y- and z- directions found by CMS procedure compared with analytical mode shapes of an Euler-Bernoulli cantileer.
The CMS procedure was repeated using different numbers of plate modes, and hence different numbers of constraints. There were larger errors in the predicted natural frequencies for modes 3 and 4 if fewer than eight constraints were used, while mode 5 was not accurately predicted unless at least 12 constraints were used. However, there was virtually no improvement in the natural frequencies if the number of constraints was increased from 12 to 16. If more than 16 were used then one begins to have only a few modes left in the system (i.e. 24 – 18 = 6), so the results begin to degrade. 4. Finite Element Modeling of Substructuring Analysis for Plate+Beam A finite element model was created to estimate the rigid body mode shapes, as mentioned previously, and also to determine whether the discrepancies between the CMS and analytical modes observed in Section 3 were due to noise in the experimental results or whether modal truncation was the culprit. The FE model was constructed with simple shell and beam elements, as summarized in Table 4 below. It was validated by comparing the FEM modes and natural frequencies with those obtained experimentally, the results of which are shown in Table 7 in the Appendix. Model Beam
Number of Nodes 21
Plate
405
Plate + Beam
418
Mesh Meshed with 0.95-in long 3node beam elements Meshed with 1-in x 1-in 8-node shell elements. Combination of the meshes described above.
Table 4: Details regarding finite element models for each system. Each node has six degrees of freedom.
Table 7 in the Appendix shows that the correlation between the experiment and FEA model is excellent, suggesting that the FE model is an accurate representation of the real system. The Modal Assurance Criterion (MAC) [4] between the experimentally measured mode shapes and the FEA mode shapes are all above 0.92,
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indicating good correlation. This is a simple system and easy to model, so one would expect the FEA model to be highly accurate and to show how well the proposed CMS procedure would work with near perfect measurements. The FEA models for the plate and plate+beam were used in conjunction with the proposed procedure to estimate the fixed-base modes of the beam. Only the mode shapes at the measurement points defined in Figure 2 were used to facilitate the comparison with the experimental results that were presented previously. The results, shown in Table 5, are qualitatively very similar to those found experimentally (Table 3). As observed in the experimental results, only the first five modes are well predicted by CMS, and the even natural frequencies (zdirection bending) are less accurate than the odd ones. Surprisingly, the 2nd and 4th natural frequencies found here using the FEA modes are less accurate than those found using the experimental modes. The mode shapes predicted by CMS, shown in Figure 7, match almost perfectly with the analytical ones, although if one looks closely some small residual rotation is visible at the base of the cantilever.
ANSYS Percent FEA Plate + CMS Prediction Analytical FixedBeam Frequency with 12 MCFS Base Beam Error in CMS (Hz) Constraints Freq. (Hz) Prediction 126.34 207.2 630.96 688.82 899.33 1207.9 1349.2 1641.3 1786.1 1895.3 2316.9 2996.3
103.25 183.8 647.53 1180.5 1810.4 2431 2930.6 3072.3 3345.6 3389.2 3536.2 4173.8
107.35 214.7 672.77 1345.5 1883.4 3691.4 3766.8 6102.2 7382.9 9115.6 12204 12732
-3.8% -14.4% -3.8% -12.3% -3.9% -34.1% -22.2% -49.7% -54.7% -62.8% -71.0% -67.2%
Table 5: CMS predictions of fixed-base natural frequencies for cantilever beam using FEA modes of the plate and plate+beam.
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2.5 FEA-CMS y FEA-CMS z Analytical
2 1.5 1 0.5 0
0
2
4
6 8 x-position (in)
10
12
14
6 8 x-position (in)
10
12
14
3 FEA-CMS y FEA-CMS z Analytical
2 1 0 -1 -2
0
2
4
Figure 7: Mode shapes of fixed-base beam predicted by CMS using FEA-derived mode shapes, compared with the analytical mode shapes.
As mentioned in Section 2, it is advisable to check the motion of the fixture to assure that the constraints were adequate to reduce its motion to a negligible amount. This was investigated by plotting the mode shapes of the plate after applying the constraints in eq. (6). Those plots do show a marked rotation of the plate in the bending direction of the beam for each of the beam’s bending modes, as illustrated for mode 4 in Figure 8. The plate motion is less than 0.03 kg-0.5. Before the constraints were applied the maximum displacement in the plate was typically about 1.0 kg-0.5 for each of the modes of the plate. The plate motion shows a nonzero rotation in the bending direction of the beam (rotation in the z-direction, about the y-axis), which would reduce the effective stiffness of the beam somewhat. However, the rotation is less than a degree, so one would expect it to be negligible.
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Mode 4 at 1181 Hz
Beam Deformation 2
y disp
0.03 0.02
0 -2
x disp
0.01 -4
0
0
5 10 Beam Axial Pos (-x)
15
0
5 10 Beam Axial Pos (-x)
15
-0.01 2 -0.03 10 15 5
0 -2
10 5
10" side (y)
z disp
-0.02
0
0
-4 12" side (z)
Figure 8: Deflection of Plate and Beam in 4th CMS estimated mode (after constraining plate motion to zero). In the left plot, the beam (not shown) is located midway between the red markers on the plate.
Another way of checking whether modal truncation is important is to check whether the plate’s free modes span the space of the observed plate+beam modes. If this is not the case, then the approximation in eq. (3) will not be accurate. To check this, the FEA predicted modes for the plate+beam were projected onto the plate modes using I. The largest difference between the projected shape and the measured shape was found and divided by the maximum absolute value of any coefficient in that shape. This gives the maximum percent error in each expanded shape, given in Table 6. Modes 1-16 of the plate+beam are accurately captured with 12 plate modes, but the higher modes are not. Modes 19 and above were not used in the CMS predictions shown here, so the errors in modes 17 and 18 are our focus. Those errors could be reduced to 10-15% by increasing the number of constraints from 12 to 16, but it was mentioned previously that the predicted natural frequencies did not change noticeably if that was done. Max Error in Expansion Mode Nat Freq Max Err 12 Modal Constraints 15 1786.1 4.3% Mode Nat Freq Max Err 16 1895.3 5.4% Zero for modes 1-6 17 2316.9 35.1% 7 126.34 0.6% 18 2996.3 58.0% 8 207.2 1.6% 19 3119.9 78.4% 9 630.96 1.6% 20 3255.6 61.4% 10 688.82 1.3% 21 3383.8 63.1% 11 899.33 0.3% 22 3552.2 50.9% 12 1207.9 6.2% 23 3836.6 63.2% 13 1349.2 1.9% 24 4455.3 68.6% 14 1641.3 1.4% Table 6: Error in expansion of the FEA mode shapes for the plate+beam onto the 12 plate modes.
4.1. Discussion The fact that the CMS predictions have about the same level of error as the experimental predictions suggests that the errors in both methods are dominated by modal truncation. This is reassuring, especially considering the large errors that are sometimes encountered in substructuring predictions due to cross-axis sensitivity and such [16, 17]. The experimental modes presented here were obtained using a standard, inexpensive technique, yet they were adequate for use in CMS. The same cannot necessarily be said if one uses the conventional CMS or frequency based substructuring approach, which requires an estimate the rotations and
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moments at the connection points, since those are difficult to measure. On the other hand, it is quite disappointing that the natural frequencies are not more accurately estimated. The plate motion shown in Figure 8 suggests that the modal basis used to describe the plate was not adequate to completely constrain its motion to ground. This problem arises because the plate deforms near the point where the beam is connected. In order to understand the effect of this localized deformation on the CMS predictions, another model was created where this region was stiffened. In this model, a 4 inch by 4 inch square of the plate surrounding the point where the beam connects (outlined with a dashed red line in Figure 2) was made to be three times as thick (1.875 inches) as the rest of the plate (0.625 inches). This significantly increases the stiffness of the plate near the connection point and should reduce the localized kinking shown in Figure 8. This model was used in the CMS procedure as described above and the fixed-base natural frequencies of the beam were estimated. The errors in the CMS predicted frequencies for y-direction bending, the 1st, 3rd and 5th natural frequencies, were only -0.3%, -0.5% and -0.8% respectively using this modified plate model, as compared to -3.8%, -3.8% and -3.9% for the regular plate as was previously shown in Table 5. Similarly, the errors in the zdirection bending frequencies reduced from -14.4 and -12.3% to -1.3 and -2.1% respectively. This suggests that localized bending of the connection point was responsible for the errors observed in the natural frequencies in Tables 3 and 5. Fortunately, one can address this difficulty by designing the fixturing to minimize localized bending whenever using the proposed CMS approach. 5. Conclusions This work presented a new method of estimating the fixed-base modes of a structure from measurements on the structure and a flexible fixture, based on modal substructuring with modal constraints. The proposed approach was evaluated using experimentally measured modes of a simple plate-beam system. A finite element model was also created to determine how the method would perform if the experiments were perfect. The comparison between the finite element and the experimental results suggests that the proposed method is not very sensitive to the errors that are inherent to experimental modal analysis. The method approximates the motion of a fixture as a sum of contributions from its free modes, and then constrains each modal motion to ground. For the system considered here, the results were always qualitatively reasonable even if far too few constraints were used, and the fidelity of the prediction increased as the number of constraints increased. Even though the plate+beam system had pairs of modes where the beam and plate motion was highly coupled, the beam acting as a vibration absorber, those modes merged smoothly to a single mode near the true natural frequency as the number of constraints increased. However, one drawback of this approach is that each constraint eliminates a mode of the fixture+subsystem, so the experimental modal database limits how many constraints can be used. Also, for the system studied here, there were still moderate errors in the predictions of the fixed-base natural frequencies near this limit. The analysis revealed that those errors were caused by modal truncation, which was exacerbated by the fact that the plate deforms locally near the point where the beam is connected. This can and should be addressed by designing the experimental fixturing to spread out the load near the connection point. The metrics presented here did help to reveal these issues, and so they should be used to check the validity of the CMS predictions when applying this technique to real systems. Also, in many applications of interest there is not so much coupling between the modes of the system and fixture, but in either case the results presented here suggest that this is a simple and effective method for estimating the fixed-base modes of real structures from experimental measurements. 6. Acknowledgements This work was supported in part by Sandia National Laboratories. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under Contract DE-AC04-94AL85000. 7. Appendix
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1098 Mode Experiment FEA Model MAC MSF 1 130.66 126.34 0.9966 1.03 2 224.0 207.2 0.9960 1.01 3 628.3 631.0 0.9928 0.98 4 693.8 688.8 0.9942 1.04 5 902.4 899.3 0.9655 1.06 6 1254.4 1207.9 0.9676 0.91 7 1350.2 1349.2 0.9266 1.05 8 1657.8 1641.3 0.9675 1.06 9 1770.7 Torsion? 10 1797.8 1786.1 0.9586 0.74 11 1906.1 1895.3 0.9893 0.98 12 2311.9 2316.9 0.9765 1.08 13 2995.7 2996.3 0.9713 1.10 14 3107.7 3119.9 0.9800 1.03 15 3233.0 3255.6 0.9617 0.99 16 3424.5 3383.8 0.9681 1.48 17 3522.8 3552.2 0.9752 0.91 18 3845.5 3836.6 0.9704 0.73 19 4455.3 20 4866.9 21 5102.3 22 5395.4 23 5717.8 24 5827.2 Table 7: Comparison of experimentally measured natural frequencies with those from the finite element model.
References [1] M. A. Blair, "Space station module prototype modal tests: Fixed base alternatives," Kissimmee, FL, USA, 1993, pp. 965-971. [2] R. L. Mayes and L. D. Bridgers, "Extracting Fixed Base Modal Models from Vibration Tests on Flexible Tables," in 27th International Modal Analysis Conference (IMAC XXVII) Orlando, Florda, 2009. [3] D. de Klerk, D. J. Rixen, and S. N. Voormeeren, "General framework for dynamic substructuring: History, review, and classification of techniques," AIAA Journal, vol. 46, pp. 1169-1181, 2008. [4] D. J. Ewins, Modal Testing: Theory, Practice and Application. Baldock, England: Research Studies Press, 2000. [5] A. P. V. Urgueira, "Dynamic Analysis of Coupled Structures Using Experimental Data," in Imperial College of Science, Technology and Medicine London: University of London, 1989. [6] M. S. Allen and R. L. Mayes, "Comparison of FRF and Modal Methods for Combining Experimental and Analytical Substructures," in 25th International Modal Analysis Conference (IMAC XXV) Orlando, Florida, 2007. [7] M. S. Allen, R. L. Mayes, and E. J. Bergman, "Experimental Modal Substructuring to Couple and Uncouple Substructures with Flexible Fixtures and Multi-point Connections," Journal of Sound and Vibration, vol. Submitted Aug 2009, 2010. [8] R. L. Mayes, P. S. Hunter, T. W. Simmermacher, and M. S. Allen, "Combining Experimental and Analytical Substructures with Multiple Connections," in 26th International Modal Analysis Conference (IMAC XXVI) Orlando, Florida, 2008. [9] Q. Zhang, R. J. Allemang, and D. L. Brown, "Modal Filter: Concept and Applications," in 8th International Modal Analysis Conference (IMAC VIII) Kissimmee, Florida, 1990, pp. 487-496. [10] J. H. Ginsberg, Mechanical and Structural Vibrations, First ed. New York: John Wiley and Sons, 2001. [11] T. G. Carne, D. Todd Griffith, and M. E. Casias, "Support conditions for experimental modal analysis," Sound and Vibration, vol. 41, pp. 10-16, 2007. [12] M. S. Allen and R. L. Mayes, "Estimating the Degree of Nonlinearity in Transient Responses with Zeroed Early-Time Fast Fourier Transforms," in International Modal Analysis Conference Orlando, Florida, 2009.
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[13] [14] [15] [16] [17]
M. S. Allen and J. H. Ginsberg, "A Global, Single-Input-Multi-Output (SIMO) Implementation of The Algorithm of Mode Isolation and Applications to Analytical and Experimental Data," Mechanical Systems and Signal Processing, vol. 20, pp. 1090–1111, 2006. M. S. Allen and J. H. Ginsberg, "Global, Hybrid, MIMO Implementation of the Algorithm of Mode Isolation," in 23rd International Modal Analysis Conference (IMAC XXIII) Orlando, Florida, 2005. M. S. Allen and J. H. Ginsberg, "Modal Identification of the Z24 Bridge Using MIMO-AMI," in 23rd International Modal Analysis Conference (IMAC XXIII) Orlando, Florida, 2005. P. Ind, "The Non-Intrusive Modal Testing of Delicate and Critical Structures," in Imperial College of Science, Technology & Medicine. vol. PhD London: University of London, 2004. M. Imregun, D. A. Robb, and D. J. Ewins, "Structural Modification and Coupling Dynamic Analysis Using Measured FRF Data," in 5th International Modal Analysis Conference (IMAC V) London, England, 1987.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
How Bias Errors Affect Experimental Dynamic Substructuring 1
De Klerk, Dennis1 ,2 M¨ uller-BBM VibroAkustik Systeme B.V., [email protected] 2 Delft University of Technology, [email protected]
Abstract As it turns out experimental Dynamic Substructuring is very sensitive to all kinds of measurement and modeling errors. In [1, 9] a statistical method was formulated to investigating the effect of random errors, due to measurement noise for example, on experimental dynamic substructuring. Shown that problems can especially occur at eigenfrequencies of lightly damped substructures, it doesn’t really explain why coupled FRF are so often completely off in a broad frequency range. This paper adds by investigating the effect of bias errors in driving point FRFs at experimental substructure interfaces. These bias errors get into existence from a different impact location from the hammer on the substructure
Introduction It was found in [9] that especially FRFs in the interface flexibility matrix (IFM) 1 are paramount in getting accurate coupling results, as their uncertainties spread to all other FRFs of the coupled system. The numerical investigation showed that small random errors can already introduce high inaccuracies in lightly damped systems. The experimental investigation on a complex, yet highly damped, vehicle [1] however showed that random noise only partly causes erroneous coupling results. Indeed, it’s mostly bias errors which affect the coupling results, e.g. systematic measurement errors and / or errors in the model description (here one can for example think of the number of coupling DOF used on the continuous experimental interface). As the IFM matrix is the most important success factor in experimental DS, it is worthwhile investigating the effect of erroneous driving point measurement at the substructure’s interface. When measuring them, one easily excites aside from the acceleron meter with an impulse hammer. If the structure permits, sometimes one can excite from the other side in-line with the acceleron meter, minimizing errors. Yet in general one does not excite at a true driving point location in experiments, hence introducing a bias error on the subsystem FRF. In any case one can question himself how accurately one has to position the excitation? When individual interface nodes get closer, probably position accuracy will become of more importance. In general one will not be able to determine the bias error on the driving point FRF. As such a compensation method is hard to develop, especially when using experimental data. It is therefore proposed to carry out a sensitivity analysis. As such one can investigate how sensitive the coupled FRF are upon excitation location. Here the investigation is twofold: 1
See section for an explanation of the IFM matrix
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_97, © The Society for Experimental Mechanics, Inc. 2011
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• If one excites the structure with the impulse hammer by hand, a repeatability of 1mm would probably be challenging to achieve. If one averages multiple FRF, reducing influences of measurement noise, these misalignments introduce an additional uncertainty on the FRF. In this paper it is investigated how these misalignment between excitation position and driving point acceleration measurement affect the coupled system FRF analytically. The uncertainty method used is described in [9]. • Due to the uncertainty on the excitation location one might well decide to use only one (not averaged) FRF in de DS coupling. Still one can investigate the coupled FRF’s sensitivity for the impact position with one additional impact measurement In section 1 the experimental DS algorithm used for the investigation is briefly introduced. The method carries the name Lagrange Multiplier Frequency Based Substructuring (LMFBS) [3]. In section 2 the uncertainty propagation method used in this investigation is briefly introduced. More details can be found, for example, in [9]. In section 3 both methods are used to formulate a sensitivity analysis for erroneous driving point excitations. A numerical validation of the method is found in section 4, where the analytical sensitivity analysis is compared with a Monte-Carlo simulation.
1
Lagrange Multiplier Frequency Based Substructuring
First assume the subsystems are assembled, block diagonal, in the receptance matrix of the uncoupled systems:
u = Yf Y
diag ⎡
⎡
Y
(1)
⎤
,..., Y ⎡
(n)
⎢ =⎣
Y (1)
⎤
· ·
· .. ·
· .
·
⎤
(1)
⎥ ⎦
Y (n)
u(1) f (1) ⎢ ⎥ ⎢ ⎥ u ⎣ ... ⎦ , f ⎣ ... ⎦ u(n) f (n)
(2)
After defining a boolean matrix B which expresses the compatibility between the subystems2 , the coupled FRF are found as:
Y tot = Y − Y B T (BY B T )−1 BY
(3)
Where (BY B T ) describes the substructures’ interface flexibilities.
2
Uncertainty Propagation Method
To analyze the effects of small random errors in experimental Dynamic Substructuring in a systematic way, the uncertainty propagation algorithm derived in [1, 9] is briefly introduced. This method allows to quantify the uncertainty on the FRF matrix entities of experimentally obtained substructure models and, more importantly, allow to determine 2
Please refer to appendix A for more details.
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how these uncertainties propagate in the DS calculations. As such it will be possible to determine the accuracy of the assembled system based on the measured time data. For the development of the uncertainty propagation method the efficient “moment” method in combination with the Lagrange Multiplier Frequency Based Substructuring (LM FBS) method described in the previous section [2, 8]. Assume now a set of n input variables xi assembled in vector x [x1 · · · xn ]T , which have known mean value x ¯i and standard deviation Δxi . Let g be a function of the variables in x. The moments of the function g(x) can then be calculated from a truncated Taylor series expansion about the mean value of the input variables. In this thesis the input uncertainty are small, obey a Gaussian distribution and all functions considered ¯ are continuous and can be linearized around the mean value of the input variables x. Therefore a first order Taylor series expansion suffices to obtain approximations for the first and second moments.3 Approximating the function g(x) by a first order Taylor series around the mean values of the input variables then gives n ∂g ¯ + g(x) ≈ g(x) ¯i ) . (4) (xi − x ∂xi ¯ x
i=1
Using the statistical rules [4] to compute the first moment (or, equivalently, the mean or expected value denoted by E[...] of function g(x) yields n ∂g (xi − x ¯ +E E [g(x)] = E [g(x)] ¯i ) , (5) ∂xi x¯ i=1
in which the second term is equal to zero, since n n ∂g ∂g (xi − x E [(xi − x E ¯i ) = ¯i )] = 0. ∂xi x¯ ∂xi x¯ i=1
(6)
i=1
The first moment of the function g(x) therefore yields: ¯ E[g(x)] = g(x).
(7)
The second moment of the function g(x), its variance, can be found from [4, 10] Var[g(x)] E[(g(x) − E[g(x)])2 ].
(8)
Application of this expression to (4) gives: ⎡ 2 ⎤ n ∂g ¯ + Var[g(x)] = E ⎣ g(x) (xi − x ¯i ) − E[g(x)] ⎦ ∂xi i=1 ⎡ ⎞⎤ ⎛ n n ∂g ∂g = E⎣ (xi − x ¯i ) ⎝ (xj − x ¯j )⎠⎦ ∂xi ∂xj i=1
=
n n i=1 j=1
3
j=1
∂g ∂g E[(xi − x ¯i )(xj − x ¯j )] ∂xi ∂xj
(9)
These approximations are usually called first order, first moment and first order, second moment (FOFM and FOSM) approximations, respectively.
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¯ is omitted for clarity. where the indication that the Jacobian of g is to be evaluated at x The expression now simplifies with the knowledge of the covariance [4] Cov[xi , xj ] E[(xi − E[xi ])(xj − E[xj ])]
(10)
to Var[g(x)] =
n n ∂g ∂g Cov[xi , xj ]. ∂xi ∂xj
(11)
i=1 j=1
For simplicity it is assumed that the variables are uncorrelated (or statistically independent), so they can vary fully independently and the covariance is equal to zero. This will hold in for the excitation location offset, as they are independent events for multiple driving points. By assuming no correlation the expression for the variance (second moment) of g(x) now simplifies to: n ∂g 2 Var[g(x)] = Var[xi ]. (12) ∂xi i=1
In practice it is convenient to express the spread in terms of the standard deviation, which has the same unit as the function itself. Note that the standard deviation is the positive square root of the variance and the standard deviation in the input variables was defined as Δxi , so that the second moment of the function g(x) is found to be n 2 ∂g Δg(x) = Δxi . (13) ∂xi i=1
This last equation will form the starting point for the uncertainty propagation method for the LM FBS calculations, although it first needs to be generalized in case the function g is a matrix function. Since all matrices are linear with respect to their entries, the derivative of a matrix with respect to any of its entries may be written as: ∂G Pij , ∂Gij
(14)
where matrix Pij is a “Boolean” type of matrix with the same size as matrix G itself. The elements of Pij are all zero except for entry (i, j), which equals one. With the definition of Pij one can now write the standard deviation of a matrix function as ! $ n m " #2 $ ! m n ∂G ΔGkl = {Pkl ΔGkl }2 , (15) ΔG = ∂Gkl k=1 l=1
k=1 l=1
where G has dimension n-by-m. Here the curly-bracket notation {· · ·} is introduced to indicate that the square and square root operations must be performed elementwise.4 Another helpful result needed for the upcoming uncertainty propagation analysis is the derivative of the inverse matrix G−1 to the elements Gij [7] ∂G−1 ∂G −1 = −G−1 G = −G−1 Pij G−1 . ∂Gij ∂Gij 4
(16)
Notice that (15) thus states that the uncertainty on a matrix is the sum of the uncertainties of its entries.
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3
Sensitivity of Impulse Hammer Excitation Location in Experimental Dynamic Substructuring
Assume one measures the subsystems’ FRF with an impulse hammer with fixed acceleron meters on the structures interface. Apart from mass loading, added stiffness and damping from the sensors cabling and others, these sensors are fixed during the experiment. Although they do not measure the acceleration on the subsystem’s interface, e.g. in the heart of the sensor usually a (few) millimeter above the surface. Ideally we would hit the sensor in line with the axis of measurement to determine the driving point FRF (the author actually experimented in this way and had good success with it...). However, standard procedure is to hit aside from the sensor, hence introducing a bias error. It is unclear how much this error will affect the coupled system FRF and one would like to have an idea how much this could be. In this study a sensitivity approach is therefore introduced to investigate its impact. First assume one does not excite the structure at a true driving point node, but for example a few millimeters aside. Then one introduces a systematic / bias error in the uncoupled FRFs which can be expressed by:
Y bias = Y true + ΔY
(17)
where Y true is the theoretically correct FRF with true driving point excitation and ΔY is the bias error introduced by the wrong impulse hammer excitation. In general we will not be able to identify ΔY exactly, yet if so, one obtains the true receptance matrix. If the error is not known though, the question rizes how much the bias error affects the coupled system FRF. In order to examine this affect, a sensitivity analysis is proposed. Indeed, a second hit just a little bit next to the previous point will introduce a different FRF. This difference is a function of the distance between both points. This difference can be expressed as a sensitivity, e.g.
Y meas = Y true + ΔY + S(x)x
(18)
where x is a vector with the distance between the impulse hammers excitation point and the (biased) center point. Notice that in general this function will be nonlinear if the distances become larger. However, the sensitivity matrix S(x) can be linearized about the best driving point position possible. Notice the error in the measured receptance matrix is now a function of hitting position of the impulse hammer x. This vector has as many entries as the number of driving points measured on the substructure. Notice therefore that if one would have 2 driving point FRF for example and a total of 9 combined interface and internal nodes, the receptance matrix would have 18 entries. The errors in each column are described however only with two position variables. Using the theory from the previous section one can now express the error in the coupled system as follows:
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ΔY tot
N ∂Y tot = { Δxii }2 ∂Δxii
(19)
i
where {. . .} denote elementwise matrix multiplication, xii are elements of diagonal matrix x and Δxii denote the standard deviation or confidence interval on x. Furthermore ΔY tot expresses the standard deviation or confidence interval on the coupled system FRF. With the derivative of a vector to one of its elements defined by:
Pii
∂Δx ∂xii
(20)
and the derivative of an inverse matrix to itself as: ∂Y −1 ∂Y −1 = −Y −1 Y ∂Yij Yij
(21)
substitution of (3), (17) and (18) in (19) yields after some manipulation:
ΔY tot =
N
{(SPii − SPii E1 + E2 SPii E1 − E2 SPii )Δxii }2
(22)
i
With:
E1 = B T (B(Y bias )B T )−1 BY bias E2 = Y bias B T (BY bias B T )−1 B (23) Please observe that Y bias should be the best measured FRF, e.g. where one minimized the unknown bias error as best as possible in the experiment. This might well mean one uses only one single measured FRF. One can determine the sensitivity matrix using an additional FRF where the distance between both excitation points (x) is measured. From (22) one can observe that a more sensitive structure, e.g. high valued S, yields higher uncertainty / errors on the total system FRF, as well as a high standard deviation in Δxii . The latter can be influenced by proper measurements, yet the sensitivity matrix is defined by the system properties and therefore not influenceable. If one uses only one FRF, a normalized standard deviation should be used. When the standard deviation is determined experimentally with multiple hits, measured standard deviations could be used.
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4
Numerical Validation of Sensitivity Analysis with MonteCarlo Simulation
For the numerical validation of the analytical sensitivity method formulated in section 3, the method is compared to a Monte-Carlo simulation. The numerical model used describes the bending of Thimoshenko beams with one side fixed. The model is based on [6, 5] where the beams were investigated experimentally as well. In subsection 4.1 the model is briefly described. In subsection 4.2 one thereafter finds the validation.
4.1
FE model of slender beams
The free bending of an Euler-Bernoulli beam in the x,z plane can be analytically defined as:
EIyy
∂4z ∂2z + ρS =0 ∂ 4 x4 ∂t2
(24)
with E module of elasticity, Iyy moment of stiffness, ρ material density and S the cross section area. Here assumptions of linearity, small displacements and uniform material properties are assumed to hold. The solution of this differential equation looks like:
z(x) = α1 cos(αx) + α2 sin(αx) + α3 cosh(αx) + α4 sinh(αx) where α =
%
ω β
% and β =
(25)
EIyy ρS .
Using the boundary condition at both sides of the beam, one finds α1 . . . α4 as: ⎡
⎤ ⎡ z(x = 0) 1 0 1 0 ⎢ −z,x (x = 0) ⎥ ⎢ 0 α 0 α ⎥ ⎢ un = ⎢ ⎣ z(x = L) ⎦ = ⎣ cos(αL) sin(αL) cosh(αL) sinh(αL) −z,x (x = L) α sin(αL) −α cos(αL) −α sinh(αL) −α cosh(αL)
⎤⎡
⎤ α1 ⎥ ⎢ α2 ⎥ ⎥⎢ ⎥ ⎦ ⎣ α3 ⎦ (26) α4
One now has the interpolation function since z(x) = N (x)un . Using the variational form of the equation of motion, one thereafter finds the dynamic stiffness matrix as: ⎡
Z11 EIyy α ⎢ ⎢ Z11 Z= 2 L γ ⎣ Z11 Z11 with γ = 1 − cosh(αL) cos(αL) and
Z11 Z11 Z11 Z11
Z11 Z11 Z11 Z11
⎤ Z11 Z11 ⎥ ⎥ Z11 ⎦ Z11
(27)
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Z11 = Z33 = (αL)2 (sinh(αL) cos(αL) + sin(αL) cosh(αL)) Z22 = Z44 = L2 (sin(αL) cosh(αL) − cos(αL) sinh(αL)) Z12 = −Z34 = −αL2 sinh(αL) sin(αL) Z13 = −(αL)2 (sinh(αL) + sin(αL)) Z14 = −Z23 = αL2 (cos(αL) − cosh(αL)) Z24 = L2 (sinh(αL) − sin(αL)) A perfect match was obtained with this model and a real experiment in [6, 5], where following specifications were used:
4.2
Validation with Monte-Carlo
Various coupling were performed with both the analytical and experimental beam. Here the beam was excited with impulses with an shaker, and the velocities were measured by laser, both at 5 (interface) driving points at the tip of the beam. The driving point had an equidistant spacing of 5 mm and each point was measured in the middle of the beam, see figure 1. Although the measurements almost seemed perfect, coupling results L
S,Iyy
z
5mm
x
Figure 1: Schematic presentation of the analyzed beam with excitation points and used variables.
did show quite large differences. It is now analyzed if these differences could be caused by (already small) errors in the driving point measurements. Now coupling is performed with either 2 or three nodes though, for ease of interpretation. The coupling nodes are defined 5, 10 and 15 mm from the beam’s end respectively. When coupling the system with 2 nodes, the middle interface node is left from the coupling. Figure 2 shows a typical interface FRF where the hammer excitation has a relative offset as denoted in the legend. The figure shows that anti-resonances change in frequency, and if zoomed in, the amplitude of the resonances change. It shows that a wrong excitation location results in a different interface stiffness and hence will change the coupled system FRF. In (19), the sensitivity matrix S is linearized about a configuration. It was already noted that the sensitivity matrix is nonlinear with respect to (larger) excitation offsets. In order to examine whether a linearization is omitted, figure 3 shows the FRF entries in the complex plane for both resonances frequencies (158 and 922 Hz) and anti-resonance E ρ S Iyy L f η
= = = = = = =
210.500.000 [Pa] 7.800 [kg/m3 ] 0,0039 x 0,0005 [m2 ] 4,0625 x10−10 0,1644 [m4 ] 1 . . . 1.600 [Hz] 0,016
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Magnitude [-]
1109 DP with different impact location
-2 10
Legend: -3 mm -2 mm -1 mm 0 mm +1 mm +2 mm +3 mm
-4 10
-6 10
-8 10
0
500
1000
1500 Frequency [Hz]
Figure 2: Typical Driving Point FRF with different excitation point offsets.
(511 Hz). It can be seen from this figure, that the changes are quite linear in the range of a few millimeters offset. If an offset, e.g. linearization distance, of 1 millimeter is used, the sensitivity matrix obtains amplitudes as shown in figure 4. It can be seen that the changes are highest at resonance frequencies (in absolute form). Now, using the uncertainty propagation method introduced in section 3, equation (19), with a standard deviation of 1 mm, the uncertainty on the coupled FRF are examined. Here the experimental beam is coupled to it’s numerical / analytical equivalent. After the coupling one therefore obtains the elongated beam, resulting in the FRF displayed in figure 5. The thick green FRF is the analytically correct FRF, having no bias error. The thick red line shows the uncertainty of that FRF when a standard deviation of 1mm in the excitation location offset is used. Note the additional “resonance” at 158 Hz. The coupled FRF has a large chance containing the eigenfrequency of its subsystem. Furthermore the first eigenfrequency of the coupled system is very sensitive to the excitation offset as well. The additional FRF in figure 5 show the Monte-Carlo simulation with 50 samples using the same standard deviation of 1 mm. Both the simulations resemble one another reasonably, although the nonlinear uncertainty propagation algorithm [9] might be in place. Figure 6 shows the same results when 3 coupling DoF are used. The figure shows that the more interface nodes are used, the problems get more severe. This is caused by the additional uncertainty
References [1] de Klerk, D. Dynamic Response Characterization of Complex Systems through Operational Identification and Dynamic Substructuring. Phd thesis, Delft University of Technology, http://www.3me.tudelft.nl/live/pagina.jsp?id=73932050-a4ca-46b5-bd9bd98b8a52dbe6, March 2009. [2] de Klerk, D., Rixen, D., and Voormeeren, S. A General Framework for Dynamic Substructuring. History, review and classifcation of techniques. AIAA Journal (2008). [3] de Klerk, D., Voormeeren, S., and Rixen, D. A General Framework for Dynamic Substructuring: History, Review, and Classification of Techniques.AIAA 46, 5 (2008), 1169– 1181. ¨, H., and Meester, L.Probability and Statistics [4] Dekking, F., Kraaikamp, C., Lopuhaa for the 21st Century. Delft University of Technology, Delft, The Netherlands, 2004.
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Imag Part
1110 -4 -7.2 x10
FRF Sensitivity @ 158 Hz
-7.3 -7.4 +1
-7.5
mm -1m
m
-7.6 -7.7 -7.8 1.86
1.88
1.90
1.92
1.94
-4 x10 1.96
Imag Part
Real Part
-9 -5.4 x10 -5.6 -5.8 -6.0 -6.2 -6.4 -6.6 -6.8 -7.0 -7.2 -1.185
FRF Sensitivity @ 511 Hz
-1m +1
-1.18
m
mm
-1.175
-1.17
-1.165
-6 x10 -1.16
Imag Part
Real Part
-7 -0.95 x10
FRF Sensitivity @ 922 Hz
-1.00 -1.05 +1
-1.10
mm -1m
-1.15
m
-1.20 -1.25 -1.30 -1.35 4.5
5.0
5.5
6.0
6.5
7.0
-7 x10 7.5
Real Part
Figure 3: Typical Driving Point FRF with different excitation point offsets. [5] Pagnacco, E., Gautrelet, C., Paumelle, J., and Lambert, S. Frequency Based Substructuring without R-Dof Measurements: A Two-Beam Test Case.In 20th International Congress of Mecahnical Engineering November 2009 . [6] Paumelle, J. Investigations in Assembly of Measured Substructures Characterized by Flexural Behavior using the FBS Method. Master’s thesis, INSA de Rouen, Department de Macanique, filiere mecanique des structures, 2009. [7] PlanetMath. http://www.planetmath.org. [8] Voormeeren, S.Improvement of Coupling Procedures and Quantification of Uncertainty in Experimental Dynamic Substructuring Analysis; Application and Validation in Automotive Research. Master’s thesis, TU Delft, 2007. [9] Voormeeren, S., de Klerk, D., and Rixen, D. Uncertainty Quantification in Experimental Frequency Based Substructuring. Mechanical Systems and Signal Processing (MSSP) 24, 1 (Januar 2010), 106–118. [10] Wikipedia. The free encyclopedia. http://www.wikipedia.org.
5
Appendix: Construction of Boolean Matrix in DS framework
This appendix illustrates the construction of the Boolean matrices B. To this end, the general system shown in figure 7 is considered, this figure schematically shows
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Magnitude [-]
1111 -4 10
Sensitivity Matrix Entries Legend: S11 S22 S33 S44
-5 10 -6 10 -7 10 -8 10 -9 10 -10 10 0
200
400
600
800
1000 1200 1400 1600
Frequency [Hz]
Figure 4: Sensitivity of FRF in complex plane at different frequencies. The two eigenfrequencies and the anti-resonance are displayed. Note that the changes behave quite linear in the millimeter range.
Magnitude [-]
Error Propagation & Monte-Carlo Legend: Std Analytical Aver Analytical *** Monte-Carlo
-4 10 -6 10 -8 10 -10 10 0
200
400
600
800
1000 1200 1400 1600
Frequency [Hz] Figure 5: Uncertainty on coupled system FRF using both the analytical uncertainty propagation method and a Monte-Carlo simulation. As can be seen, the coupled FRF change heavily when two interface coupling nodes are used.
the coupling of two general substructures. Both substructures consist of 3 nodes; substructure A has 4 degrees of freedom while substructure B holds 5 DoF. In this example, nodes 2 and 3 of substructure A are coupled to nodes 5 and 6 of substructure B, respectively. So, three compatibility conditions should be satisfied: ⎧ ⎨ u2x = u5x u2y = u5y (28) ⎩ u3x = u6x To express this condition one can define Bu = 0, where B is a signed Boolean matrix that must be constructed. The total vector of degrees of freedom u is: T u = u1y u2x u2y u3x u4x u4y u5x u5y u6x The signed Boolean matrix B is now found as: u1y u2x u2y u3x u4x u4y u5x u5y u6x ⎡ ⎤ 0 1 0 0 0 0 −1 0 0 0 1 0 0 0 0 −1 0 ⎦ B = ⎣ 0 0 0 0 1 0 0 0 0 −1
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Magnitude [-]
Error Propagation & Monte-Carlo Legend: Std Analytical Aver Analytical *** Monte-Carlo
-4 10 -6 10 -8 10 -10 10 0
200
400
600
800
1000 1200 1400 1600
Frequency [Hz] Figure 6: Uncertainty on coupled system FRF using both the analytical uncertainty propagation method and a Monte-Carlo simulation. As can be seen, the coupled FRF change heavily when three interface coupling nodes are used. u4y u4x
(b) u5y u5x u2y
u2x u6x
y
(a)
u1y
u3x
x
Figure 7: Coupling of two arbitrary substructures to illustrate the formulation of the Boolean matrices B.
Every coupling, or equivalently, every compatibility condition, corresponds to a line in the Boolean matrix B. Therefore, in the general case where the coupled substructures comprise n degrees of freedom of which m are coupled interface DoF, the matrix B has size m-by-n. In this example, n = 9 and m = 3; the size of B is 3-by-9. It can easily be seen that the condition Bu = 0 is equivalent to the three compatibility equations in equation (28).
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
A Benchmark Test Structure for Experimental Dynamic Substructuring
P.L.C. van der Valk; J.B. van Wuijckhuijse; D. de Klerk Delft University of Technology, Faculty of Mechanical, Maritime and Materials Engineering Department of Precision and Microsystem Engineering, section Engineering Dynamics Mekelweg 2, 2628CD, Delft, The Netherlands [email protected]
Abstract In this paper a benchmark test structure for experimental dynamic substructuring is presented. The benchmark is a simple structure designed to gain insight into difficulties experienced in experimental dynamic substructuring (DS). First a brief introduction of dynamic substructuring is presented, followed by a summary of current bottlenecks in experimental DS. From these difficulties a set of requirements for the benchmark is formulated. Thereafter the design is presented and its numerical model is validated by a measurement on the fabricated benchmark. Finally a DS analysis is performed on the benchmark structure to show it’s ability to quickly verify or falsify a DS analysis.
1
INTRODUCTION
Dynamic substructuring is a very useful tool in structural dynamics and is becoming increasingly popular in the engineering society. Dividing large structures into smaller substructures has several advantages in structural dynamic analysis; first and foremost it allows the coupling of numerical substructures with experimentally obtained components in order to compute the dynamic behavior of the total system. Another big advantage is that the dynamic behavior of systems that would otherwise be too large, complex or time consuming to measure and/or simulate can be determined. More benefits of (experimental) dynamic substructuring are described in [5]. There are three domains in which dynamic substructuring can be performed; the ‘physical’ domain (using physical DoF u), the modal domain (using reduced substructures) and the frequency domain (using FRFs). In addition two types of DS can be identified; numerical dynamic substructuring and experimental dynamic substructuring. In a numerical DS analysis, only numerical substructures are coupled, these substructures can either be full or reduced FEM models (using CMS methods [1, 8, 2, 13]). Numerical DS is already well developed and accepted within the engineering community; for instance Guyan reduction and the method of Craig-Bampton are already integrated in many FE packages. Experimental DS involves both numerical substructures and substructures obtained through measurements (’experimental’ substructures). This is done in either the modal domain using CMS methods [11] or in the frequency domain using FRF coupling methods [5, 9, 3]. Experimental DS was first developed in the 1980’s and is still an interesting research field: There remain difficulties which one could encounter when performing an experimental DS analysis, these are described in section 2. There was a need for a simple and versatile benchmark test structure to investigate these difficulties, but also other phenomena (such as non-linearity, high damping, etc.) which could be present in a (sub)structure. The goal of this paper is to present this benchmark test structure for experimental dynamic substructuring. The design will be shown in Sec. 3. The validation of the FEM model will be done in Sec. 4, where also the results of the first DS analysis will be presented.
2
DIFFICULTIES IN EXPERIMENTAL DYNAMIC SUBSTRUCTURING
When performing an experimental DS analysis, there are some important issues that have to be dealt with in order to achieve accurate results. If these difficulties are not properly dealt with, significant errors might be present in the coupled system representation. All these issues are caused by the fact that it is virtually impossible to (properly) measure all properties of the different substructures. The different methods require their own steps, each step with their own difficulties. For a coupling analysis with CMS methods, a modal analysis has to be done in order to acquire the first m modes of the subsystem. When coupling techniques in the frequency domain are used, one couples directly with the obtained FRF’s,
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_98, © The Society for Experimental Mechanics, Inc. 2011
1113
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1114 hence no modal analysis is required [5]. A brief overview of the different issues often encountered in dynamic substructuring is given below.
Truncation errors A problem encountered when using CMS for experimental DS is modal truncation. Since only the first m modes of the (sub)structure are extracted from the measurements, the information from the higher modes is lost. This leads to a stiffer system with less degrees of freedom to deform in. The result can be improved by adding residual flexibility, this is one deflection shape approximating all higher modes. Even though shifts in resonance frequencies of the coupled system and a better result of the DS analysis can be expected, an error will still be made.
Modal Analysis Modal analysis might not always be possible, since the structure could, for instance, be non-linear, have high damping or have a very high modal density. In such cases, direct coupling using the measured FRF’s can be more appropriate.
Rigid Body Modes Rigid body mode information is essential in coupling (unrestrained) substructures using CMS. If the rigid body modes are not included, the coupling of substructures will give erroneous results. Since the structure will always move in a combination of rigid and flexible modes, this effects the entire frequency range. This is only a problem when using CMS, since measured FRF’s will always contain all the rigid body mode information.
Rotational Degrees of Freedom One of the classical problems encountered in experimental DS is the measurement and excitation of the rotational degrees of freedom. The rotational information is essential in order to obtain the full receptence matrix (1). Due to the coupling between the rotational and translational DoF, both have to be determined to give an accurate representation of the interface, neglecting this will give erroneous results on the coupled system representation, see also [6, 4].
ut uθ
=
Ytt Yθt
Ytθ Yθθ
ft fθ
(1)
Displacement vectors are denoted by u, force vectors by f and the transfer functions by Y . Translational information is denoted by the subscript t and rotational information by the subscript θ. There are different approaches to obtain rotational information. One can either try to measure them or reconstruct the rotational information from measured translational data. One way to do the latter is to assume that the interface has only six rigid body motions, which can be reconstructed from a minimum of six DOF on three nodes (see [16]). This method will show good results if the assumption of the rigid interface is valid.
Continuity of the interface In practical applications all interface connections are continuous surfaces, when usually only a limited number of points are measured. The continuous behavior of the interface is than approximated from this discrete number of measurement points, thus creating a truncated description of the interface motions.
Dynamics of joints Usually connections are either modeled as rigid or with linear flexible joints (see [9, 10]). In engineering practice, however, it shows that a lot of connections show non-linear dynamic behavior. An example of this non -linear behavior is the dry friction which occurs between bolted parts.
Experimental errors As the name experimental dynamic substructuring already implies, experimentally acquired data is used to describe the coupled system. It is obvious that measurement errors, can directly affect the accuracy of the DS analysis. Numerous errors can be made while performing measurements; a selection is briefly discussed below. • Measurement noise: Since some random measurement noise is practically unavoidable, measures have to be taken to reduce it as much as possible. One effective way of minimizing random noise is to average over a large set
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1115 of measurements. The effect of random noise is described in [7]; • Sensor positioning and alignment: Anti resonances can be very sensitive to the exact location of the excitation; • Unmeasured side forces from the stinger: By misalignment or due to bending of the stinger, unmeasured side forces are introduced into the system, thus falsifying the FRF estimates; • Added mass effect: Mass will locally be added due to attached measurement equipment. This mass loading effect will alter the FRF’s; • Signal processing errors: Several sorts of signal processing errors could occur, for example due to leakage and errors from converting signals from analog to digital; • Local non-linear behavior in the (sub)structure: This is for example due to frequency dependent behavior (e.g. rubber) or dry friction between bolted connections; • Influence of the suspension: Measurements of lightly damped systems can be heavily affected due to damping of the suspension. It is also possible that the lower eigenfrequencies will shift due to the suspension stiffness.
3
DESIGN OF THE BENCHMARK STRUCTURE
From section 2 it can be concluded that there are two major challenges in improving the accuracy of the experimental DS analysis. On the one hand it is to reduce the experimental errors as much as possible in order measure the‘true’ behavior. On the other hand it is to identify and take into account all the different phenomena which occur in the total structure. Taking into account all the difficulties described in section 2 as well as the fact that structure should be very versatile, a set of requirements for the benchmark structure has been formulated. This set will be presented in section 3.1. In section 3.2 the design is presented and the design choices are explained.
3.1
Set of requirements
Connecting different substructures and elements Since the benchmark structure should be able to connect to different substructures but also different kinds of passive and active components (e.g. springs, actuators), a broad range of possible connections should be available. These interfaces have to be able to accommodate different types of connections, which also have to be detachable without causing damage to benchmark setup. From these requirements and [15] the following set of connection methods are chosen: bolted, blind riveted, glued, soldered and clamped connections.
Eigenfrequencies In order to make sure that the flexible modes of the system are well separated from its rigid modes, the first flexible mode of the system should not be below 50 Hz. Since a modal analysis of the benchmark has to be possible, it must have very low damping and eigenfrequencies should be well spaced.
Obtaining rotational information In sec. 2 the importance of rotational information is discussed. In the design there should be a section which can be assumed rigid or can be made rigid, to enable use of the EMPC method for determining rotations. In this method the interface is assumed to be a rigid section with six DoF (three translational and three rotational), which are constructed by measuring the translational DoF on three points [16]. Reproducibility An important feature of the setup is the reproducibility of results. Materials that show temperature or environmental dependent behavior or are susceptible to aging should be avoided (i.e. rubber). It is also important that the setup itself is reproducible, meaning the benchmarks should differ no more than 0.1 mm from each other. Cost In order to allow ’destructive’ testing (e.g. welded connections) or modifications to the benchmark, the cost per piece should be kept low, therefore one can choose to order them in batches.
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1116 From these requirements it can be concluded that the DS benchmark should be a relative simple setup which shows linear behavior, so it can easily be measured and modeled. It should also be versatile and adaptive so all the phenomena mentioned above can be investigated.
3.2
Final design
Figure 1 shows the design of the benchmark structure.
Figure 1: Benchmark structure
The benchmark system is a simple but versatile structure because it is accessible in three dimensions and expandable by using multiple structures. All connections issued in section 3.1 are applicable using the pre-drilled holes and geometric features. The versatility of the structure lies not only in the fact that a single structure is in many ways adaptable and expandable with new elements and can be measured and excited in many directions, but also that multiple benchmark structures can be combined to form different geometries. Two examples are shown in figure 2.
(a)
(b)
Figure 2: a and b show two possible combinations
In figure 2a two structures are connected in such a way that a cavity forms between them, in which items can be placed (e.g. springs, dampers). The dynamic effects of these components on the experimental DS analysis than can be investigated easily since the dynamics of the main body (in this case two benchmark structures) are well known dynamically. In figure 2b another possible configuration using two benchmark structures is shown. The most important features of the benchmark are:
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1117 • A first eigenfrequency at 61.3 Hz, assuring that the flexible modes are well separated from the rigid body modes; • Well separated eigenfrequencies up to 1 kHz, with a minimum of 3 Hz separation, so all the modes can easily be found by performing a modal analysis; • Double (but distinctive) modes, due to the single symmetric geometry. By attaching other structures the setup can be made totally unsymmetrical or double/triple symmetrical; • Sections can locally be ”rigified” by attaching a rigid part to the plate, thus allowing for the implicit measurement of rotations using multiple point connections; • Linear behavior and very little damping (e.g. damping ratios less than 0,01% and mainly due to the suspension). The simplicity of the structure however is a certainty since it is made out of plain 5 mm stainless steel and can be manufactured by use of laser cutting and machinal bending. Since the whole process is governed by numerical controls the reproducibility is very high and also non expensive, enabling destructive testing at little cost. The accuracy of the manufacturing process lies within 0.1mm. 4
MEASUREMENTS
In this section two different measurements will be handled. In section 4.1 the model will be validated by modal analysis on the structure. In section 4.3 a first DS analysis will be performed by attaching a simple mass to the benchmark structure and the result will be compared to a measurement.
4.1
Model validation
In this section the numerical model of the benchmark structure will be validated by comparing it with a measurement on the structure. In order to validate the model, a few properties of the material used (stainless steel AISI 304) are given: • Isotropic material model; • Young’s modulus, E = 200 GPa; • Poisson’s ratio, ν = 0.29; • Density, ρ = 8000 kg/m3 . The first 50 eigenfrequencies and eigenmodes of the model were obtained by FEM analysis in COMSOL using the CADmodel as seen in figure 1. The measurement was performed using Laser Doppler Vibrometry (LDV). The benefit of this measuring method is that no additional mass loading effects of sensors (accelerometers) is seen in the measurement results. In figure 3 the measurement setup is displayed. In order to validate the FEM model of the benchmark, a roving hammer test was performed on the structure using a grid of 45 points. While measuring velocity at one point, the system was excited at all the 45 points in the z-direction. In order to isolate the setup from environmental disturbances, it was suspended using low-stiffness elastic bands. Averaging of the measurements was applied to reduce the random noise on the measurement results. The driving point (DP) FRF is shown in figure 4, where it is compared to the equivalent FRF synthesized from the modes obtained from the FEM model. Eigenfrequencies, eigenmodes and modal damping were obtained by modal analysis from the measurements. A Modal Assurance Criterion (MAC) analysis was performed between the measured modes and the modes obtained from the FEM model. The results are shown in table 1, together with the measured eigenfrequencies and the difference between the measured and computed eigenfrequencies. From table 1 it can be seen that within the frequency range of interest, the deviation of the eigenfrequencies of the FE analysis with respect to the measurement are within 1.4%. The MAC analysis, a more detailed and objective analysis, showed that of 17 modes, 13 have a correlation of 99% or higher. All modes have a correlation of 93% or higher.
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Figure 3: Measurement setup
Measurement DP FRF Model DP FRF
20
0
−20
Mag
−40
−60
−80
−100
−120 0
100
200
300
400
600 500 frequency (Hz)
700
800
900
1000
Figure 4: DP FRF of model and experiment
4.2
Rigidity
In section 2 the importance of rotational information was discussed. In order to use the EMPC method (and assume parts of the benchmark as rigid), the rigidity of the design was checked [16]. The DoF on the interface are projected onto the rigid body modes and compared to the original interface DoF (2).
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1119 Mode 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
feig,meas. (Hz) 61.3 71.3 113 189 200 237 278 370 410 539 590 648 651 900 906 1067
feig,FE vs. feig,meas. (%) -1.4% -0.7% -1.7% -0.4% -0.9% -0.5% 0.3% 0.1% 0.1% -1.4% -0.9% 0.0% 0.0% -0.7% -0.7% -0.9%
MAC (FE vs. Meas.) (%) 100 100 100 100 99 99 99 100 100 99 99 99 100 96 93 99
Table 1: Model validation results
rigidness =
R(RT R)−1 RT uc uc
(2)
100%
The most rigid part of the benchmark was found to be as situated in figure 5a. Figure 5b shows the rigidity of this part over the desired frequency range.
100
Rigidness (%)
95
90
85
0
100
200
(a)
300
400
500 600 frequency (Hz)
700
800
900
1000
(b)
Figure 5: Most rigid section (a); rigidity (b)
Up to 400 Hz the rigidity is higher than 97%. Between 400 Hz and 650 Hz dips can be found down to 93%. From 650 Hz up to 1 kHz, the rigidity is again higher than 97%.
4.3
Numerical DS analysis
After validation of the model a first numerical DS analysis was performed and compared to a measurement on the assembled system (figure 6). After attaching a simple mass, a roving hammer test was performed using the same setup as in section 4.1. The mass was attached to the benchmark structure with 4 bolts (M5, 12.9) torqued to 9 Nm (figure 6), according to [12]. Since the bolted connections will only locally enforce compatibility between the substructures, a diameter around the bolt centerline has been computed in which the substructures are assumed to be rigidly connected. This diameter was determined to be 13.3 mm [14]. The FEM models are coupled at these interfaces and the first fifty eigenfrequencies
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Figure 6: Benchmark structure (substructure 1) with added mass (substructure 2)
and eigenmodes were determined. From these eigenmodes and -frequencies a driving point FRF has been synthesized, in figure 7 this synthesized FRF is compared to the driving point FRF from the measurement. In table 2 the eigenfrequencies computed from the FEM model are compared to the measured eigenfrequencies.
measurement DP FRF substructered model DP FRF
0
−20
Mag
−40
−60
−80
−100
0
100
200
300
400
500 600 frequency (Hz)
700
800
900
1000
Figure 7: NDS, DP FRF of model and measurement
In figure 7, the small peaks at 195 and 655 Hz are also eigenfrequencies. Their amplitude is small due to the driving point measurement being at a modal node of that specific mode and therefore the laser vibrometer practically cannot detect the motion. An interesting observation is that the eigenfrequency at 535 Hz is highly dependent on the radius of the interfaces. A bigger radius will mean a stiffer connection to the (rigid) mass, this will lead to ”stiffning” of the mid-section of the structure. Since mid-section of the structure will highly influence the 10th eigenmode, its eigenfrequency will shift upwards for larger interface radii. In other words, a larger interface area will result into a higher (local) stiffness and thus a higher eigenfrequency. In figure 8 the influence of the coupling radii on the 10th eigenfrequency are shown.
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1121 Mode 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
feig,meas. (Hz) 59.7 79.4 108 183 195 242 276 367 412 535 620 637 654 900 905 1020
feig,FE vs. feig,meas. (%) 0.2% 0.l% -2.3% -1.0% -1.5% -1.1% -0.7% -0.9% -0.2% -0.3% 0.7% -1.5% -0.5% -1.7% -1.8% 3.4%
Table 2: DS results
1 node/bolt r= 3.9 mm Measurement DP FRF r= 4.6 mm r= 4.9 mm r= 6.9 mm
−10
−20
Mag
−30
−40
−50
−60 500
510
520
530
540 550 frequency (Hz)
560
570
580
Figure 8: DP FRF’s for different radius (r) of joint stiffness area per bolt
5
CONCLUSION
From the measurements the following conclusions can be drawn: • The FEM model of the DS benchmark was validated by measurements and showed exceptionally good results, with MAC values higher that 93% and a maximum difference of 1.4% between the measured and calculated eigenfrequencies. This means the FEM model is accurate in the frequency range of interest (40 Hz - 1000 Hz) and can therefore be used as a valuable research tool. • Because the rigidness of the most rigid section of the benchmark displayed in figure 5 remains well above 93% for the complete frequency range of interest, the section can be assumed rigid. This allows for methods to ‘measure’ the rotations (eg. EMPC). • A numerical substructuring analysis was performed and compared with a measurement of the coupled system (4.3). It shows a very good correlation up to 600 Hz and allows for ”tuning” the bolt connection model. The differences at higher frequencies could be due to the assumed rigid bolt connections between the DS benchmark and the attached mass. At higher frequencies non-linear behaviour due to dry friction on the interface could become more dominant.
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1122 References [1] Craig, R. Coupling of Substructures for Dynamic Analyses – An Overview. In Proceedings of AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference and Exhibit (April 2000), pp. 1573–1584. [2] Craig, R., and Bampton, M. Coupling of Substructures for Dynamic Analysis. AIAA Journal 6, 7 (1968), 1313–1319. [3] Crowley, J. R., Klosterman, A. L., Rocklin, G. T., and Vold, H. Direct structural modification using frequency response functions. In 2nd International Modal Analysis Conference, Orlando (1984). [4] de Klerk, D. Dynamic Response Characterization of Complex Systems through Operational Identification and Dynamic Substructuring. PhD thesis, Delft University of Technology, Delft, the Netherlands, October 2008. [5] de Klerk, D., Rixen, D., and Voormeeren, S. General Framework for Dynamic Substructuring: History, Review and Classification of Techniques. AIAA Journal 46, 5 (May 2008), 1169–1181. [6] de Klerk, D., Rixen, D., Voormeeren, S., and Pasteuning, F. Solving the RDoF Problem in Experimental Dynamic Substructuring. In Proceedings of the Twenty Sixth International Modal Analysis Conference, Orlando, FL (Bethel, CT, February 2008), Society for Experimental Mechanics. Paper no. 129. [7] de Klerk, D., and Voormeeren, S. Uncertainty Propagation in Experimental Dynamic Substructuring. In Proceedings of the Twenty Sixth International Modal Analysis Conference, Orlando, FL (Bethel, CT, February 2008), Society for Experimental Mechanics. Paper no. 133. [8] Guyan, R. Reduction of Stiffness and Mass Matrices. AIAA Journal 3 (February 1965), 380. [9] Jetmundsen, B., Bielawa, R., and Flannelly, W. Generalized frequency domain substructure synthesis. Journal of the American Helicopter Society 33 (January 1988), 55–65. [10] Liu, W., and Ewins, D. Substructure synthesis via elastic media. Journal of Sound and Vibration 361-379 (2002), 8. [11] Morgan, J. A., Pierre, C., and Hulbert, G. M. Forced Response of Coupled Substructures Using Experimentally Based Component Mode Synthesis. AIAA Journal 35 (1997), 334–339. [12] Muhs, D., and Wittel, H. Roloff/Matek Machine onderdelen. Academic service, 2005. ISBN: 90 395 23215. [13] Rixen, D. J. A dual Craig-Bampton method for dynamic substructuring. Journal of Computational and Applied Mathematics 168 (2004), 383–391. [14] van Beek, A. Machine Lifetime Performance & Reliability. Delft University of Technology, 2004. ISBN: 9037002080. [15] van den Boom, I. R. Leidraad verbindingen. TU Delft, 2007. [16] Voormeeren, S. Coupling procedure improvement & uncertainty quantification in experimental dynamic substructuring. Master’s thesis, TU Delft, 2007.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Modal Test and Suspension Design for the Orion Launch Abort System Douglas Osterholt Daniel Linehan ATA Engineering, Inc. 11995 El Camino Real, Suite 200 San Diego, California 92130
ABSTRACT This paper documents ATA Engineering, Inc.’s modal test on Orbital Sciences Corporation's (OSC) Launch Abort System (LAS) for the Orion crew vehicle and ARES launch system. The modal test was performed in Dulles, Virginia, at OSC’s facility from December 9 through 11, 2008. One configuration was tested: full motors with the crew module (CM) simulator attached. The LAS was suspended with a twopoint airbag system to isolate the rigid body modes of the LAS. This paper will discuss the challenges in designing and implementing the airbag suspension system as well as document the modal test setup and execution. INTRODUCTION The Advanced Programs group of Orbital Sciences Corporation (OSC) is supporting Lockheed Martin (LM) in the development of the Orion Launch Abort System (LAS). ATA Engineering, Inc., (ATA) was contracted to perform the ground vibration test (GVT) for the LAS. The GVT was successfully completed December 9 through 11, 2008. One configuration was tested: the inert motor and crew module (CM) simulator. In preparation for the GVT, OSC provided ATA with a finite element model (FEM) of the LAS. ATA developed a test-analysis model (TAM) and selected measurement locations. The TAM contained a reduced-model mass and stiffness matrix and a back-expansion matrix for visualization of the test mode shapes. ATA used this information to plan and conduct the GVT, and the resulting data was used to perform the test-to-FEM model correlation. To simulate free-free boundary conditions, the LAS was isolated atop an airbag suspension system that was designed by ATA engineers prior to the GVT. Adjustments to the airbag suspension system at the test site are documented in this paper and may aid future suspension designs. ATA conducted the test using multiple-input random excitation and sine-sweep excitation methods. ATA provided the data collection equipment and instrumentation. The entire process of test setup, test execution, and test teardown was completed in three days, with a total of 182 measurement locations. ATA has developed unique methodologies and custom software to quickly and efficiently execute GVTs, and this paper will discuss these processes. DESCRIPTION OF LAS TEST ARTICLE AND SUSPENSION The LAS is a single-stage solid rocket used to safely project the Orion crew module from the Ares I launch vehicle in the event of an emergency. The test article includes the crew module simulator, adapter cone, inert abort motor, interstage, inert jettison motor, canard, inert attitude control module, and nose cone. The LAS in its final test configuration is shown in Figure 1. The LAS assembly is approximately 53 feet long, including the crew module simulator. The predicted weight of the final test configuration, which was derived from the pretest FEM, is 30,866 lbm. The test article and test equipment used for data
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_99, © The Society for Experimental Mechanics, Inc. 2011
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acquisition and on-site analysis was set up in the LAS low bay at the OSC facility in Dulles, Virginia. The LAS was tested while suspended by air bags in order to separate the rigid body modes from the structural modes of the LAS and to simulate a free-free boundary condition. The design of this suspension is discussed in the following section.
Figure 1. LAS in final test configuration. SUSPENSION DESIGN AND CHALLENGES Typical free-free suspension designs used by ATA Engineering for modal testing include bungee cords, airbags, air canisters, foam, or rubber pad systems. The style is usually defined by the required isolation frequency, the overall weight to support, and physical constraints. After a preliminary analysis, airbags were determined to be the most efficient isolation system for this test article. Airbags provide a compact high-load rating spring. The airbags chosen for this application based on load ratings were Firestone #21 and #20-2. Two support locations along the length of the LAS were required, one at the crew module simulator and one under the interstage. For the interstage support, a cradle was designed so that the proper margins of safety were met for several load cases including modal test-induced loads, road transport loads, and forklift and hoisting loads. The overall design of the interstage support system with four Firestone #20-2 airbags is shown in Figure 2. The CM simulator design integrated four Firestone #21 airbags. The airbag configuration is shown in Figure 3 and was chosen to balance the loads carried by the airbags. Custom pedestals were designed for the underside of the airbags to span the gap between the airbags and ground. The pressure of each airbag was controlled independently for maximum flexibility.
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Cradle support
Airbag locations I-beams for static support Support base
Figure 2. Interstage support and airbag design for LAS suspension.
Pads for airbag attachment
Figure 3. CM Simulator airbag configuration for LAS suspension. View from underside of simulator, looking up. The LAS was on a fixed rail system to keep the axial alignment of the components during assembly, and each component’s support system was on this rail and could not be removed. As the airbags were inflated and the vehicle suspended, the lateral movement of the LAS was controlled using ratchet straps. Lateral stabilization using the straps proved more effective than airbag pressure variation. A picture of the ratchet straps configured for the test is provided in Figure 4. The rigid body modes, provided in Table 1, were not adversely affected by the straps. In fact, the suspension system was very effective at isolating
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the rigid body modes from the flexible modes being sought in the modal survey. The highest rigid body mode was the roll mode at 2 Hz. Table 1. Rigid body frequencies of the LAS with airbag suspension. Direction Z (fore-aft) X (lateral/roll) Y (vertical) Roll (RZ) Pitch (RX) Yaw (RY)
Frequency (Hz) 0.98 0.39 1.71 2.06 1.92 0.85
Strap used for lateral support
Airbag Suspension
Figure 4. Lateral restraint was added to the airbags to prevent contact with support cradles. TEST PLANNING AND SETUP Prior to arriving at the OSC facility, ATA performed the pretest analysis necessary to determine an optimized set of response measurement locations. ATA created a test model representative of the LAS to develop a set of test measurement degrees of freedom (DOF). This test model and the associated DOF, or accelerometer locations, are depicted in Figure 5.
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Figure 5. LAS test model. The arrows represent a test DOF where an accelerometer was placed. The total for the final set of measurement locations determined from the pretest analysis was 127 fixed DOF at 116 node locations. This number does not include drive point accelerometers or load cells. It also does not include accelerometers which were added on the nozzles, internal components, and axial accelerometers on the forward and aft ends of the abort motor. All measurement channels totaled 182. The geometry and response measurements were defined using both Cartesian and cylindrical coordinate systems. One location at the nose of the LAS and several on the CM simulator used the global Cartesian coordinates, while all other measurement locations used a cylindrical coordinate system. The two coordinate systems are illustrated in Figure 6.
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Figure 6. The cylindrical and Cartesian coordinate systems. TEST EXECUTION The GVT was performed at the OSC facility in Dulles, Virginia, December 10 through 11, 2008. The test article was located in a designated testing area in the low bay at OSC’s facility. ATA used an internally developed spreadsheet-based software suite called Automated Test Setup (ATS) [1] to quickly and efficiently assemble the test channel table. Barcode labels were used to map measurement locations to accelerometer serial numbers. Using ATA’s data acquisition system, which is TEDS [2] capable, TEDS information mapped channel locations to accelerometer serial numbers. A cascaded view of ATS is presented in Figure 7. ATA was responsible for the installation and removal of all transducers and cabling necessary for the GVT. All DOF locations on the LAS were measured and marked on the surface by ATA. These locations matched those in the test display model and were selected for visualization of the primary modes. After all of the measurement locations were mapped on the test article, the accelerometers and cables were installed. Two portable electrodynamic shakers were used to excite the LAS for the GVT. These were initially positioned at the LAS nosecone orthogonal to each other. For one data collection run, the shakers were repositioned to the rear of the test article, attached to the CM simulator, and rotated by 45° from their original nosecone “clock” orientations, as shown in Figure 8. Two shaker excitation methods were used. First, multiple-point random (MPR) excitation used two shakers simultaneously to excite the test article from 1 to 50 Hz. To study the dynamic linearity of the LAS, the MPR testing was performed at three force levels, varying by a factor of 3 from the lowest to highest force level. Second, sine sweep testing was conducted using one shaker at a time. To supplement the shaker testing, impact testing, using an instrumented modal impact hammer, was performed at the end of the GVT to observe the transmissibility of the CM joint.
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Figure 7. Overview of ATS software, which automates GVT channel table setup.
Figure 8. Shaker locations. First set at nose (left) and second set on CM (right). TEST RESULTS The primary target modes for the LAS GVT were the first three orthogonal bending mode pairs. Additionally, the first torsion mode was successfully extracted and two significant shell modes were observed in the curve-fitted data. The rigid body modes of the LAS were either observed manually or extracted from the acquired data. These modes were of interest for the correlation effort that followed the GVT. Modal parameter estimation was performed using Alias Free Polyreference (AFPoly™) [3] [4] and
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ATA Engineering’s MATLAB curve-fitting utility. The alias-free polyreference technique, in ATA’s AFPoly™ tool, employs a Laplace-domain, curve-fitting algorithm capable of extracting lightly or highly damped modes in a single pass. Examples from a two-shaker burst random run are presented. The stability plot for pole selection is shown in Figure 9. This figure shows the power spectrum mode indicator function (PSMIF) and multivariate mode indicator function (MMIF) superimposed on the stability diagram, with the selected poles indicated by the solid squares.
Figure 9. Two-shaker burst random run – stability diagram and pole section. The pole selections were verified by comparing the synthesized PSMIF, MMIF, and FRF to the corresponding measured functions. Figure 10 shows the fit of the PSMIF, Figure 11 shows the fit for the MMIF, and Figure 12 shows the drive point FRF for one shaker. This process was performed for each burst random run and each sine sweep or impact test to verify the quality of the pole estimations. Final modes were selected from these runs and compared to the analysis predictions.
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Figure 10. Pole selection verification – PSMIF.
Figure 11. Pole selection verification – MMIF.
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Figure 12. Pole selection verification – drive point FRF. The final set of test results is listed in Table 2. The last column in the table shows the percent difference between the measured test frequencies and the predicted analysis frequencies. The test frequencies were consistently higher than the predictions. Table 2. Pretest analysis and test results comparison table.
Priority S S S S S S P P P P S O O S S P - Primary S - Secondary O - Other
Mode # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Mode Description Lateral with yaw Fore/Aft Yaw Roll Bounce Pitch 1st bending 1st bending 2nd bending 2nd bending torsion shell shell 3rd bending 3rd bending
Analysis Results Freq (Hz) 0.04 0.04 0.05 0.98 1.49 2.26 7.19 7.27 17.09 17.32 26.36 NA NA 30.30 30.55
Test Results Freq (Hz) Damp (% crit) 0.39 0.46 0.97 2.11 1.73 1.95 7.77 2.0% 7.87 1.5% 19.69 0.8% 20.32 0.8% 29.32 1.1% NA NA 39.44 1.7% 42.27 1.3%
Test to Analysis % diff, freq
7% 8% 13% 15% 10%
23% 28%
The modes acquired from the test measurements were checked for orthogonality using the analytical mass matrix. This mass matrix resulted from the FEM being reduced to the test-measurement locations. The orthogonality check is obtained from the following matrix product.
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>O12 @ >)1 @T >M AA @>) 2 @
Equation (1)
where >)@>)@ = mode shapes (normalized to [MAA] to produce 1.0 on the [O12] diagonal). For selforthogonality, >)@and >)@ are the same. [MAA] = reduced model analytical mass matrix [O12] = orthogonality matrix Table 3 presents the test mode self-orthogonality using the pretest reduced test-analysis model mass matrix. The self-orthogonality shows independent test shapes with minimal off-diagonal terms in the matrix. In addition to orthogonality, modal assurance criteria (MAC) was also used to verify test mode shape independence. Table 3. Test self-orthogonality, demonstrating good mode uniqueness. This table includes five rigid body modes extracted from the test data. Values less than 0.10 are not displayed. Test Self Orthogonality Table Test Shapes
Test Shapes
Ott
1st
1st
2nd
2nd
torsion
3rd
3rd
1
2
3
4
5
6
7
8
9
10
11
12
0.5
1.0
1.7
1.9
2.1
7.8
7.9
19.7
20.3
29.3
39.4
42.3
1
0.5
1.00
0.05
0.06
0.05
0.01
0.07
0.13
0.02
0.11
0.02
0.59
0.07
2
1.0
0.05
1.00
0.01
0.00
0.07
0.02
0.01
0.00
0.01
0.07
0.01
0.05
3
1.7
0.06
0.01
1.00
0.02
0.01
0.00
0.01
0.00
0.01
0.01
0.05
0.04
4
1.9
0.05
0.00
0.02
1.00
0.03
0.00
0.01
0.00
0.02
0.01
0.02
0.06 0.05
5
2.1
0.01
0.07
0.01
0.03
1.00
0.02
0.02
0.02
0.02
0.01
0.05
1st
6
7.8
0.07
0.02
0.00
0.00
0.02
1.00
0.09
0.00
0.01
0.03
0.06
0.03
1st
7
7.9
0.13
0.01
0.01
0.01
0.02
0.09
1.00
0.03
0.01
0.01
0.01
0.04
2nd
8
19.7
0.02
0.00
0.00
0.00
0.02
0.00
0.03
1.00
0.03
0.10
0.02
0.01 0.00
2nd
9
20.3
0.11
0.01
0.01
0.02
0.02
0.01
0.01
0.03
1.00
0.05
0.08
torsion
10
29.3
0.02
0.07
0.01
0.01
0.01
0.03
0.01
0.10
0.05
1.00
0.10
0.02
3rd
11
39.4
0.59
0.01
0.05
0.02
0.05
0.06
0.01
0.02
0.08
0.10
1.00
0.10
3rd
12
42.3
0.07
0.05
0.04
0.06
0.05
0.03
0.04
0.01
0.00
0.02
0.10
1.00
The linearity of the LAS test article was studied using three levels of MPR excitation. Figure 13 displays the combined two-reference power spectrum mode indicator function for the three MPR forcing levels. Minimal frequency shifts are observed as excitation force is varied, indicating a linear structure.
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Figure 13. PSMIF demonstrating relative linearity of LAS test article. The last test data analysis was performed to compare the mode shapes from different shaker locations. This was done to see if the mode shapes change when the excitation position changes; in this case, the shakers were oriented at 45° from the original location and moved from the nose cone to the rear CM simulator. Table 4 provides the test cross-orthogonality of the three bending mode pairs, comparing the nosecone shaker tests to the 45° rotated CM shaker test. This shows that the second and third bending mode pairs do not change with the new shaker orientation. However, the bending planes of the first bending mode pair were slightly affected by the shaker repositioning, indicating slight principal axis sensitivity. The corrected root-sum-square (CRSS) value proved that all modal mass was captured in this comparison. The final mode shapes for the first three bending mode pairs are shown in Figure 14. These mode shapes are back expanded to the display set using the analytical back expansion matrix. All of the test results were used for model correlation and updating. The model correlation is beyond the scope of this paper, but the final comparison table is presented in Table 5.
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Table 4. Test mode cross-orthogonality of the three bending mode pairs comparing the nosecone shaker orientation to the CM simulator shaker orientation. Test/Test Cross Orthogonality Table Run 14 - shakers at CM simulator, vertical/lateral 1st Run 10 - shakers at nose, 45° angles
Otg
1st
2nd
2nd
3rd
7.9
8.0
19.8
20.5
3rd 39.4
Test 42.5 CRSS
1st
7.8
0.97
0.24
0.00
0.00
0.05
0.05
1.00
1st
7.9
0.31
0.94
0.02
0.01
0.00
0.02
1.00
2nd
19.7
0.00
0.04
1.00
0.01
0.02
0.02
1.00
2nd
20.3
0.02
0.01
0.04
1.00
0.07
0.01
1.00
3rd
39.1
0.04
0.00
0.01
0.05
1.00
0.13
1.00
3rd
42.3
0.04
0.02
0.01
0.00
0.16
0.98
0.98
1.00
1.00
1.00
1.00
1.00
0.98
Test
CRSS
Figure 14. First, second, and third bending mode shapes of the LAS.
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Table 5. Final posttest analysis and test results. Test and analysis frequencies are compared before and after the model update.
Priority Mode # S 1 S 2 S 3 S 4 S 5 S 6 P 7 P 8 P 9 P 10 S 11 S 14 S 15 P - Primary S - Secondary O - Other
Mode Description Lateral with yaw Fore/Aft Yaw Roll Bounce Pitch 1st bending 1st bending 2nd bending 2nd bending torsion 3rd bending 3rd bending
Initial FEM Final FEM Analysis Results Analysis Results Freq (Hz) Freq (Hz) 0.04 0.57 0.04 0.46 0.05 0.99 0.98 2.04 1.49 1.71 2.26 1.91 7.19 7.86 7.27 7.92 17.09 19.04 17.32 19.34 26.36 30.16 30.30 40.84 30.55 41.07
Test to Test to Test Initial FEM Final FEM Results Freq (Hz) Damp (% crit) % diff, freq % diff, freq 0.39 89.0% -47.4% 0.46 90.5% -0.9% 0.97 95.2% -1.7% 2.11 53.3% 3.2% 1.73 13.7% 1.1% 1.95 -15.8% 2.0% 7.77 2.0% 7.4% -1.2% 7.87 1.5% 7.7% -0.6% 19.69 0.8% 13.2% 3.3% 20.32 0.8% 14.7% 4.8% 29.32 1.1% 10.1% -2.8% 39.44 1.7% 23.2% -3.6% 42.27 1.3% 27.7% 2.8%
DISCUSSION The LAS GVT program was successfully completed as part of the Orion development activities. Using automated instrumentation and data collection processes, this test was completed in a minimal amount of time. This included the time required to install and verify the newly designed suspension system. Aside from lateral stability of the airbags, which was easily remedied, the modal survey suspension system was very effective at isolating the rigid body modes from the flexible modes of the test article. This led to excellent modal test results and, ultimately, good model correlation. Effective use of automated test software tools allowed ATA and OSC engineers to quickly evaluate test results. Since mode shapes could be produced within minutes of data collection, clear understanding of the test article behavior was achieved and comparison to model predictions was possible. Successful model correlation has followed. REFERENCES [1]
Brillhart, R., and Dillion, M., “Automated Test Setup in Modal Testing,” 10th International Modal Analysis Conference, Los Angeles, CA, 02, 2002.
[2]
Brillhart, R., “Improving Test Efficiency and Accuracy with TEDS, 20th Aerospace Testing Seminar,” Manhattan Beach Marriott, Manhattan Beach, CA, 03, 2002
[3]
Brillhart, R., Napolitano, K., and Osterholt, D., “Utilization of Alias Free Polyreference for Mixed Mode Structures,” 26th International Modal Analysis Conference, Orlando, Florida, 2, 2008.
[4]
Vold, H., Richardson, M., Napolitano, K., and Hensley, D., “Aliasing in Modal Parameter Estimation - A Historical Look and New Innovations,” IMAC, January 2008.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Modal Survey Test and Model Correlation of the CASSIOPE Spacecraft Viresh Wickramasinghe, Yong Chen and David Zimcik National Research Council Canada, Institute for Aerospace Research 1200 Montreal Road, Building U66A, Ottawa, Ontario, Canada K1A 0R6 Phillippe Tremblay MAYA HTT 4999 Ste. Catherine Street West, Suite 410, Montreal, Quebec, Canada H3Z 1T3 Harold Dahl and Ian Walkty Magellan Aerospace 660 Berry Street, P.O. Box 874, Winnipeg, Manitoba, Canada R3C 2S4
ABSTRACT A comprehensive modal survey test based on multi-input multi-output experimental modal analysis techniques was conducted on the CASSIOPE spacecraft. This paper describes the details of the methodology used to perform the successful experimental modal test to efficiently extract the critical modes of the spacecraft. Results from the modal test have been used to validate the analytical finite element model and to provide confidence in the structural integrity of the spacecraft design. The test was performed on the flight model of the CASSIOPE spacecraft in the final stages of integration, which included all of the payload and bus instruments and electronics boxes. The multiple-input excitation for the spacecraft was generated using two portable electrodynamic modal shakers installed on the top and bottom of the spacecraft to distribute the excitation energy and the response was measured using 81 miniature accelerometers. A digital multi-channel data acquisition system was used to record the time domain data and calculate the frequency domain spectra. Advanced modal analysis software was used to extract modal parameters from the measured data and critical modes were compared with predictions from the finite element model. Most modes identified through the experimental data compared favorably with the predictions. Nevertheless, some differences were large enough to require iterative update of the finite elelement model. The structural dynamics information from the updated finite element model was used to plan the mechanical vibration qualification test and predict the response of the spacecraft to the launch vehicle environmental loads through coupled loads analysis.
1.0
INTRODUCTION
The integrated CASSIOPE spacecraft shown in Figure 1 is a culmination of a hybrid satellite development project of the Canadian Space Agency designed to carry a proof of concept commercial communications system as well as a scientific experiment package. It is necessary to accurately understand the dynamic behavior of the satellite in order to ensure that it will survive the vibration loads during the launch. Unaccounted behavior in the satellite structure may lead to catastrophic failure in structural components or damage to the payload during launch. The CASSIOPE spacecraft is a hexagonal structure of nominal dimension 1.6 m, height 1.4 m and launch mass of approximately 1060 lb. Successful development and operation of precision systems such as satellites require very accurate analytical models that need to be validated using experimental data. Traditionally, ground vibration test (GVT) based on Experimental Modal Analysis (EMA) techniques is used to extract the modal characteristics such as frequency, damping and shapes of critical modes of the structure [1]. These experimentally acquired modal data are correlated with the finite element analytical results in order to update the numerical model. Spacecraft structural dynamics information gained in both finite element modeling and ground vibration testing will
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be used to plan the qualification mechanical vibration tests and to predict the response of the spacecraft to the launch vehicle environmental loads through the coupled loads analysis. Due to the limited availability of the satellite as well as time and cost constrains in a spacecraft development program, it is important to perform ground vibration test in an efficient and timely manner. The objective of this paper is to describe the detailed multi-input and multi-output experimental modal test methodology and data analysis techniques used to perform the successful modal test of the CASSIOPE spacecraft in order to extract the modal parameters efficiently. Careful planning of the test, close coordination during the setup and the use of advanced hardware and software enabled this modal survey test to be accomplished in four days with the modal data available in near real-time. In Figure 1: Integrated CASSIOPE spacecraft. contrast, a similar test in the past could have taken a few weeks or months to perform and the date needed to be post analyzed. Reducing the time required for the test to only a few days had an enormous impact on the success of the satellite development program to avoid scheduling delays, and the test was carried out within the budget constraints. In order to obtain the most realistic modal information, the test was performed on the flight model of the CASSIOPE spacecraft in the final stages of integration which included all of the payload and bus instruments and electronics boxes. In contrast to modal test of simple structures, multiple-input techniques are generally used for large structures in order to distribute the excitation energy throughout the structure to ensure adequate excitation of all the modes simultaneously [2]. Furthermore, multiple reference measurements mitigated the likelihood of missing a mode during the curve fitting process, particularly when modes are closely coupled or modes are in fact repeated [3]. A multi-channel digital data acquisition system was used for signal generation as well as response measurements while calculating multi-referenced Frequency Response Functions (FRF). The analysis software used advanced curve fitting algorithms known as PolyMAX to estimate frequency, damping and mode shape from the experimental test data. PolyMAX is an advanced polyreference lease-squares complex frequency algorithm that generates very clear stability diagrams for modal parameter estimation [4, 5]. This advanced curve fitting method allowed a large frequency band containing a high number of modes in a single analysis run. It resulted in near-real time availability of the modal parameters at the end of each test segment to verify the quality of the data and provide guidance in planning subsequent test segments. This paper provides details of the ground vibration test procedure including the spacecraft configuration, test setup, instrumentation, data acquisition and modal analysis techniques. More importantly, it includes the test procedure followed to perform the multi-input multi-output experimental modal test in order to accurately extract the modal parameters efficiently. Experimentally extracted modal frequencies, damping ratios and mode shapes of critical CASSIOPE spacecraft components are compared with the finite element model data. The paper also provides the details of the iterative procedure to update the finite element model in order to correlate more closely with the experimental data.
2.0
TEST SETUP AND INSTRUMENTATION
The modal survey test was conducted in the Class 100,000 clean room at Bristol Aerospace Limited in Winnipeg, Manitoba and the CASSIOPE spacecraft was in a flight-like configuration. Major components were installed in the spacecraft platform including the top solar panel, but the side solar panels, launch vehicle interface brackets and the thermal multi-layer insulation blankets were not installed in order to provide access to the inner components during the test. The spacecraft was attached to the handling fixture and oriented such that its Z axis was in the vertical direction as shown in Figure 2. Major components considered in the modal test configuration are shown schematically in Figure 3, namely, top solar array, CX panel, mid deck, ePOP panel, GAP antenna interfaces and RAM boom interface. These components were included in the test configuration in order to ensure that there was no modal coalescence between these major components. The details of the test setup were agreed by all stake holders involved in the test in order to make necessary preparations. Detailed planning in advance helped to mitigate challenges that generally occur during similar tests and allowed the modal survey test on the CASSIOPE spacecraft to be completed efficiently within the allotted time frame.
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2.1 Excitation Input The spacecraft was excited using two portable electrodynamic modal shakers capable of generating a dynamic maximum force of 100 lb(f) and 25 lb(f) respectively, as shown in Figure 2. The top shaker with 100 lb(f) capability was suspended above the spacecraft and attached to the CX panel to impose a free boundary condition in order to minimize the mass loading effects on the panel. The bottom shaker with 25 lb(f) capability was placed below the spacecraft and attached to the yoke side of the handling fixture. This relatively fixed boundary condition imposed on the 25 lb(f) shaker helped to provide sufficient force to excite the handling fixture while mass loading at this location was not a concern. The two shakers were employed to excite the spacecraft from the top and bottom simultaneously in order to distribute the excitation forces throughout the test article so that all modes were excited adequately [6]. Furthermore, multiple excitation references were essential to identify modes that were close to each other. A force transducer was mounted on each of the adaptor bracket in series with the stinger in close proximity to the structure to measure the input driving force applied to the spacecraft from each electrodynamic shaker. 2.2 Response Accelerometers The response of the major components on the CASSIOPE spacecraft and the yoke of the handling fixture were measured using 79 high sensitivity miniature accelerometers bonded to the structure. Use of low weight miniature accelerometers mitigated the effect of mass loading on the test article. The high sensitivity of the the accelerometers was ideal to obtain good signal-to-noise ratio measurements even when the response amplitudes were relatively low. The finite element model was used to determine the placement and quantity of the accelerometers required to clearly identify the important dynamic response of the spacecraft [7]. These include fundamental mode shapes of the major panels schematically shown in Figure 3 and vibration response of critical equipment installed in the satellite. In addition, two accelerometers were placed on the stinger adaptor brackets to measure the shaker drive point response. The drive point measurement included the input force and the output response at the same point on the structure in the same direction. The drive point measurement was generally used to evaluate the data quality through reciprocity and phase change. The FRF of the drive point is viewed as a single measurement that shows the Top integrated response of excited modes of the structure. Shaker CASSIOPE Spacecraft 2.3 Data Acquisition System A total of 81 accelerometers and 2 force transducers were divided into 2 batches for data recording purposes using a 48-channel digital data acquisition system. The LMS SCADAS 305 mobile front-end was configured with 4 cards of the V12 modules for CX simultaneous data recording of the 48-channels during Panel the modal test [8]. For the given input excitation condition, data was recorded for both batches of sensors prior to changing the input excitation condition. The sampling frequency of the data acquisition system was set at about 10 times above the maximum frequency of interest in order to ensure the high quality of the acquired data. Handling Fixture Bottom Shaker
Figure 2: Modal survey test setup.
3.0
TEST PROCEDURE
The modal test of the CASSIOPE spacecraft structure was performed in accordance with a test procedure collaboratively prepared by all parties involved in the test and approved in advance. The test procedure clearly identified the details of the testing process and the responsibility of each member of the team involved in the test. This clear understanding helped the testing team from several different organizations to perform the test in a cohesive manner to achieve all
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Top Solar Array GAP Antennas CX Panel Mid deck RAM boom ePOP panel Handling Fixture
Figure 3: Schematic of major components. a. b. c.
required objectives of the modal survey test effectively. Several Important aspects of the testing methodology are discussed below. 3.1 Excitation Technique In modal testing, no single excitation technique is considered to be superior for every modal test application. Therefore several excitation techniques were used in order to excite the structure properly and extract the modal frequencies, damping ratios and mode shapes of the CASSIOPE spacecraft. The two electrodynamic shakers were driven independently and simultaneously using the LMS SCADAS 305 mobile front-end output channels using uncorrelated signals. Post analysis showed that simultaneous use of both shakers improved the quality of the data which enabled extraction of more details of the structural dynamics such as decoupling of closely spaced modes. Response data were collected using several different dynamic excitation signals:
Burst random signal with the burst duration of 80% with a uniform windowing function Sine chirp signal with the chirp duration of 80% with a uniform windowing function Continuous random with a Hanning windowing function
The uniform windowing function or no windowing was applied to the data for periodic chirp and random burst signals. Allowing sufficient time for the response signals to decay by the end of the measurement duration avoided the leakage problem experienced by FFT analysis. However, only processed data could be recorded by the LMS Test.Lab data acquisition system during burst and chirp excitations. Therefore, a continuous random excitation was used with a Hanning windowing function in order to record time domain data for re-processing in the future if necessary. The nominal excitation frequency range of each excitation signal was set to cover a minimum of the first 10 modes of the spacecraft structure. It is important to ensure that uncorrelated signals are used when driving both shakers simultaneously. During burst random excitation, this was achieved by using two uncorrelated random signals. In the case of the sine excitation, the top shaker was provided a chirp signal while the bottom shaker was driven by a random signal and vice versa. However, significant improvement in the coherence function was obtained at low frequencies when the top shaker attached to the CX panel was driven by a sine chirp signal while the bottom shaker attached to the handling fixture was excited with a random burst signal. This excitation condition provided the best signal-to-noise ratio as well as the value close to unity in coherence calculations for most of the channels across a large frequency band. This may be due to the fact that high apparent mass attched to the bottom shaker is more capable of providing a background random excitation rather than exciting modes of the spacecraft structure using the sine chirp. Typical excitation input recorded by both the force transducers and combined output of a response accelerometer are shown in Figure 4. The shaker force input signals were monitored using two load transducers to ensure that the maximum applied load limits were not exceeded. The appropriate dynamic force generated by the shakers was determined to avoid any potential damage to the sensitive equipment installed on the satellite and depended on the shaker location and excitation signal type. The maximum dynamic force amplitude of the top shaker was 45 N rms and 25 N rms for random burst and sine chirp excitations, respectively. A higher maximum dynamic force of 50 N rms was generated by the bottom shaker for both random burst and sine chirp excitations. The bottom shaker could be operated at a higher force level because it was connected to the handling fixture instead of a direct interface with the spacecraft. This indirect connection of the bottom shaker to the spacecraft required higher force level in order to adequately excite the top end of the satellite structure.
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Random Excitation
Sine Chirp Excitation
Accelerometer Response
Figure 4: Typical excitations and responses.
Multi-Referenced FRFs
Multi-Referenced Coherence
Figure 5: Typical FRF and coherence.
3.2 Data Processing The commercially available LMS Test.Lab Spectral Testing software package was used to record and calculate the PSD (Power Spectrum Density), FRF (Frequency Response Function) and Coherences from the time-domain data in real-time [9]. It calculated the FRF as the ratio of the cross-spectrum between input and output and the power spectrum of the inputs based on H1 formulation which assumed noise-free input measurements. Signals from the load transducers attached to the shaker inputs were used as the reference signals in calculating the FRFs. Multiple referenced FRFs were calculated when both shakers were used simultaneously. The total of 81 response accelerometers generated a total of 162 FRFs when both references were used. Typical FRFs calculated for a given sensor using both references are shown in Figure 5 along with the resultant coherence. High quality data obtained from this modal survey testing approach was verified by the coherence which was close to unity in the complete frequency range of interest. A frequency resolution of 0.1 Hz was selected for data analysis in order to distinguish any closely spaced modes. To improve the coherence and mitigate the measurement noise in the data, 30 linear averages were taken during each test run. The calculated FRFs were assigned to the appropriate nodes of the geometry model in order to animate the identified mode shapes for proper visual display and evaluation purposes. 3.3 Modal Parameter Estimation The LMS Test.Lab modal analysis software package was used to extract modal parameters from the measured FRF data. The FRFs from each batch of sensors recorded separately during data acquisition were combined and analyzed together for modal parameter estimation. The software used an advanced curve fitting algorithm called PolyMAX to estimate frequency, damping and mode shape from the experimental test data [4]. PolyMAX is an advanced polyreference lease-square complex frequency algorithm that generates very clear stability diagrams for modal parameter estimation as shown in Figure 6. The algorithm was able to process all measurement degrees of freedom within the complete frequency range of interest without much difficulty. The identification of the modal parameters using all FRF’s in the calculation was aided by the Multivariant Mode Indicator Functions (MMIFs). The MMIFs are frequency domain functions that exhibit local minima at the natural frequencies of real normal modes to identify the resonance frequencies of the structure. For this multi-referenced data set, the primary MMIF showed natural frequencies while local minima of the secondary MMIF shows any repeated or closely spaced modes of the structure as shown in several
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Secondary MMIF (green)
Stable poles (s)
(g/N) Amplitude
Primary MMIF (blue)
Local minima
Sum of FRF (red)
ss ss ds vs vs ds ss s s d s ds ss ds s s vs f s f s f s f s s s v s s s d s d s d s f s v s v s f s f s d s d s f s d s f s f s f s f s f s f v o s s s s o s s d d s
s s s s s s s v v f v f s v s s d d d f f v v o
s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s
s s s s s s s s s s s s s s s s s s s s v s v v s s s s v s v s v v v s v d f v s v v o f o
s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s d s s s s s f f v s
s s s f s v s v d s d f v f f d f f f f o
sd sd sf dd sf ss sf ff fs sf ss sf ff sf ss sf ff dd sf sf ff fd sf ff ff ff sf sf sf sf ff s f f v f s f s d s s d s f v f v
F B B
Sum FRF SUM Multivariant Mode Multivariant Mode s o v o
s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s Indicator s Indicator s s s s s s s s s d
1 2
s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s f f s s f v f f o
s s s s s s s s s s s s s s s s d s s s s s d d f d f f d s v s d f v f v f d d s d f v f o
ss s ss v ss v ss v vs v ss v ss v ff o sf fs sv sv ss sv ss ss ss sv ss ss sv sv sv sv sv sv sv s f s v s f v f s s s f v f s v s v s v s v s o s s s s s s v v f
s f v d f s d f f f f f v f f f f f f d v f f f f d f o
f f o
64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17
189e-6 Linear
Hz
Figure 6: Stability diagram for the complete frequency range of interest. frequencies in Figure 6. Once the modal parameters were extracted by selecting stable poles marked as “s” on the stability diagram, these parameters were used to synthesize the FRFs for each channel in order to compare with the FRFs generated from the experimental data [10]. Comparison of synthesized FRFs with the measured FRFs was important to determine the level of accuracy of the estimated modal parameters that represented the overall dynamics of the structure. The comparison showed that synthesized FRFs compared well with the measurement at most locations of the structure and most FRFs correlated above 80% as shown in Figure 7. Extracted mode shapes were compared to each other using the Modal Assurance Criteria (MAC), which is an orthogonality check of the extracted parameters. When compared with all extracted mode shapes, the MAC matrix showed a value of near zero in the off-diagonal terms suggesting that the identified mode shapes were independent from each other.
4.0
MODAL SURVEY TEST RESULTS
Selected estimation methods enabled modal parameters from each of the major components of the CASSIOPE spacecraft structure, namely, natural frequency, damping ratio as well as the modal shape to be identified with ease. A total of 16 modes were identified during the modal test that included the spacecraft modes as well as modes induced by the spacecraft handling fixture. Most of the lower frequency modes were related to the handling fixture, and the symmetric nature of the fixture design also generated closely spaced modes. For example, cradle rocking modes around the X and Y axes as shown in Figure 8 were only 0.1 Hz apart. These closely spaced modes in two orthogonal directions were clearly resolved as two distinct modes due to the use of dual shakers to generate multi-referenced data.
Figure 7: Comparison of synthesized multireferenced FRFs with measured FRFs.
Most of the critical modes of the spacecraft components were identified from the modal survey test without much difficulty. The top solar array and the CX panel modes are shown in Figure 9 and two mid-deck modes are shown in Figure 10. Modal survey tests clearly showed that modal coalescence between the top solar panel and the CX
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panel is not possible due to the large difference in the resonance frequencies of the fundamental modes. Furthermore, these results validated that the minimum natural frequency requirements imposed by the launch vehicle in the thrust direction had been achieved. These extracted modal parameters were used to plan the environmental testing to qualify the structure for launch and update the finite element model in order to predict the response of the spacecraft to the launch vehicle environmental loads through the coupled loads analysis.
5.0 Figure 8: Closely spaced modes of handling fixture.
Figure 9: Top solar array and CX panel modes.
FINITE ELEMENT MODEL UPDATE
Prior to the test, the finite element model was modified to reflect the test configuration by removing the side solar arrays and mounting the spacecraft onto the handling cradle in the vertical position. However, only the top part of the cradle that rotates on the trunnions was modeled. The bottom portion was assumed to be sufficiently stiff, in the vertical position, to act as a rigid structure. The modal survey test results validated the applicability of this assumption. A modified NX NASTRAN normal modes solution was used to optimize the accelerometer locations that best identify the target modes of each major component. Once the mode shapes were extracted experimentally, they were compared to the analytical mode shapes using animation, modal frequency and the modal assurance criterion (MAC). For the pre-test model, mid-deck primary modes and the cradle mode of the spacecraft showed very good correlation as shown in Table 1. Data also showed that the top solar array, ePOP and CX panel modes did not correlate to predictions from the analytical results. Therefore, a review of the finite element model was undertaken in order to improve the correlation of the finite element model with the test results.
An iterative process was used to update the finite element model for improved correlation with the modal Figure 10: Mid deck (Mode 1) and Mid deck (Mode 2). test results. First, the mass properties of the CX and ePOP panels were reviewed in order to more closely match the “as built” configuration tested. It was found that a difference of 55 lbs could be attributed to the estimated mass of the harnessing system. A similar difference of 48 lbs was found between the measured CX panel mass and the mass used in the FEM model. This difference is attributed to the mass budget having been conservative for the payload units, payload harness and bus harness. The Update 1 column in Table 1 shows the improvement in the correlation due to the reduction in non-structural mass as well as the removal of rotational stiffness of the solar array panel bracket bearings. Further modifications were made to the CX and ePOP panels in the Update 2 such as reduction in the effective modulus of the top solar array material and relocation of several brackets from the ePOP panel manhole to the edge of the panel. In addition, ePOP panel manhole brackets were completely re-meshed to transfer the load directly from the manhole to the ePOP panel. As a result of Update 2, the ePOP panel and top solar array mode correlated better with the experimental data while the CX panel frequency did not show a significant shift. In Update 3 as shown in Table 1, the modulus of the cradle material was increased to improve the correlation between the analytical and test Z cradle modes. As a result, the second analytical cradle mode frequency in the Z direction correlated with the test mode frequency within 2.4%. The increase in the cradle stiffness also helped to separate the CX panel mode from the lateral cradle mode in its
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vicinity. After Update 3, the iterative model update was halted because all model predicted target mode frequencies were within 10% of the modal parameters extracted from the model survey test. The rationale for updating a known material modulus is to account for mesh stiffness uncertainties in the vicinity of structural joints.
Mode Description
Table 1: Correlation between test and analytical modal frequencies. Mid-Deck (Mode 1) Mid-Deck (Mode 2) Top Solar Array ePOP panel CX Panel Cradle Z (Mode 1) Cradle Z (Mode 2) Cradle Rock X
Pre-Test -1.8% 5.9% -10.9% 25.1% 16.8% 17.7% 11.7% 9.0%
Update 1 -2.1% 5.5% -8.6% 20.7% 11.9% 14.9% 11.6% 5.3%
Update 2 -2.2% 5.6% 0.0% 9.7% 11.4% 14.8% 11.6% 5.3%
Update 3 -2.4% 5.3% -0.01% 9.6% 8.7% 5.3% 2.4% -2.2%
After updating the finite element model, the important mode shapes for the spacecraft and the handling fixture based on Update 3 were compared with the mode shapes extracted from the modal survey tests. A value of zero for the MAC result shown as Table 2 indicated no correlation between mode shapes while perfect correlation is indicated by the value of unity. The top solar array, ePOP panel, mid-deck (Mode 1) and both cradle Z mode shapes correlated well with MAC values higher than 0.75. However, the CX panel and mid-deck (Mode 2) showed relatively good corelation while cradle rocking mode in the X direction showed very poor correlation. The poor modal correlation in the X direction modes may be due to the fact that the excitation for this modal survey test was provided only in the Z direction. This lack of excitation in the X direction may have led to low signal to noise in FRFs, making it difficult to extract modes in the orthogal direction. Table 2: MAC calculations between analytical and test modes.
ePOP panel
CX Panel
Mid-Deck (Mode 1)
Mid-Deck (Mode 2)
Cradle Z (Mode 1)
Cradle Z (Mode 2)
Cradle Rock X
Top Solar Array ePOP panel CX Panel Mid-Deck (Mode 1) Mid-Deck (Mode 2) Cradle Z (Mode 1) Cradle Z (Mode 2) Cradle Rock X
Top Solar Array Test Modes
Analytical Modes – Update 3
0.88 0.00 0.02 0.00 0.01 0.02 0.02 0.00
0.04 0.86 0.01 0.00 0.11 0.08 0.09 0.06
0.06 0.01 0.66 0.00 0.10 0.23 0.27 0.14
0.00 0.03 0.15 0.94 0.49 0.08 0.15 0.08
0.00 0.00 0.05 0.01 0.28 0.01 0.00 0.05
0.00 0.05 0.21 0.08 0.11 0.68 0.70 0.17
0.00 0.08 0.26 0.15 0.15 0.59 0.78 0.10
0.03 0.06 0.07 0.60 0.26 0.07 0.08 0.07
The updated finite element model of the CASSIOPE spacecraft was then used to introduce components that were not included in the modal survey tests configuration such as the side solar arrays and load cells for the launch vehicle interface stacks. This updated model was also used to analyze the vibration test configuration as well as provide input to the coupled launch load analysis. The updated model confirmed that the potential coalescence between the top solar array and CX panel modes was no longer an issue because these modes were separated by almost 20 Hz. In addition, the lowest axial mode occurs at a much higher frequency than the limit imposed by the launch vehicle dynamics.
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6.0
CONCLUSIONS
This paper described the experimental modal test performed on the CASSIOPE spacecraft structure to identify the critical modes in order to validate the analytical finite element model. The spacecraft modal survey was conducted in a flight-like configuration with a high level of integration to obtain the most realistic modal information. The test results clearly showed that potential modal coalescence between the top solar panel and the CX panel was not an issue and that the minimum natural frequency requirements imposed by the launch vehicle in the thrust direction had been achieved. More importantly, the test results show that the experimental modal test methodology developed for the CASSIOPE spacecraft modal survey is an efficient approach that can be extended to the modal test of other large structures. The multi-input multi-output modal test using two shakers simultaneously not only distributed excitation energy throughout the structure but also generated multi-referenced FRFs to resolve closely spaced modes. Furthermore, the advanced curve fitting algorithm was capable of generating clear stability diagrams so that modal parameters could be extracted without much difficulty. The updated finite element model can serve as a powerful technical tool to predict the dynamic response of the spacecraft to the launch vehicle environmental loads. This modal testing methodology, data analysis and finite element model update approach developed for this modal survey test can be extended to ground vibration tests of other large complex structures such as aircraft or spacecraft.
ACKNOWLEDGEMENTS The authors would like to acknowledge the invaluable contribution and participation from Mark O’Grady and Yvan Soucy from the Canadian Space Agency, Andrew Woronko, Mike Grigorian and Lubomir Djambazov from MDA Corporation as well as Luc Hurtubise and Brent Lawrie of Aeroacoustics and Structural Dynamics Group in planning and performing the modal test of the CASSIOPE spacecraft structure.
REFRERENCES [1] Ewins, D. J., “Modal Testing: Theory, Practice and Application,” Research Studies Press, 2000. [2] Avitable P., Singhal, R., Peeters, B., and Leuridan, J., “Modal Parameter Estimation for Large Complicated MIMO Tests,” Sound and Vibration, Vol. 40, No. 1, pp. 14-20, 2006. [3] Shye, K., VanKarsen, C., and Richardson, M., “Modal Testing using Multiple References,” International Modal Analysis Conference V, 1987. [4] Peeters, B., Van der Auweraer, H., Guillaume, P., Leuridan, J., “The PolyMAX Frequency-Domain Method: A New Standard for Modal Parameter Estimation?” Shock and Vibration, Vol. 11, pp.395–409, 2004 [5] Mevel, L., Benveniste, A., Basseville, M., Goursat, M., Peeters, B., Van der Auweraer H., and Vecchio, A., “Input/output versus output-only data processing for structural identification—Application to in-flight data analysis” Journal of Sound and Vibration, Vol. 296, pp. 531-552, 2006. [6] Peeters, B., Hendricx, W., Debille, J., and Climent, H., “Modern Solutions for Ground Vibration Testing of Large Aircraft,” Sound and Vibration, IMAC Show Issue, pp. 8-15, 2009 [7] Kammer, D. C., “Sensor Placement for On-Orbit Modal Identification and Correlation of Large Space Structures,” AIAA Journal of Guidance, Control and Dynamics, Vol. 14, No. 2, pp. 251-259, 1991. [8] LMS Instruments, LMS SCADAS III Data Acquisition Front-End, Breda, The Netherlands, www.lmsintl.com, 2009. [9] LMS International, LMS Test.Lab Spectral Testing Rev 9A, Leuven, Belgium, www.lmsintl.com, 2009. [10] Cooper, J. E., “Comparison of Modal Parameter Estimation Techniques on Aircraft Structural Data” Mechanical Systems and Signal Processing, Vol. 4, No.2, pp 157-172, 1990.
BookID 214574_ChapID FM_Proof# 1 - 23/04/2011
BookID 214574_ChapID 101_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Assessment of Nonlinear Structural Response in A400M GVT Javier Rodríguez Ahlquist, José Martinez Carreño, Héctor Climent, Raúl de Diego and Jesús de Alba Airbus Military John Lennon s/n, 28906 Getafe (Madrid) – Spain e-mail: [email protected], [email protected] Nomenclature: Aeroelastics, Ground Vibration Test, GVT. ABSTRACT As part of the flutter clearance and aircraft certification process, a Ground Vibration Test (GVT) was performed on the first Airbus A400M out of production line. The A400M is a military transport aircraft with a maximum take-off weight of 141 Tm and capable of take off and landing on unprepared runways. Its powerplant is formed by four TP400 turboprop engines with a combined power of 44,000 SHP. With the A400M being a relatively large four-engine aircraft, its structural dynamic response is characterized by a considerably rich modal density in the frequency range of interest of the test, up to 30 Hz. The pylonmounted turboprop engines are more flexible that alternative motorizations, contributing to an even denser modal base. Following usual practice, most relevant modes, including those of powerplant and control surfaces, were appropriated (tuned) at different excitation levels. This allowed assessing the magnitude and character of structural nonlinearities inherent to real structures. Selected results are presented in this document together with some of the lessons learned.
Figure 1: A400M MSN001 during GVT at the Flight Test Centre in Seville (Spain) T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_101, © The Society for Experimental Mechanics, Inc. 2011
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1148 1
INTRODUCTION
The A400M Ground Vibration Test was carried out in autumn 2008 on premises of the A400M Final Assembly Line in Seville (Spain). Five different aircraft configurations (light A/C, open ramp, refueling pods ON, extended flaps and heavy A/C) were tested in the frame of five weeks. EADS Military Transport Aircraft Division (MTAD), now Airbus Military, with headquarters in Getafe (Spain) was responsible of the test, with technical assistance of LMS International and Alava Ingenieros. Airbus Military has its roots in the former Construcciones Aeronáuticas S.A. (CASA), founded in 1923 and since then the largest Spanish aircraft manufacturer. In 1999 CASA merged together with DASA and Aerospatiale-Matra to form the EADS (European Aeronautic Defence and Space) company. In April 2009 military transport aircraft activities were reorganized and integrated with the name of Airbus Military into Airbus, the commercial aircraft division of EADS. With an experience in dynamic testing of more than 35 years, activities of the former CASA in this field have seen a dramatic increase in the last years with four fullscale GVT’s: A310 Boom Demo (2006), A330 MRTT (2007), C295 underwing pods (2008) and A400M (2008). These tests justified the considerable investment made in state of the art test instrumentation, including an acquisition system of 768 channels, and the development of the required competence in GVT preparation, execution and analysis. Compared to other GVT’s, the high-wing and T-tail configuration of the A400M represented a challenge in terms of robustness and stability for the platforms required for shaker installation. Excitation points are high relative to floor level, while important loads were to be applied because of the aircraft size. This motivated selecting a significantly more robust design for powerplant and tail group platforms, far from the conventional temporary scaffolding commonly used in other tests.
Figure 2: A400M GVT scaffolding outline (top) and details (bottom)
BookID 214574_ChapID 101_Proof# 1 - 23/04/2011
1149 A second distinctive feature in the GVT setup is the suspension system. Bungee suspensions are usually superior compared to pneumatic suspensions, as they allow lowering the frequency of aircraft rigid body modes (RBM). This is desirable, as the aircraft dynamic response is less affected by support conditions. The limitation is normally aircraft weight. The specially developed bungee suspension for this GVT, makes the A400M one of the largest aircraft ever tested on bungees. The system allowed a RBM range comprised between 0.3 Hz and 1.3 Hz, 40% lower than the first flexible mode.
Figure 3: A400M GVT suspension system: MLG (left) and NLG (right) In parallel with test hardware design and manufacture, a dynamic finite element model (FEM) of the aircraft was adapted to reproduce GVT conditions including suspension system and mass states of the various GVT aircraft configurations. The engines were introduced in the model using dynamic condensation of the detailed engine FE model.
Figure 4: A400M GVT pre-test GVT FE model 500 accelerometers were distributed over the entire aircraft structure and additionally 200 on specific aircraft systems. Accelerometers were positioned on selected structural hard points at pre-defined locations ensuring perfect correspondence between test and FE model.
Figure 5: Test wireframe model with accelerometer distribution in the A400M GVT
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1150 57 different excitation points were used during the test. Different shaker types covered various requirements in terms of maximum load and stroke. Maximum levels up to 1000 N and 25 mm stroke were reached during the test. Of the 256 acquisitions performed for the five aircraft configurations, 60% of them corresponded to sweep sine excitation (Phase Separation Method, PSM), 33% to modal appropriation of 25 different modes (Phase Resonance Method, PRM), and the rest was dedicated to rigid body mode determination (hammer and manual excitation) and random with multiple shakers.
Figure 6. Excitation points used during A400M GVT 2
RESULTS
In total 70 aircraft flexible modes were identified in Configuration 1 up to 45 Hz. This figure does not include local modes corresponding to propeller, landing gear and others. The aircraft architecture, with T-tail and fuselage rear door and delivery ramp, resulted in empennage modes at lower frequency than for other aircraft of comparable size. Powerplant was source of an important number of modes in the range of interest. These included vertical lateral and roll of each powerplant and at least two types of engine bending modes. As expectable in fourengine aircraft, each mode type derived in series of four modes as the different engines coupled together. Propeller blade bending modes turned out to be located towards the higher end of the range of interest, producing an important number of low-damped modes in a relatively narrow frequency range. Structural nonlinearity observed throughout the test can be freely categorized in: 1) Conventional nonlinearity 2) Nonlinearity affecting modal shapes 3) Combined nonlinearity with multiple mechanisms 1) Conventional nonlinearity Real structures usually present nonlinear behavior up to some extent. Significant sources of nonlinearity in aerospace structures include riveted metallic construction, nacelle latches, hydraulic actuators of control surfaces and elastomeric engine mounts. The degree of nonlinearity for a given structural resonance will depend on the modal displacements at the locations where nonlinearity is originated. The pylon-based engine mounting system is known to be more flexible than alternative truss designs. On top of it, turboprop engines are significantly more flexible than turbofan engines with comparable size. This originated a considerable number of engine and Engine Mounting System (EMS) modes below 30 Hz. Localized nonlinear damping is mainly attributable to high performance elastomeric engine mounts meant to decouple the engines from the rest of the aircraft.
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1151 The main footprint of nonlinear effects affecting a vibrating structure is the variation in terms of frequency and amplitude of its structural response when excited with different levels of energy. This can be immediately observed in the resulting Frequency Response Functions (an example is shown in Figure 7).
Figure 7. Driving point FRF’s corresponding to engine hub excitation along Y-axis with various force levels: 250 N (red …); 500 N (green --..); 750 N (blue ___). Apart from frequency shift, nonlinear response can be associated to significant FRF peak skewness at affected resonances. This skewness can in turn produce equivocal pole stability plots using conventional modal identification algorithms in frequency domain. During the A400M GVT, certain highly damped, highly force-dependent structural resonances showing positive skewness were systematically fitted by default by means of two poles: a low frequency pole with low damping fitting the left flange, and a higher frequency and higher damping fitting the right flange. This two-pole decomposition, which from lower level acquisitions was known to be unphysical, was avoided by narrowing the analysis range around the affected resonance and reducing considerably the size of the modal model. A graph comparing a measured FRF with a fitted double pole and a single pole modal model is shown in Figure 10. Synthesized modal models derived from FRF’s show difficulties in reproducing nonlinear peak skewness. Measured
a/F
Synthesized
a/F
Frequency
10 dB
1 Hz
Frequency
Figure 8. Measured driving point FRF vs. fitted modal model for highly damped force-level dependent resonance (EMS roll mode). Top: two-pole model (default); bottom: single-pole model.
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1152 Linearity plots show modal properties (frequency, damping) as functions of excitation force and vibration amplitude, which can all be accurately measured when using harmonic excitation. The divergence of measured data with respect to a linear model can be used to assess the character and magnitude of the dominating sources of structural nonlinearity. A few examples as obtained in the A400M GVT are shown in Figure 9 through Figure 11. All reported examples correspond to homogenous excitation, this is, maintaining exciter arrangement. For these cases, variation of modal frequency ranged within 5% for a force level ratio (max/min) of ca. 4. Damping shows larger variations, especially for the case of elevator antisymmetric rotation (Example 3), where damping more than doubled when testing with different force levels. When comparing results for nonidentical exciter arrangements, the variation of modal frequencies is larger with maximum variation in excess of 10%. Modal properties derived from different excitation arrangements are generally more difficult to assess. These results can be compared with earlier studies [3]. It becomes apparent that the number of excitation levels needed to characterize structural nonlinearities varies depending on each mode, and that force-level dependence can be difficult to predict. Given that modal appropriation (PRM testing) is considerably time demanding, proper planning is required ahead of the GVT to ensure that nonlinear response is properly characterized at representative force and structural deformation levels.
1.00
1.0
0.95
0.9
1.00
0.90
u/umax
f/fmax ( __ )
d/dmax ( …. )
0.75
0.50
0.8 0.25
0.85 0.00
0.25
0.50
0.75
0.7 1.00
0.00 0.00
0.25
u/umax
0.50
0.75
1.00
p/pmax
Figure 9. Engine roll mode: modal frequency and damping vs. displacement at resonance (left) and displacement vs. excitation level (right). 1.00
1.0
0.95
0.8
1.00
0.90
u/umax
f/fmax ( __ )
d/dmax ( …. )
0.75
0.50
0.6 0.25
0.85 0.00
0.25
0.50 u/umax
0.75
0.4 1.00
0.00 0.00
0.25
0.50
0.75
1.00
p/pmax
Figure 10. Rudder rotation: modal frequency and damping vs. displacement at resonance (left) and displacement vs. excitation level (right).
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1.0
0.95
0.7
1.00
0.90
u/umax
f/fmax ( __ )
d/dmax ( …. )
0.75
0.50
0.4 0.25
0.85 0.00
0.25
0.50
0.75
0.1 1.00
0.00 0.00
0.25
u/umax
0.50
0.75
1.00
p/pmax
Figure 11. Elevator antisymmetric rotation: modal frequency and damping vs. displacement at resonance (left) and displacement vs. excitation level (right). 2) Nonlinearity affecting modal shapes In the previous section it was shown how modal frequency and damping can depend on excitation level, now it will be seen how modal shapes may also change. This was especially evident in the case in the A400M in a frequency range comprising the inner engine pitch and yaw. These modes presented considerable difficulty when attempting appropriation at high force levels. The reason can be inferred from Figure 12. The representation of modal frequencies as function of excitation force reveals multiple intersections as force levels are increased. This is produced as different modes show different sensitivity to changing excitation level and exciter arrangement. 1.00 E MS Y a w modes
LLLL
Relative Modal Frequency f/fmax
0.90 0.85
DDDD
E MS P itc h modes
0.95
DDUU
LRRL
DUUD
0.80
L--R
DUDU 0.75
DUDU
-LR-
0.70
DDUU
0.65
DUUD
DDDD
0.60 0.55 0.00 Random
0.25
0.50
0.75
1.00
Excitation Level p/pmax
-LR-
L--R
LRRL
LLLL
Figure 12: Force level dependence of EMS pitch and yaw modes.
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1154 While modal identification of these modes turned out straightforward using very low excitation levels (random excitation), higher forces reached using sweep sine excitation produced occurrence of lateral and vertical modes in a very narrow frequency range. This does not just affect modal shapes, as it is illustrated in Figure 13, but also test quality. Combined vertical and lateral displacement, whose ratio varies as force levels change, make difficult to ensure perfect alignment between shaker and structure. Out-of-excitation-axis displacement is not only difficult to avoid, its amplitude is also considerable, as a result of high force levels being exerted at low frequency. A large number of acquisitions were devoted to powerplant characterization at low frequency, using different exciter arrangements. In spite of all attempts, phase purity of the modes in this range, as quantified by means of Modal Phase Collinearity, resulted lower than average for the rest of the test.
Figure 13. Engine pitch mode DUUD as derived from acquisition using random excitation (a), and with sweep sine excitation at engine hubs along Z-axis (complex mode shapes). Observe difference in lateral powerplant component between both cases. 3) Combined nonlinearity with multiple mechanisms For certain modes combining control surface and powerplant response, it turned out a priori difficult determining the dominant source of nonlinearity from the list of usual suspects (hydraulic actuators, elastomeric engine mounts). This issue is a key consideration when selecting the optimum excitation point(s). Testing provided the necessary insight (see Figure 14). Representing the modal frequency of one of these modes as function of the excitation force turned out that for much lower excitation levels, modal frequency is lower when exciting directly at the control surface. On the other hand, engines could be excited at much higher levels than the rest of the structure during Sustained Engine Imbalance testing, a side activity of the GVT during which maximum constant levels of 600 to 1000 N were reached. Aileron excitation is probably more representative, but questions arise whether true convergence can always be reached within admissible excitation levels. Appropriation of control surface modes was performed exciting them directly.
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f / fmax
1.00
0.95
0.90 0.00
0.25
0.50
0.75
1.00
p / pmax
Figure 14: Left: Modal shape of structural resonance involving aileron rotation and outer engine aft bending. Right: Dimensionless modal frequency as function of excitation force: engine hub excitation (red diamond) and aileron excitation (blue triangle) 3
SUMMARY
For the A400M GVT a state-of-the-art test setup was used to characterize the aircraft structure in terms of modal frequency, damping, stiffness and mass. Extremely robust support structures allowed reaching considerably high excitation levels without perturbing aircraft response. This allowed characterizing structural nonlinearities for relevant aircraft modes. Challenges derived from high modal density and modal coupling increased the complexity of modal identification, with the relatively flexible engine and engine mounting system playing a key role. Different aspects have been reviewed surrounding the topic of nonlinear behavior: modal shape modification and combined nonlinear mechanisms. Every GVT is different, and this is likely to remain so as new materials and designs are incorporated in future aircraft. In any case, the A400M GVT has turned out to be one of most complex GVT’s performed in recent times. The aircraft size and number of test configurations, the flexibility of the turboprop engines, high modal density and structural nonlinear response represented altogether a significant technical challenge that had to be addressed within a tight test schedule. The quality of the A400M GVT test results owes to more than two years of preparation activities and the fruitful collaboration between Airbus Military, its technical partners LMS and Alava Ingenieros and the close collaboration with Airbus, with special mention to its Aeroelasticity Department in Bremen.
4
REFERENCES
[1] Ewins D.J., “Modal Testing: Theory and Practice”, Research Studies Press Ltd., 1984. [2] Worden K. and Tomlinson G.R., “Nonlinearity in Structural Dynamics”, Institute of Physics Publishing, 2001. [3] Göge D., Sinapius M. and Füllekrug U. (DLR), “Non-Linear Phenomena in GVT of Large Aircraft”, Proc. of IFASD (International Forum on Aeroelasticity and Structural Dynamics), Amsterdam, Netherlands, 2003. [4] Peeters, B., Climent, H., de Diego, R., de Alba, J., Rodriguez-Ahlquist, J., Martinez-Carreño, J., Hendricx, W., Rega, A., García, G., Deweer, J., and Debille, J., “Modern Solutions for Ground Vibration Testing of Large Aircraft”. Proceedings of IMAC 26, the International Modal Analysis Conference, Orlando (FL), USA, 2008. [5] Oliver M., Rodríguez Ahlquist J., Martinez J., Climent H., de Diego R. and de Alba J., “A400M GVT: The Challenge of Nonlinear Modes in Very Large GVT’s”. Proc. of IFASD (International Forum on Aeroelasticity and Structural Dynamics), Seattle, USA, 2009.
BookID 214574_ChapID FM_Proof# 1 - 23/04/2011
BookID 214574_ChapID 102_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Application of PVDF foils for the measurements of unsteady pressures on wind tunnel models for the prediction of aircraft vibrations W. Luber1 and J. Becker2 EADS-MAS (Military Air System) 85077 Manching, Germany [email protected]
Abstract An experimental investigation of unsteady pressures on military aircraft wind tunnel models has been performed using Polyvinylidenfluorid (PVDF) sensors. The unsteady pressures from the wind tunnel measurements are a prerequisite for the prediction of dynamic loads and vibrations during the development and design of new fighter aircraft structures. In the past the wind tunnel unsteady pressure measurements had been performed using wind tunnel models equipped by Kulite pressure sensors. This technique resulted in very high cost wind tunnel models. In contrast the application of the PVDF foil Smart Sensor & Signal Processing Technology would lead to improvements through the application of an affordable test technique which could be also applied in flight with the benefit of more accurate and locally detailed pressure information. The present investigation includes the measurement of unsteady pressures with PVDF foils on a wind tunnel model of a trainer aircraft configuration in the high speed transonic wind tunnel of the NLR Amsterdam. The investigated Mach numbers were M0.5, M0.7 and M0.9 and the incidence range was 0 to 45 degrees. The results of the experiments are discussed using the analysed time histories and power spectral densities of the stochastic unsteady pressure and comparisons to results using direct unsteady pressure measurements. Main aspect is the evaluation and validation with respect to technical industrial applicability of PVDF foils.
1 2
Wolfgang Luber, Chiefengineer Structural Dynamics and Aeroelasticity Dr. Ing. Jürgen Becker, Retired from EADS
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_102, © The Society for Experimental Mechanics, Inc. 2011
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1. Introduction During the development of military aircraft structural design dynamic loads have to be established which arise from unsteady aerodynamics caused by separated flow on the aircraft at high incidences, so called buffet forces which are essential components of the design and fatigue loads. These periodic and stochastic buffet forces excite the elastic vibration modes of aircraft structural components, leading for instance the wing-, the fin- and horizontal tail buffeting. Fin-buffeting is an aero-elastic phenomenon occurring on various high performance fighter aircraft. Flying at high angles of attack vortices originate from the leading edges of wing and fuselage. These unsteady vortices which contain fluctuating flow components burst drastically near the vertical tail of the aircraft and exciting the structure of the vertical tail in its natural modes. The resulting buffet fatigue loads can become an airframe fatigue and maintenance problem and might either require a structural design including dynamic buffet loads or heavier structures, excessive inspection or active measures to reduce dynamic structural loads. In the past buffet forces have been predicted using an experimental technique based on the measurement of wind tunnel models instrumented with a certain number of unsteady pressure pick ups. For instance the wing and fin buffet loads of several military aircraft have been experimentally derived by application of the unsteady pressure measurement technique, ref. 1 - 10. This technique resulted in very high cost wind tunnel models, for both the development and maintenance of the model. Another promising technique for the measurement of unsteady pressures has been proposed by several authors, described for example in, ref. 11-13 which is based on the application of the PVDF foil Smart Sensor & Signal Processing Technology. This technology is especially investigated here on a wind tunnel model of a military trainer aircraft configuration of EADS Military Air System Deutschland. 2. Description of the wind tunnel model tests 2.1 Description of the wind tunnel model The wind tunnel model is a 1:15 scaled configuration of a military combat aircraft. The design of this Aircraft is driven by the strategic target to combine both a Trainer and a light combat aircraft together. The layout of the Aircraft shows that it is a two seat airplane with one engine and the capability to fly supersonic speeds. The basic configuration contains beside the two missiles, mounted on the tip position of the wings also some interface points for carrying underwing external stores like fuel tanks and reconnaissance pods. For aerodynamic purpose the model is similar to the full version and therefore the air intake is built in the scale. This Aircraft has two air intakes on the left and right side of the centre fuselage. Flying in high angle of attack is main point of the design criteria of this aircraft. Due to the fact that this aircraft should fly a very long time with more flight hours compared with a pure fighter aircraft the structure should be designed to withstand all the buffet excitation. The better design would be that the buffet excitation will be minimized. The configuration is shown in Fig. 1.
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Figure 1: Wind Tunnel Model AT2000 2.2 Description of the wind tunnel model instrumentation The instrumentation of the fin is shown in figure 2. The instrumentation was performed by Mirow Systemtechnik GmbH, Berlin. A PVDF foil has been attached at the port and starboard side of the fin. Twelve pressure locations at the port and starboard side have been installed and instrumented at 4 span-wise sections.
MP 1 – MP 12 Port Fin
MP 13 – MP 24 Starboard Fin
Figure 2: Location of the port and starboard pressure measuring points The following different configurations of the aircraft can be changed during wind tunnel trials, by using components of the trailing flaps to simulate different deflection angles. Wing trailing edge flap: Vertical Tail:
K rudder
Horizontal Tail:
K HT
K flap
20q; 10q; 0, 10q; 20q
0, 10q; 20q 20q; 15q; 10q; 5q; 0, 5q; 10q; 15q; 20q
The on board measurement equipment is installed behind the cockpit in the front fuselage. The cockpit can be removed to maintain the equipment.
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Figure 2 shows the complete wind tunnel model installed in the NLR High Speed Tunnel in Amsterdam. The pitch up position of the port vertical tail can be seen very well on the figure. Figure 4 depicts the wiring system of the installed piezos in more detail.
Figure 3: Model in NLR HST wind tunnel – Fin instrumented with PVDF foil
Figure 4: Location of pressure measuring location on port and starboard fin
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2.3 Description of the PVDF foil Smart Sensor & Signal Processing Technology The measuring technique is based on the application of a piezoelectric foil with a discretised sensor structure which is glued onto the airfoil model. The piezo foil is an extremely thin (9P < h < 100 Pm) plastic foil made of PVDF films and is metalized on both sides. The piezo effect results from the partially crystallized structure of the foil material, which is polarized by high field intensity while solidifying. For this reason the foil reacts with a change of electrical charge proportional to stress which is gripped from the metalized surface and is registered by means of a charge amplifier. The sensors are well suited to be used on airfoils in wind tunnel as well as in free flight test, since their applicability is within a range of temperature between –40 and +150 °C. The high sensitivity of the foils and the small attenuation of the piezo material allow an almost inertia less measurement of unsteady forces. In general several stress factors (shear, pressure and temperature fluctuations) occur in the piezo foil’s measuring signal. The superimposed components can be isolated to a great extent by an appropriate separation technique, see ref. 7. Some PVDF properties are demonstrated in table 1 below. Symbol t d31
Parameter Thickness Piezo Strain Constant
PVDF 9,28,52,110 23
Copolymer <1 to 1200 11
d33
Piezo Strain Constant
-33
-38
g31
Piezo Stress Constant
216
162
g33
Piezo Stress Constant
-330
-542
Y
Young’s Modulus
2-4
3-5
Units μm (micron, 10-6) 10-12 m/m/V/m or C/ m2/ N/m2 10-12 m/m/V/m or C/ m2/ N/m2 10-3 V/m / N/m2or m/m/C/ m2 10-3 V/m / N/m2or m/m/C/ m2 109 N/m2
Table 1: Typical Properties of piezo film- further details see ref. 7 2.4 Description of the wind tunnel model test program The test have been performed at the NLR Amsterdam high speed wind tunnel (HST) Wind tunnel conditions are described in table 2: Flow parameters Tunnel speed Dynamic pressure Tunnel pressure Temperature Mach number Reynolds number Angle of attack Side slip angle
U q p T Ma Relμ Į
169 232 290 9900 13200 16000 ~950. ~20. 0.5 0.7 0.9 0.68 x 106 0…15…45 10 < Į < 45; ǻĮ = 5 0.0
m/s Pa Pa Deg. Celsius deg
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Signal parameters Sampling rate Low pass frequency Test time
fM fT tM
2000 256 20
Hz Hz s
Table 2: Wind tunnel test conditions 3. Test results 3.1 Evaluation of time histories, rms values and power spectral densities of unsteady fin buffet pressures 3.1.1 Unsteady pressures Time histories of the surface pressure p(l, t) at the different locations are recorded during the tests. Non-dimensional pressure coefficients are defined as: p (1, t ) p f c p (1, t ) qf and the mean pressure values cp(l) at the location l is: 1 c p (1) c p (1, t ) dt T³ T is the test duration. The mean square value is defined by 2 1 c p2 (1) c p (1, t ) c p (1) dt ³ T
>
@
and the rms (root mean square) value is
c p2 (1)
3.1.2 Time histories Figures 6 - 9 show typical time histories of measuring point MP9 for different angles of attack. 3.1.3 RMS pressures RMS pressures in [Pa] for the locations MP1, MP2, MP6, MP8, MP9, MP11 and MP12 are shown in figures 5 and non dimensional rms pressure coefficients are compared as function of angle of attack in figure 5.
Figure demonstrate the comparison of the rms pressure coefficients of the locations MP1, MP2, MP8, MP9, MP11, MP12 versus incidence.
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MP1
0.3
>@
MP12
cP rms
MP8
MP6 MP2
MP11
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0.1 M = 0.5
0.05 0.02 0
5
10
cP rms of buffet pressures
15
20
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30
35
40
45
D >·@
Figure 5: RMS values of pressure coefficients
3.1.4 Power spectral densities In the figures power spectral densities, the power spectra are depicted. 3.1.4.1 Peak frequencies in the power spectra densities The comparison of the spectra of different location show very similar behaviour of peaks. Dominant peaks always occur around 380 and 405 Hz and around 920 Hz. The first two peak frequencies correspond to the well know vortex shedding phenomenon behind the wing of an aircraft configuration at higher incidences, known for example from the Eurofighter configuration, which is characterized by a periodic process as also known as von Karman vortex street behind cylinders. It is interesting to detect two similar frequencies which indicate a fine resolution of the signals by the PVDF foil technique. The peak frequencies are found to be identical for all locations and for all incidences. Reduced frequencies k = flȝ/U of the two first peak frequencies are: k =0.54 and 0.575 (lȝ = 024 m). The amplitude of the first peak is always significantly higher. The peak frequency at 920 Hz might result from another vortex shedding phenomenon, the origin of which is not known but might be caused by the front fuselage. Peaks from motion induced unsteady pressures are believed to be of minor importance. 3.1.4.2 Effect of pressure signal location The peak amplitudes in the spectral densities and the power spectra vary to some extent for the different locations. This was already previously demonstrated by the rms values in figure 5. The effect of location is shown through the figures 6 to 9 for the locations MP8 at 37.5 degrees, for location MP8, MP9 and MP11 at 10 degrees.
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Figure 6: Power spectral density, power spectrum variance and time history of pressure MP8 at 37.5 degrees
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Figure 7: Power spectral density, power spectrum variance and time history of pressure MP8 at 10 degrees
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Figure 8: Power spectral density, power spectrum variance and time history of pressure MP9 at 10 degrees
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Figure 9: Power spectral density, power spectrum variance and time history of pressure MP11 at 10 degrees
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3.1.4.3 Effect of model incidence The power spectra densities and power spectra and rms values of the buffet pressures increase with incidence. From 0 to 10 degrees the increase is moderate. Beyond 15 degrees an increasingly stronger magnitude development is found, as shown in the figures 10, 11 .The increase with incidence is seen especially at the frequency of about 400 Hz corresponding to a reduced frequency of ~ 0.6.
Sp[Pa2]
Prms[Pa] 1.5x107
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1000 f [Hz@
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45 D [deg]
Figure 10: Power Spectra Sp [Pa2] and rms of pressure signal position MP8 at Mach 0.5 as function of incidence
Sp[Pa2]
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Figure 11: Power Spectra Sp [Pa2] and rms of pressure signal position MP9 at Mach 0.5 as function of incidence
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3.2 Conclusion of PVDF foil measurements All expected trends of fin buffet pressures which are known from other aircraft configurations are found through the measurement with PVDF foils. Especially the trend of the rms values with incidence, the characteristic peaks in the power spectra and the development of the rms values and the power spectra are similar to the results on other configurations. The resolution of the signals with amplitude and frequency is judged to be accurate.
4. Validation of the PVDF foil Smart Sensor & Signal Processing Technology
It is intended to validate the PVDF foil Smart Sensor & Signal Processing Technology for the measurement of unsteady aerodynamic fin buffet pressures through a comparison with results from classical wind tunnel model measurements of unsteady pressures using Kulite pressure pick ups on the same model. For the comparison of the PVDF unsteady fin pressures described in chapter 3 wind tunnel measurements with the same model but with pressure pickups (Kulites) installed on the fin have been performed using the low speed wind tunnel of the Technical University of Munich – Institute of Aerodynamics. The measurement was initiated and supported by EADS Military Aircraft. The model, see fig. 12 the instrumentation and the test results are described below. 4.1 Description of the wind tunnel tests using pressure pick ups on the fin (TUM test) In figure 12 the wind tunnel model which was used at Technical University Munch for validation of the measurements are shown with the three unsteady pick up position.
Figure 12: TUM test Location of pressure pick ups
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4.1.1 Description of the experimental technique and the test program The following test parameters characterize the pressure measurements for the 1:15 scaled AT2000- high speed model in figure 12 and described in table 3 Flow parameters Tunnel speed Dynamic pressure Tunnel pressure Temperature Mach number Reynolds number Angle of attack
U q p T Ma Relμ Į
40.0 ~890. ~950. ~20. 0.118 0.68 x 106 0…15…30 0 < Į <10 , ǻĮ = 5 10 < Į <20 , ǻĮ = 2 20 < Į <30 , ǻĮ = 1 0.0
Side slip angle Signal parameters Sampling rate Low pass frequency Test time Block samples
fM fT tM N
2000 256 15 30000
m/s Pa Pa Deg. Celsius deg
Hz Hz s
Table 3: TUM wind tunnel measurement test parameters
4.1.2 Unsteady pressures Time histories of the surface pressure p(l, t) at the different locations l are recorded during the tests. Non-dimensional pressure coefficients are defined as: p (1, t ) p f c p (1, t ) qf and the mean pressure values cp(l) at the location l is: 1 c p (1) c p (1, t ) dt T³ T is the test duration. The mean square value is defined by 2 1 c p2 (1) c p (1, t ) c p (1) dt ³ T
>
@
and the rms (root mean square) value is
c p2 (1)
The power spectral density of the fluctuating pressure coefficient is defined by the conjugate complex multiplication of the complex values of the Fourier transform F (l,Ȧ) of the fluctuating pressures coefficients at the location l: 2 S cp (1, Z ) lim F (1, Z ) F (1, Z ) T f c introducing the reduced frequency k Uf
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the non-dimensional form of the power spectral density S cpN Uf S cp (1, k ) c The non-dimensional power spectral density can be transformed into an amplitude spectral density 'f c c pA (1, k ) 2 S cpN (1, k ) 'k where 'k is. Uf S cpN (1, k )
4.2 Results of the TUM measurement The classical measurement of buffet pressures using Kulites has already been validated in the past by the Technical University of Munch (TUM) Institute of Aerodynamic, Ref. 4 – 6. The time histories of the fin buffet pressures from the TUM wind tunnel measurement with the military trainer aircraft configuration have been evaluated for the angle of attack region 0 to 30°. A low pass filter has been applied with a cut off frequency of 256 Hz. Root mean square values of the non dimensional pressure coefficients for the filtered signals at the locations P1, P2 and P3 and amplitude power spectra have been generated. Fig. x shows the development of the cprms values of the signals at P1, P2 and P2. A strong increase of the rms values is present beyond 20°. All three signals show a very similar trend with incidence, this well known trend is also present at the fin of different aircraft configurations.
Figure 13 TUM test - Root mean square values of the unsteady fin pressures at the location P1, P2 and P3 as function of angle of attack
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The amplitude power spectra of the signal at P1 is depicted in fig. xx below for the different angle of attacks. A pronounced peak occurs at a reduced frequency of ~ 0.6. This peak increases with incidence and reaches a maximum value at 30 °. The development beyond 30° is not known from these measurements. This reduced frequency corresponds to the vortex shedding frequency of the wing.
Figure 14: TUM test - Amplitude power spectra of the pressure at P1 as function of angle of attack
4.3 Comparison of the test results from PVDF foil measurements and pressure pick up measurements 4.3.1 Comparison with results of a different aircraft configuration Results of buffet measurements on the fin of a canard delta configuration are shown below. The trend of the rms values versus incidence is similar to the results of the NLR trainer tests. The power spectra of a fin signal (P13) show peak values at a reduced frequency of about 0.6 similar to the NLR trainer buffet pressure spectra. This comparison gives confidence in the trends of the PVDF foil test results.
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^
Figure 15 Delta Canard Configuration – Fin buffet pressures
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4.3.2 Comparison of PVDF test results with results from TUM experiments for the trainer configuration Experimental buffet investigations performed in the past on the Eurofighter configuration have demonstrated that low speed buffet measurements at Mach ~ 0.1 can be applied for the prediction of buffet pressures at Mach number up to Mach 0.8, Ref. 4-6. On this basis low speed fin buffet experimental results from measurements performed in the low speed wind tunnel of the TUM at 40 m/s on the military trainer aircraft configuration can be applied for the validation of PVDF fin measurement results at Mach 0.5 from NLR HST wind tunnel as derived from the same mode, fig. 15.
As demonstrated below the locations MP9 and MP8 of the PVDF measurement are very close to the Kulite location P1 of the TUM experiment, see fig. 16. The location of MP11 is close to P2 and MP12 is near P3. Therefore for validation of the PVDF measured pressures comparison is performed using signals from similar locations.
Figure 16: Comparison of pressure signal locations NLR (MP1-MP12) to TUM P1, P2 and P3 A comparison of cprms values of NLR test signals MP9, MP8, MP11 and MP12 with TUM test signals P1 and P2 is depicted in the figure 17 below. The comparison in the range up to 30 degrees demonstrates that both the magnitude and the trend with incidence between the two different tests are in close agreement.
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MP1
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MP8
MP6 MP2
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M = 0.5
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D >·@
Figure 17: Comparisons of cprms values of signals at P1 and MP9 and MP8 and between P2 and MP11 and P3 with MP12 are depicted in the figures below.
0.1
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5
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Figure 18: Comparison of cp rms from TUM test signal P1 with NLR test signals MP8 and MP9
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Figure 19: Comparison of cp rms from TUM test signal P2 with NLR test signal MP11
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Figure 20: Comparison of cp rms from TUM test signal P3 with NLR test signal MP12 Good correlation is found for the cprms buffet pressure values for all angles of attack up to the limit of 30 ° which is present due to the limited TUM measurement program, see figures 18, 19 and 20. Small deviations might be due to calibration technique and different Mach number and cut off frequency. As demonstrated by the variance and power spectra 8, 9 and 10, the PVDF PSD results show besides the peak at k~0.6 also a peak at k~1.2. Both peaks contribute to the overall rms value.
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From this result of the comparison of the two different measurements it can be concluded that the PVDF measured buffet pressures, which are showing very similar magnitudes, are accurate enough to be used for prediction purposes. Comparison of amplitude power spectra of the comparable signals from MP9, MP8 to P1, MP11 to P2 and MP12 to P3 lead to a similar conclusion. In the table below peak pressure signals from power spectra are compared for the TUM signal P1 and NLR signal MP8. The amplitudes are similar. The difference results mainly from lower smoothing of the TUM signal which results in higher values. The rms pressure comparisons give a more precise picture.
NLR Trainer configuration TUM Trainer configuration
Peak Pressure Reduced signal frequency MP8 ~0.6
Incidence [deg]
S cp max
10,0 30,0
0.000065 0.0040
P1
10,0 30,0
0.00010 0.0096
~0.6
Table 4 5. Recommendations PVDF methods need special attention during calibration. Application of the PVDF method for the derivation of unsteady buffet pressures during the development and design process of military structures is strongly recommended. 6. Conclusions From the result of the comparison of the two different measurements it can be concluded that the PVDF foil technique is adequate for the application of the buffet prediction. This could be demonstrated through the validation of PVDF measured unsteady buffet pressures.
Furthermore it is concluded that the application of PVDF buffet pressure measurement technique leads to strong cost reductions compared to the classical approach during the design and certification of military aircraft structures including buffet dynamic loads. This is due to the fact that for the PVDF measurement the existing aerodynamic wind tunnel model for the derivation of stationary aerodynamic coefficients can be applied and it is not necessary to built an additional wind tunnel model for buffet as in case of the classical method.
7. References [1] Luber W., Becker J., Sensburg O. Impact of Dynamic Loads on the Design on European fighter 43rd Structures and Material Panel Conference, AGARD Florence 1996 [2] Becker J., Dau K. Evaluation of vibration levels at the pilot seat caused by wing flow separation. 44th Structures and Materials Panel of AGARD, Lisboa, April 1977.
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[3] Becker J., Gravelle A. Some results of experimental and analytical buffeting investigations on a Delta wing. Second International Symposium on Aeroelasticity and Structural Dynamics, AGARD SMP Aachen April 1985. [4] Breitsamter C. and Laschka B. Turbulent Flow Structure associated with Vortex Induced Fin Buffeting. AIAA Journal of Aircraft, Vol. 31, No. 4, July-Aug. 1994. [5] Breitsamter C., Laschka, B. Aerodynamic Active Control For EF-2000 Fin Buffet Load Alleviation. Proceedings of the 38th Aerospace Meeting &Exhibit, AIAA 2000-0656, 2000. [6] Breitsamter C., Laschka, B. Fin Buffet Load Alleviation Using An Actively Controlled Auxiliary Rudder At Sideslip. Proceedings of the 22nd Congress of the International Aeronautical Council of the Aeronautical Sciences (ICAS), 2000. [7] Schmid A., Breitsamter C., Laschka, B. Charakteristik von Seitenleitwerk-Buffetlasten im Stall- und Poststall Bereich. Deutscher Luft- und Raumfahrtkongress / DGLR-Jahrestagung, Paper: DGLR-2001-068, Hamburg, Sept. 2001. [8] J.K. Dürr, U. Herold-Schmidt, H. W. Zaglauer, and J. Becker Active Fin - Buffeting Alleviation for Fighter Aircraft SPIE`s 6th Annual International Symposium on Smart Structures and Materials, Conference 3326, March 1999, San Diego, USA [9] Luber W., Becker J. Comparison of Piezoelectric Systems and Aerodynamic Systems for aircraft Vibration Alleviation SPIE`s 5th Annual International Symposium on Smart Structures and Materials, Conference 3326, March 1998, San Diego, USA [10] Becker J., Luber W. The Role of Buffet in the Design of European Fighter 44th AIAA Structural and Materials Conference April 2003, Hampton, USA [11] W. Nitsche, M. Swoboda and P. Mirow Shock dectection by means of piezofoils Z. Flugwiss. Weltraumforsch. 15 (1991), 223-226, Springer Verlag [12] Lee, I., Sung, H.J. Development of an array of pressure sensors with PVDF film Journal Experiments in Fluids, January 1999 Springer Berlin/Heidelberg, Volume 26, Number 1-2, [13] R. Danz, B.Elling, A.Büchtemann and P. Mirow Preparation, characterization and sensor properties of ferroelectric and porous fluoropolymers 11th International Symposium on Electrets, 2002
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Constrained viscoelastic damping, test/analysis correlation on an aircraft engine Etienne Balmes1,2 , Mathieu Corus3 , St´ephane Baumhauer4 , Pierrick Jean4 , Jean-Pierre Lombard4 1 SDTools, 2 Arts et Metiers ParisTech, 3 Ecole Centrale Paris, 3 SNECMA ABSTRACT Constrained viscoelastic damping treatments are fairly common in many applications but have not been industrially applied to aerospace engines. New engine designs tend to use BLISK (integrally bladed disks) machined from a single part and thus showing less friction damping than earlier designs. Constrained viscoelastic treatments are considered as a possible technology to enhance damping. The paper presents results of a modal test performed in an environmental chamber to highlight temperature sensitivity effects which are characteristic of viscoelastic treatments. In a second part the computational methodology used for simulations is presented. Meshing, model reduction and post-treatment issues will be addressed. The significant challenge is this problem is the computation of accurate strain levels in the viscoelastic layer for a large model with different symmetry for the disk and the constraining layer. Test analysis correlation at various temperatures will be presented showing good correlation for both the overall levels and the prediction of the influence of temperature. INTRODUCTION In turbomachines have traditionally relied on friction in the blade roots to achieve sufficient damping levels in the rotors. With the extension of BLISK (integrally bladed disks) machined from a single part, interfaces between parts are now limited to stage connections and no longer induce sufficient damping. Friction joints remains the main damping treatment technology for BLISK, but has its limitations and trade-offs.
Figure 1: Sample modes with axial and tangential bending of the rim
Constrained viscoelastic treatments are well known to induce significant levels of damping[1, 2] . They have been used in a wide range of industrial applications for decades. The principle proposed in [3] is an adaptation of the concept using the flexibility of the inter-stage rims. Figure 1 shows radial deformation for two sample modes. It clearly appears that rim
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_103, © The Society for Experimental Mechanics, Inc. 2011
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bending occurs in the axial direction (first mode shown) and/or tangential direction (second mode). Constrained layer damping can thus be considered as a damping mechanism. The work presented here considers the base design shown in figure 2. The titanium BLISK has 36 blades and the carbon/epoxy constraining layer is cut in 4 segments. The two axial segments of the constraining layer correspond to developable surfaces, which eased manufacturing. The shaft thickness is close to 2 mm, the constraining layer 1 mm and the viscoelastic material 0.2mm. The localization of the constraining layer within the main shaft, avoids any interaction with the air flow and imposes very little shear in the constrained layer thus limiting the risk of delamination. In this design, the shaft and constraining layer shapes were only optimized in thickness but other shape optimisation can clearly enhance performance.
Figure 2: Sector of bladed disk, and sector of damping treatment
Section 1 discusses results of the experimental modal test that was performed at 3 temperatures to illustrate damping sensitivity to temperature. Section 2 presents the finite element model considered for simulations and briefly outlines computational methodologies needed to deal with a potentially huge problem: 950 000 nodes, 220 modes and in the full model that needs to account for the frequency/temperature dependence of the viscoelastic material. The viscoelastic material, SMACTANE 50, was selected to demonstrate the design (but is not suitable for actual engine operation). Correct prediction of the drop in damping associated with temperature increase, shown in section 2.2, was used to validate the model. Throughout the paper, frequencies and damping ratio are shown in an adimensional unit system to preserve confidentiality of the prototype details. 1 1.1
MODAL ANALYSIS OF A DEMONSTRATOR Experimental setup and modal results
Measurements were performed at Ecole Centrale Paris (MSSMat). Input loads were measured with a B&K load cell and velocities with a Polytec vibrometer. Data acquisition was performed using an LDS Dactron portable analyser. Handling of measurements was performed using a prototype data acquisition environment based on the MATLAB and SDT [4] environment. Tests were performed within an environmental chamber thus allowing a fairly precise control of the temperature. Temperature stabilization was obtained by waiting several hours and an automated procedure for measurement rejection during fan operation was introduced. In the future, automated measurement quality assessment would be very much needed to perform long acquisitions more reliably.
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Obtaining a clean input implied many attempts. The structure is very lightly damped and one seeks to excite a band of several kHz. The configuration eventually retained, and shown in figure 3, is a load cell impact generated using a soft pulse in an electrodynamic shaker. The final configuration used a load cell aligned with the shaker axis, horizontal offsets allowed dynamic amplification but gave lesser quality measurements. To avoid damping induced by exponential windows typically used for impact tests, the measurements were not windowed at all. This absence of windowing deteriorates estimated transfers away from resonances, so that identification should only be performed using data around resonances. Outside problems with the harmonics of 50 Hz, measurements are rather clean at 25 C and less so at other temperatures.
Figure 3: General view of measurement points. Excitations configuration #1.
1.2
Mode extraction
In figure 4, the complex mode indicator function of the overall test gives a very clear indication of the first five blade modes, visible trough the accumulation of resonances shown as vertical dotted lines in the plot. In those bands, the number of modes is very large (up to the number of blades, that is 36).
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Figure 4: CMIF of test at 25C
The frequency resolution considered in the test was quite high (45 000 points), but this is still insufficient to properly separate all modes. Figure 5left thus shows two isolated modes (with frequencies indicated as vertical lines) at the beginning of accumulation and a large number of peaks with only a few properly identified. The associated damping estimates should thus be taken with caution. The existence of a consistent lower limit on damping within the accumulations seems however to correspond to the expected material damping (mostly due to thermoelastic effects in this case). Test 89* 89*
Test 84* 84*
IdFrf 89* 89*
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60
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Figure 5: Sample transfers around a blade mode (left) and an isolated mode with split resonance.
For modes outside the accumulations, the identification is not necessarily much easier as most modes are nearly but not exactly double. The excitation configuration retained does favor 0 diameter modes, so that almost all the significant peaks correspond to double modes. Figure 5right illustrates a particular resonance, where careful inspection clearly shows two modes. The local relative error on the Nyquist [5] shown at the bottom indicates a very accurate identification for which non-linear optimization of the pole locations [6] was necessary. Figure 6 displays a few global deformations. Figure 7 shows the influence of temperature on response levels. As expected for the selected material, performance drops very significantly as temperature increases. Measurement problems (changes in impact location laser direction due to sag of the suspension at higher temperatures) don’t really allow proper quantitative
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comparison of transfers, the correlation will thus be based on predicted modal damping levels. Global damping results will be shown with FEM predictions in 2.2.
Figure 6: A few modal deformations.
Figure 7: Blade tip response as a function of temperature. Full range and zoom
2 2.1
MODEL AND CORRELATION Modeling procedure
The fundamental mechanism of constrained viscoelastic damping is shear in the constrained layer. To obtain such predictions the sheared layer must be modeled using volume elements. While for automotive applications a shell/volume/shell layup is classical [7, 8] , typical engine models use volumes. The models considered in this project, thus introduce a volume mesh for each layer with linear or quadratic element formulations depending on the choice made for the main structural FEM. To predict shear levels getting the geometry right is critical. In particular offsets with respect to the neutral fiber need to be precise. As the thickness of the layer can be very small (0.2 mm here), the accuracy on node positions needed for proper predictions is much higher than typically found in FEM input files (often ten digits or less). This problem is illustrated in figure 8 where the initial mesh shows a significant fluctuation of energy at inter-sector boundaries and the corrected mesh does not (a slight variation to blade locations is expected and found).
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Figure 8: Variation in strain energy in the viscoelastic layer. Left : initial tetrahedral mesh, Right : corrected geometry with perfect revolution and hexahedral elements obtained by revolution. (The two different configurations shown differ from the prototype built)
When considering the actual geometry of tested structure, the BLISK has 36 blades while the constraining layer has four segments. The underlying full 3D model contains 950 000 nodes and there are 224 modes in the bandwidth of interest. To obtain accurate predictions, one uses the multi-stage cyclic symmetry method introduced in [9, 10] and industrialized in the SDT Rotor[11] module. On first computes mono-harmonic solutions for target diameters. 0,1,2 and 8 diameter modes for high and low values of the viscoelastic modulus are first computed (each computation takes about 15 mn CPU). The resulting vector set contains 473 vectors in the bandwidth of interest. The iterative maximum sequence procedure is then used to generate bases for each of the 3 stages considered in the model (the base sector with 36 blades and 276 independent vectors, the lower and upper constraining layers with 4 segments and 90 and 42 vectors respectively). From these reduced sector models, one can assemble a reduced 3D model with 34485 DOFs (of whom 80 % are internal DOFs of the second order elements in the viscoelastic layer). Figure 9 illustrates typical restitution two modes. To get a feel of the displacement shape, a coarse outline of the shaft and blades is showing all blades. To analyze energy dissipation, strain energy density in the constrained viscoelastic layer is then displayed.
Figure 9: Displacement viewed on a coarse outline of the BLISK and strain energy in the viscoelastic layer.
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For correlation, one then needs to account for the frequency and temperature dependence of the material modulus and loss factor shown in figure10. To achieve such predictions in a reasonable time, a multi-model reduction approach, implemented in the SDT Visco module [8] , is used. 250 modes of the reduced shaft model are computed for low and high modulus values and the model is reduced on the associated subspace. Poles are then computed for a range of complex moduli and interpolation is used to give precise damping estimates.
1.3
5
1.2 1
10
1.1 Loss factor
Re(G) [MPa]
5
1 0.9
25
0.8 25
0.7 Smactane 50_G max =750 uf min =530 [uf]
45 Reduced frequency
0.6 45 Reduced frequency
Figure 10: Modulus and loss factor variation for SMACTANE 50 in the target frequency range.
2.2
Correlation of modal damping ratio
Figure 11 shows, using the same scale, measured and predicted damping ratio as a function of temperature. Understanding the level of correlation requires significant comments. The blade modes are clearly visible as drops in the damping levels (and vertical lines in the computation). In the model, titanium is taken to have no material damping at all, so predicted damping drops to extremely low levels as almost all the energy is concentrated in the blades. In the test, estimating damping for those modes is very difficult due to strong modal overlap. A low value near 0.3 damping units does however exist and is expected due to thermoelastic material damping. Between the accumulations associated with blade modes, the shaft undergoes significant deformation and is thus damped by the constrained viscoelastic layer. Each correlation should be considered in detail. The first experimental mode (a 2 diameter global bending) underestimates damping. Damping for this mode is essentially located in the constrained layer just below the inter-stage rim. The model does not account for segmentation of the rim and is thus too stiff and lightly damped. In the 550 to 620 frequency unit range damping is under-predicted with no clear explanation. The first modes of the accumulation are extremely close in level. In the [630-670] range the coincidence is very good. In the 670 to 730 the test starts to pose problem with only the lightly damped modes showing (two more significantly damped modes could be found in the 25 C test but not clearly at other temperatures).
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1
10
0
10
−1
10
−2
5C 25 C 45 C
Damping
10
550
1
10
0
10
−1
650 Frequency [uf]
700
750
650 Frequency
700
750
Damping
10
600
5C 25 C 45 C 10
−2
550
600
Figure 11: Experimental (top) and computed (bottom) frequencies and damping ratio at 5, 25 et 45 C
The sensitivity to temperature is very consistently predicted with the expected trend of performance decreasing with temperature. 3
CONCLUSION
A basic design of constrained viscoelastic treatment was tested on a BLISK. The results clearly show that, outside narrow accumulation frequencies corresponding to blade modes, damping is significantly enhanced by the treatment. Overall damping levels and trends when the viscoelastic material properties are affected by temperature correlate well between test and analysis. Current work on the concept focuses on selecting materials adapted to real engine temperatures, extending design methodologies to optimize treatments and demonstrations in operational conditions. REFERENCES [1] Nashif, A., Jones, D. and Henderson, J., Vibration Damping, John Wiley and Sons, 1985. [2] Rao, M. D., Recent Applications of Viscoelastic Damping for Noise Control in Automobiles and Commercial Airplanes, Journal of Sound and Vibration, Vol. 262, No. 3, pp. 457–474, 2003. [3] Dupeux, J., Baumhauer, S., Garcin, F., Lombard, J., Seinturier, E. and Balmes, E., Movable Impeller for a Turbojet and Turbojet comprising same., Patent EP2009238, US2009004021, 2009.
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[4] Balmes, E., Bianchi, J. and Lecl` ere, J., Structural Dynamics Toolbox 6.2 (for use with MATLAB), SDTools, Paris, France, www.sdtools.com, May 2009. [5] Balmes, E., Methods for vibration design and validation, Course notes Ecole Centrale Paris, 1997-2007. [6] Balmes, E., Frequency domain identification of structural dynamics using the pole/residue parametrization, International Modal Analysis Conference, pp. 540–546, 1996. [7] Plouin, A. and Balmes, E., A test validated model of plates with constrained viscoelastic materials, International Modal Analysis Conference, pp. 194–200, 1999. [8] Balmes, E., Viscoelastic vibration toolbox, User Manual, SDTools, 2004-2007. [9] Sternch¨ uss, A. and Balmes, E. and Jean, P. and Lombard, JP., Reduction of Multistage disk models : application to an industrial rotor, Journal of Engineering for Gas Turbines and Power, Vol. 131, 2009. [10] Sternch¨ uss, A., Multi-level parametric reduced models of rotating bladed disk assemblies, Ph.D. thesis, Ecole Centrale de Paris, 2009. [11] Balmes, E. and Bianchi, J., Rotor module for SDT, User Manual, SDTools, 2008-2009.
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BookID 214574_ChapID 104_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Identification of material damping in vibrating plates using full-field measurements A. Giraudeau, F. Pierron LMPF, Arts et Métiers ParisTech, Rue St Dominique, BP 508, 51000 Châlons en Champagne, France [email protected];[email protected] ABSTRACT This study concerns the identification of complex stiffness components (ie, stiffness and damping) on isotropic vibrating plates. One of the main difficulties in damping measurements comes from the connection of the specimen to its environment (joints, clamps etc…) causing extra energy dissipation. The present procedure based on full field slope measurements from a deflectometry set up together with an inverse identification technique called the Virtual Fields Method is capable of identifying the material damping regardless of the dissipation coming from the specimen boundary conditions. This paper will present some experimental results confirming this statement. 1. INTRODUCTION The measurement of material elastic stiffness and damping parameters, essential for the prediction of the vibrating or vibro-acoustic behaviour of a large range of structures, is common in material testing laboratories. The identification of the stiffness parameters is usually performed using tension, bending or torsion tests on rectangular coupons leading to simple stress states that can be expressed as functions of the specimen geometry and the applied load through a closed-form solution of the mechanical problem. Nevertheless, these procedures exhibit certain drawbacks. First, experimental boundary conditions must comply with that of the mechanical model, which is not always easy to achieve. Then, only a small number of parameters can be retrieved from a specific test because of the very simple stress state. As a result, several tests usually have to be performed to identify the full set of material parameters, increasing the cost of the procedure. As an alternative, several authors have tried to use the resonance frequencies of bending plates to identify the full stiffness tensor. More recently, the above approach was refined by using not only modal frequencies but also mode shapes. An alternative to these methods was suggested by Grédiac et al. [1] making use of the measurement of slope fields at the surface of bent plates and performing the identification through a particular application of the principle of virtual work, the so-called Virtual Fields Method (VFM). This procedure uses the global equilibrium of the observed part of the coupon. One advantage of this technique is that stiffnesses are obtained directly (no iterations, no optimization scheme) and that restrictions on specimen geometry and boundary conditions are less critical than with other methods. The measurement of damping parameters is a more complex problem than stiffness because of all the parasitic dissipation that is usually added in a classical mechanical test. One another main advantage of the present extension of this method is to greatly minimize the parasitic effects of the boundary conditions. 2. THEORY – VIRTUAL FIELDS METHOD (VFM) 2.1. Considered case The considered coupon of the isotropic test material is a rectangular thin plate (thickness h), free on its external boundary and clamped approximately in its centre to an excitation device which imposes a sinusoidal driving movement in the out-of-plane direction (inertial excitation). The external boundary of the plate is free. Assuming linear viscoelastic behaviour of the material and thin plate theory, the vibrating response of the plate is pure harmonic bending at the same frequency [2]. This response can
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_104, © The Society for Experimental Mechanics, Inc. 2011
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be described by the actual deflection field w(x,y,t) such that: w( x , y , t ) w ( x , y ) cos( Zt I ( x , y )) or using complex notation: w( x, y, t ) e[(wr jwi ) exp( jZt )] . 2.2. Global equilibrium The plate can be virtually divided in two parts: a small zone called 3 surrounding the clamping area and the second part :which is free on its external boundary. The virtual border between these two parts 3 and : shall be called *In the case of small perturbations, the local equation of equilibrium can be written at any point M of : (using the convention for summation on repeated indices)ҏ:
V ij, j f i
U Ji
(1)
where Vis the stress tensor, f the vector of volume forces, Uthe density and Jthe acceleration vector (with i and j belong to {1,2,3}). Each term can be multiplied by a u* function which is user selected and must be continuous over :and differentiable. This function can be seen as a displacement field which is virtually imposed all over ::
Vij, ju*i fi u*i
UJ i u*i
(2)
Equation (2) can then be integrated over :and leads to:
³
:
Vij, ju*idV ³ f i u *idV :
³
UJ i u*idV
:
(3)
Using the divergence theorem for the calculation of the first term and assuming H is the virtual strain field related to the virtual displacement field u*, Eq. 3 can be rearranged in a more useful expression :
³ VijH*ijdV :
³
w:
Ti u *idS ³ f i u *idV :
³
UJ i u *idV
:
(4)
where T is the vector of boundary tractions over :It can be seen each term of Eq. 4 could be viewed as a virtual work under the action of the virtual displacement u* and the related H virtual strain. Respectively from the left to the right one can find, the virtual work of the internal forces, of the junction forces along the *border, of the volume forces and finally of the acceleration forces. Due to the assumption of the linear behaviour of the plate material, the acceleration of the harmonic response field of the plate can be simply expressed as JL Zui. Finally assuming that the volume forces can be neglected, the global equilibrium of the part :free along its external boundary, can be written as:
³ VijH*ijdV ³ Ti u *idS UZ2 ³ w.w *dV :
*
:
(5)
where w is the out-of-plane deflection (it is assumed that the inertial forces generated by the in-plane deformations can be neglected since they are at least one order of magnitude lower than w) and w* the virtual out-of-plane deflection. 2.3. Virtual Fields Method (V.F.M.) The principle of the Virtual Fields Method is to replace the stress components in the above equation by the actual elastic strains through the constitutive equations which parameters are to be identified. Then, by selecting 'appropriate' virtual fields, u*(x,y), it is possible to derive equations relating the materials constitutive parameters to integral functions of the actual strains. In the present case, the only external forces acting on : are the connection forces on the * boundary. If a virtual displacement field u* is selected such that it cancels out the virtual work of these connection forces then the second term of Eq. 5 is null and only the first and the last integrals of the latter remain. It can be shown that the first term can be expressed according to the Love-Kirchhoff theory, and Eq. 5 can then be rewritten as:
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³ (Dij jZBij )k jk *idS U hZ2 ³ w.w * dS S
(6)
S
where w and k are the actual deflection and curvatures fields to be measured on the surface S of whereas w* and the corresponding k* are the virtual deflection and curvature fields to be selected for the identification (here, i and j belong to {1,2,6} with the usual rule of contracted indices). [D] and [B] are respectively the elastic bending stiffness and viscous damping matrices, whose D11, D12, B11 and B12 components are the unknown material parameters. It must be pointed out that this relationship is valid whatever the excitation frequency, at resonance or out of resonance. By separating the real and imaginary parts of Eq. 6 it can be shown that the latter can be split into two independent equations, presented in Eq. 7, where Gp and Hp denotes combinations of products between virtual and actual curvatures. These quantities are computed using the real or imaginary parts of the actual curvature fields according to whether the p index is re or im. B
B
D11G re D12 H re Z B11G im Z B12 H im
U h Z2 ³ w re w * dS
D11G D12 H Z B11G Z B12 H
UhZ
S
im
im
re
re
2
im * ³ w w dS
(7)
S
with
Gp
³ k
p 1
k1* k p2 k *2 12 k 6p k *6 dS ; H p
S
³ k
p 2
k1* k1p k *2 12 k 6p k *6 dS
(8)
S
The introduction of two independent deflection virtual fields w1*, w2* and related virtual curvatures fields k1*, k2* and k6* leads to a linear system of four equations whose unknowns are the unknown parameters: D11, D12, B11 and B12. B
B
2.4. Virtual fields selection The selection of the virtual fields is a major issue for the identification method. It is carried out using three successive criteria. x First, using piecewise virtual fields [3], virtual deflection and curvature fields are generated such that they are null along the border leading to the cancellation of the virtual work of the connection forces. x Then, to ensure independence between the four equations issued from Eq. 3, a particular application of a work by Grédiac et al [4] is used to select the ‘special’ virtual deflection fields w(1)*, w(2)* which verify particular values of Gp and Hp and lead to very simple calculations of the parameters. x The final selection is achieved using an adaptation of the work by Avril et al [5] providing ‘optimized’ virtual fields which minimize the effects of the measurement noise on the identification results. 3. Experimental results 3.1. Set-up As presented in the previous section, the application of the Virtual Fields Method to the present case requires an excitation arrangement providing an inertial out-of-plane excitation of the plate which is free at its boundaries and a measurement set-up based on an optical method providing the curvature fields and the out-of-plane displacement fields on the whole surface of the plate. These measurements must be taken at two particular times: in-phase and at S lag with the driving movement of the plate. These two particular positions correspond respectively to the real and imaginary parts of the displacement field of the coupon. Tests have been carried out using an acrylic 200 x 160 x 3 mm3 specimen. The sinusoidal out of plane inertial excitation is provided by a dedicated device where the coupon is clamped in its centre between the end of the driving rod of the device and a rigid steel washer of the same diameter,
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tightened by an axial screw. An accelerometer mounted on the rod is used to control the imposed driving movement (see Fig. 1).
Fig. 1 – Experimental set up 3.2. Slope measurements by deflectometry To avoid noise problems on curvatures arising from a double spatial differentiation of measured deflection fields, measurement of the slope field is preferred through deflectometry [6,7]. Images are frozen using a flash triggered by the driving movement with a 0 or S lag to get the images respectively related to the real or imaginary parts of the response. Deflection and curvature fields are respectively obtained from spatial numerical integration and differentiation of polynomial fits to the slope fields (see Fig. 2). Real T1
Real T1
Imag. T2
Imag. T2
2
0.5
1
0
0
80 Hz
(mrad)
100 Hz
-0.5 (mrad)
Fig. 2 – Examples of measured slope fields at 80 Hz and 100 Hz 3.3. Identification results Identification has been carried out using Eq. 7 with a set of two displacement virtual fields selected such that they are null along the rectangular border * of the 3 zone surrounding the clamping area (central area in white on Fig. 2). The two stiffness parameters D11, D12 and the two viscous damping parameters B11, B12 of the tested material are first extracted. According to [8], material elastic properties, Young modulus E and Poisson ratio Q and their related loss factors K= tan(G) and KQ are presented in Table 1 below for five excitation frequencies. It must be noticed the 100 Hz excitation is close to a resonance of the tested plate and the four other frequencies are out of resonance. The identified values are compared with the Young modulus and its loss factor obtained from, respectively, vibrating tests performed by the authors in the same frequency range on a beam and classical DMTA tests. Coefficients of variation resulting from eight measurements at each tested frequency are noted in italic blue. B
B
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Table 1 – First identification results E (GPa) X KE=tan G KX
70 Hz 4.62 0.33 0.065 -0.024
1.8% 1.3% 2.9% 10%
80 Hz 4.63 0.33 0.052 0.019
1.3% 0.5% 1.8% 4.7%
90 Hz 4.71 0.33 0.050 -0.014
1.2% 0.8% 1.8% 6%
100 Hz 4.63 0.34 0.064 -0.018
2.4% 1.7% 7.1% 9.1%
110 Hz 4.69 0.6% 0.34 0.6% 0.068 0.3% 0.021 0.8%
Beam/DMTA
4.9
2.4%
0.054 3.5%
The results for the elastic constants seem to be consistent and no significant differences are noticeable between the results issued from measurements at resonance or out of resonance. Some improvements are needed to increase the accuracy of image freezing in the time domain. The present trigger process is probably responsible for the higher scatter in the two loss factors compared to the elastic constants. 4. Insensitivity to boundary conditions As mentioned in section 2, the selection of appropriate virtual fields leads to the cancellation of the virtual works of the junction forces along the * border. Therefore if dissipation phenomena occur in the clamping area, this does not affect the global equilibrium of the : observed part of the coupon as mentioned in Eq. 6. As a consequence, the identification process should not be disturbed. To prove this very important feature of the method, a second experimentation was carried out where the same coupon was alternatively mounted at the end of the driving rod, first rigidly clamped as previously described in section 3. Secondly the same fixture is used with two rubber washers (1.5 mm thick) inserted between each face of the plate and the rigid parts of the driving device. In the first case, due to the rigid clamping, the displacement of the rod and the driving movement of the plate are the same. In the second case, a miniature accelerometer was mounted directly on the rear face of the plate, as closely as possible to the clamping area (see Fig. 3). This transducer provides a good evaluation of the out of plane solid movement of the plate. The comparison between the two acceleration signals provided by this accelerometer and the one mounted on the driving rod shows the amplitude of the solid motion of the plate is 14% lower than the amplitude of the movement of the rod and the phase lag is 34°. These results are obviously due to the dissipative behaviour of the rubber washers.
Fig. 3 – Details of second tested attachment of the plate Identifications were carried out using images taken at 100 Hz with the two configurations. The results are presented in Table 2 in the same manner as above.
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Table 2 – Identification results at 100 Hz with the two different attachments 100 Hz
Initial clamping
E (GPa) X K( tan G KX
4.63 0.34 0.064 -0.017
2.4% 1.7% 7.1% 9.1%
Rubber washers 4.73 0.33 0.055 -0.018
1.5% 0.7% 5.2% 6.4%
These results confirm that only the material properties of the : area of interest are identified without effects of the external conditions, which is one of the strengths of the method. 5. Conclusion Previous studies have shown the principle of the Virtual Fields Method applied to vibrating plates and demonstrated the experimental feasibility. The last tests have proved that the method greatly minimizes the parasitic effects due to experimental conditions. This property obviously deserves major attention since it is of great interest for the identification of material damping. Very important potential applications of the method concern the identification of local material damping in a structure (plate or shell with stiffeners, for instance) and the possibility of obtaining local damping values at different locations of a plate or structure to be used as damage indicators. This is presently underway on composite plates. References [1] Grédiac M., (1996) Direct identification of elastic constants of anisotropic plates by modal analysis: theoretical and numerical aspects. J. of Sound and Vib., 193(3):401-415. [2] Giraudeau A. and Pierron F. (2005) Identification of stiffness and damping properties of thin isotropic plates using the virtual fields method. J. of Sound and Vib., 284:757-781. [3] Toussaint E., Grédiac M., and Pierron F. (2006) The virtual fields method with piecewise virtual fields. Int. J. of Mech. Sci., 48:256-264. [4] Grédiac M., Toussaint E. and Pierron F. (2002) Special virtual fields for the direct determination of material parameters with the virtual fields method Int. Journal of Solids and Structures, 39(10):2691-2705. [5] Avril S., Grédiac M., and Pierron F. (2004) Sensitivity of the virtual fields method to noisy data. Comp. Mech., 34(6):439-452. [6] Giraudeau A., Guo B. and Pierron F. (2006) Stiffness and damping identification from full field measurements on vibrating plates. Exp. Mech. 46(6):777-787. [7] Surrel Y., (2004) Deflectometry: a simple an efficient non interferometry method for slope measurement. In SEM Annual Congress on Experimental Mechanics, 7-10 June in CostaMesa, California, USA. [8] Pritz T. (2007) The Poisson’s loss factor of solid viscoelastic materials. J. of Sound and Vib., (306):790-802.
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
A state observer for speed regulation in rolling mill drives
Ferdinando Mapelli*, [email protected] Emanuele Ruspini*, [email protected] Edoardo Sabbioni*, [email protected] Davide Tarsitano*, [email protected] *Department of Mechanical Engineering, Politecnico di Milano Via la Masa 1, 20156 Milano (Italy) NOMENCLATURE Ji Ki ri Tm T23 T35 Tu,i-th [J] [K] [A] [B] [E] a b u d kp ki t wi wp fm H(s) L(s) y [C] [D]
[Aoss] [Boss] Cki kbl rbl Cs d bli Ti [G]
Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ Æ
Torsional moment of inertia of i-th element (motor shaft or spindles) Torsional stiffness of i-th shaft or spindle Torsional damping of i-th shaft or spindle Rotation of i-th element Motor torque Torque transmitted between end of motor shaft and upper spindle Torque transmitted between upper spindle and lower spindle Load torque on upper or lower spindle Inertial matrix Stiffness matrix State matrix Input state matrix Load state input Inertial coefficient of damping Stiffness coefficient of damping Input vector Load vector Proportional gain Integral gain Transmission ratio Rotational speed of i-th element Speed regulator bandwidth Speed regulator phase margin Transfer function Closed loop transfer function vector of observed variables Output state matrix Output input matrix Estimated state vector Extended state observer estimated state vector Extended state observer state matrix Extended state observer input state matrix Elastic torque transmitted by i-th shaft Restoring torque stiffness coefficient Restoring torque damping coefficient Regulator damping coefficient First order response time constant Clearance of i-th backlash Kp/Ki Observer gain matrix
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ABSTRACT Rolling mill drive-trains, driven by AC or DC motors, have historically experienced premature component fatigue failures even when the perceived operating load is well below the design limit. This is often related to low damped torsional vibrations, especially to self-excited vibrations in rolling slippage (especially during threading and tailing out of a rolled piece), overloading the elements of the drive-train. Long transmission shafts make these vibrations become more critical. A regulator aimed at damping the torsional vibrations of the rolling mill drive-train and thus reducing the electric motor speed fluctuations is presented in this paper. The proposed regulator relies on a reduced order state observer able to estimate the shaft torque amplifications due to the working process. The capability of the regulator to damp out torsional vibrations has been verified through simulations on a lumped parameter torsional model of a single-stand rolling mill accounting for torsional deformability of the power-train shafts.
INTRODUCTION The regulation of the rotational speed of electric motors is a widely diffused issue in industrial applications, because long transmission shaft may amplify low damped oscillations. An accurate regulating system for the motor speed taking into account the behaviour of the mechanical system is thus required. In particular, this paper focuses on the design of a regulator of a hot rolling mill characterized by long transmission shafts and a gearbox with a large torsional inertia In order to ensure high performances, i.e. a fast response to transitories, the bandwidth of feedback control system may include the first eigen-frequency of transmission system, so during working process torsional oscillations start, these are very difficult to remove or reduce taking into account classical damping methods. The difference between motor speed and working rollers speed lead to premature components fatigue failure and manufacturing defects. The control system has been developed with the aim to use the few information, available in real plants. A state observer has thus been implemented. Based on this state observer a control system aimed at regulating the speed of the electric motor powering the rolling mill and trying to reduce the torsional oscillations eventually arising during working conditions has been implemented. The performances of the implemented control system and observer have been evaluated using a lumped parameters model of a stand of a hot rolling mill. Simulation and practical application of the proposed control system in industrial electric drives are proposed.
ROLLING MILL MODEL Rolling mill is composed by several rolling stands. Stands present a different layout as the rolling process proceeds. In particular dimensions of stands closer to the tubes inlet are larger. Attention is focused on a stand. It is composed by an electric motor transmitting power to the reduction box (or gearbox) by means of a motor shaft. In Table 1 the characteristic of the electric motor are reported. Motor power Max Torque Angular speed Motor inertia Spindle + slab inertia Motor shaft stiffness Spindles stiffness t
1700 [kW] 18000 [Nm] 496 [RPM] 2 1020 [kgm ] 2 10000 [kgm ] 448240000 [Nm/rad] 2241000 [Nm/rad] 0.2834
Table 1: Characteristic of rolling mill.
In Figure 1 the scheme of considered rolling mill are reported.
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Figure 1: Scheme of the rolling mill.
The gearbox reduces angular speed passing from the motor shaft to the spindles and divides the driving torque equally on the two spindles connected with the milling rollers. The spindles thus rotate at the opposite speed. In Table 1 the characteristic of shaft, gearbox and spindles First torsional eigen-frequency of system is 2.38 [Hz]. Parameters of the model have been taken from the literature and are typical of big size plants ([6], [1]).
MECHANICAL MODEL In order to investigate the rolling mill dynamic behaviour, a lumped parameters torsional model has been implemented. The scheme of the torsional model is shown in Figure 2. It is composed by the motor shaft, the upper/lower spindles and the reduction box. Motor shaft and spindles are coupled by means of gears with backlash. Backlash in fact introduces discontinuities in the transmission of motion, thus producing self-excited torsional vibrations responsible of sudden amplifications of the torque oscillations. In order to take into account the torsional deformability of the system, each shaft/spindle has been modeled as two lumped inertias concentrated in the shaft/spindle endings and characterized only by a torsional moment of inertia (Ji, see Figure 2). Lumped masses are connected by a torsional springs (Ki, see Figure 2). Elements of the gearbox haven’t been individually modelled and their torsional moment of inertia have been merged with one of the shaft/spindle to which they are connected.
Figure 2: Lumped parameters torsional model of the rolling mill.
The equations of motion of the model can be written as: (1)
1196 where:
(2)
(3)
(4)
(5)
Being t the global transmission ratio of the gearbox, Tm the electric motor driving torque, Tu1 and Tu2 are the resistant torques applied to the rollers and due to the milling process, T23 and T35 are the torques exchanged between the gears of the transmission box between the motor shaft and the upper spindle and between the upper and the lower spindle respectively (see Figure 2). Structural damping has been introduced as a linear combination of the mass and stiffness matrices: (6)
These parameters have been tuned in order to obtain a level of damping equal to approximately 5‰ of the critical damping for each vibrating mode when clearance of backlash between gear wheels is equal to zero. In order to reproduce discontinuities in the transmission of motion due to the presence of backlash in gears (which may produce self-excited torsional vibrations), the visco-elastic restoring torque between the i-th and (i+1)-th inertias can be written as ([3]):
(7)
where:
'TL
TL W LTL
(8)
1197
7LL .EO
EOL EOL
'TL
Figure 3: Characteristic curve of backlash element: restoring torque vs relative rotation.
As it can be seen from eq (7) and Figure 3, when the relative rotation Δθi between the two gear wheels is lower than backlash clearance bli, no torque is transferred from one wheel to the other. On the contrary, when relative rotation is larger than backlash clearance, a visco-elastic restoring torque is acting between the i-th and the (i+1)-th dofs. Kbl and Rbl are respectively the stiffness and the damping coefficient characterizing the restoring torque.
0.04 Ti
0.03
WTi+1
0.02
[rad]
0.01 0 -0.01 -0.02 -0.03 -0.04 0
0.5
1 time [s]
1.5
2
Figure 4: Rotations θi and tθi+1. Sine wave imposed.
As an example, the behavior of the backlash element of eq. (7) is shown when rotation qi is imposed. The imposed motion is a sine wave with amplitude 0.03 [rad] and frequency 1 [Hz]. Figure 4 depicts the time history of rotations θi and θi+1, while in Figure 5 the relative rotation Dθi and the visco-elastic restoring torque are represented. In Figure 5, the clearance of backlash is also shown (black dashed line). As can be inferred from Figure 5, a torque is applied between θi and θi+1 only when the relative rotation is larger than the backlash clearance. Due to this, there is a time delay in the response of θi+1 with respect to rotation θi (see Figure 4). It is straightforward to notice that, if the clearance of backlash is zero (i.e. there is no backlash between the gear wheels), the dofs of the system are reduced to four, being rotations θ3 and θ5 directly a function of θ2.
1198 -3
5
x 10
4 Clearence
3 2
[rad]
1 0 -1 -2 -3 'T [rad] Ti, i+1 [MNm]
-4 -5 0
0.5
1 time [s]
1.5
2
Figure 5: Relative rotation Dθi and restoring torque Ti,i+1. Sine wave imposed.
CONTROL STRATEGY The scheme of the regulator aiming at damping the torsional vibrations of the rolling mill drive train is shown in Figure 6. The driving torque and motor angular speed are assumed to be the measured quantities.
Figure 6: Scheme of control strategy.
The proposed control system consist of 2 contributions (Tm1 and Tm2). The first one, T m1, is a feedback contribution coming from the PI speed regulation loop; the second one (T m2) is a feed forward contribution designed to damp torsional vibration (TVSS - Torsional Vibrations Suppression System in Figure 6) and it is proportional to the difference between the input gearbox shaft and the upper spindle angular speeds (wu1). (9)
Since and are not measured, a state observer is needed. In the following the design of the PI regulator and of the state observer are presented.
PI regulator design In order to design the electrical motor regulator, the model previous described has been linearized and the backlash of gears has been neglected. Thus the dofs of the system are reduced to 4 (q1 q2 q4 q6). The PI regulator gains (kp and ki) have been tuned up by neglecting the influence of the T m2. Obviously, if the u=d=0, the system is free to rotate. By applying the following set of coordinates:
(10)
1199 The states of the system has been reduced from 8 to 7. The drawback is that information about the system angular position is lost. Under the previous assumption, the eq. of motion of the system can be expressed in state-space form as: (11)
where: (12) (13)
(14)
(15)
(16)
in which (17) (18)
Based on the plant of eq. (11) through (18) and moving to the Laplace domain, a Proportional-Integrating (PI) regulator (common choice for industrial process motors) has been designed in order to control the angular speed of the motor wm. (19)
In order to account for the motor dynamics, a first order response has been introduced having the constant d. The block diagram of the feedback control system (having d and as input) is shown in Figure 7
1200
Figure 7: block diagram linearized mechanical system with PI regulator.
The PI regulator have been set up neglecting the influence of the resistant torques due to the rolling process (contained in vector d, see eq. (13)). The block diagram for PI regulator design is thus sown in Figure 8. In figure, N(s) is the transfer function between the driving torque T m1(s) and the angular speed of the motor wm(s): (20)
Figure 8: Block diagram PI regulator.
In order to guarantee a proper bandwidth (wp) and a proper phase margin (fm) for the feedback control system the following conditions must be imposed to the closed loop transfer function L(s) = H(s)/(1+H(s)), (see Figure 8): (21)
By solving the system of 2 equations and 2 unknowns of eq. (21), kp and ki can be evaluated. In particular a bandwidth of 12 [Hz] has been imposed to achieve the desired performances. The bandwidth of the feedback control system is thus higher than the first torsional frequency of the system (2,38 [Hz]). In order to avoid the integral charge phenomena, an anti-windup regulator has been added to the control system (see Figure 9) in view of practical application.
Figure 9: block diagram of PI regulator with anti-windup regulator.
1201 Angular speed 35
30
30
25
25
speed [rad/s]
speed [rad/s]
Angular speed 35
20
15
10
Motor Gearbox Lower spindle Upper spindle
5
0
0
5
10
15 Time [s]
20
25
20
15
10
Motor Gearbox Lower spindle Upper spindle
5
30
Figure 10: Simulation considering PI regulator and windup filter.
0
0
1
2
3
4
5
6
7
Time [s]
Figure 11: Simulation considering PI regulator and windup filter (detail of starting step).
Figure 10 and Figure 11 show the response of linearized system to a step input reference speed having amplitude 30 [rad/s] and applied after 1 second of simulation. A resistant torque of 50 [kNm] is instead applied after 20 seconds to the spindles in order to simulate the milling process. The angular speeds of upper and lower spindles are scaled by the transmission ratio to refer all speed to rotor speed. In Figure 11 is possible to observe the oscillations caused by first eigen-frequency of system during start transitory; the system presents a good time response, but torque oscillations which may produce failures can be seen.
State observer As anticipated, in order to damp the torsional vibrations, an additional contribution to the driving torque T m1 is needed (Tm2 see eq. (9)). Since wu1, w2 are not measured, a state observer, based on Luenberger theory ([2], [4]), has been implemented. In particular, a reduced order state observer has been implemented since only few measurements are usually available on large plants. Moreover reduced order state observer, with respect to full order state observers, do not require the estimate of the full state, thus being simpler and more suitable to be transferred on cheap RT controllers. The observer has been designed based on the plant of eq. (11): (22)
where: (23) (24) (25)
To set-up the observer it’s useful to divide the state vector x into 2 parts: x1 containing the measured variables (in this case wm) and x2 contenting unmeasured variables. (26)
The first equations of (22) can be rewritten as: (27)
1202 The following observer can be designed considering the second equation of the system (29): (28)
being [G] the observer gain matrix and e the error function. Noting that the quantity [A12]x2 is known (being the only unknown of the first equation of the system (27)), the error function can be defined as: (29)
By substituting the first equation of (27) into (28), we get (30)
The observer gain matrix [G] has been defined by using the pole placement method in order to have an observer dynamic faster than the mechanical system one ([5]). The observer of eq. (30) allows to obtain an adequate estimate of the system state only if disturbances are small. In order to increase the observer robustness, the state vector can be augmented by including the torques acting on the spindles. The new state of the system therefore is:
(31)
Assuming that the torques acting on the spindles (which a disturbance for the system) can be modeled as white noise: (32)
The system (22) can be rewritten as: (33)
where:
(34)
1203
(35)
The state observer can now be implemented exactly in the same way as described above. Vectors x1 and x2 in eq. (28) to (30) must be substituted with and and their estimates. Vector contans the motor angular speed, while contains the gearbox angular speed, the spindles angular speed, the shafts elastic torque and the resistant torque applied to the spindles.
RESULTS In order to assess the performances of the proposed control system, it has been integrated into the nonlinear model of the rolling mill. Starting transitories have been simulated with and without activating the Torsional Vibrations Suppression System to highlight the influence of this additional control loop with respect to traditional PI regulators. Figure 12, Figure 13, Figure 14 and Figure 15 show the starting transitory (same as Figure 10 and Figure 11) when only the PI regulator is applied to the rolling mill nonlinear model. From Figure 16 to Figure 19 are reported the same transitory when the Torsional Vibration Suppression System is added. As can be seen, when Torsional Vibration Suppression System is introduced, oscillations of the spindles and of motor shaft are significant reduced and the rise time is maintained. Z model
Z model
60
60
Zm Z2
50
50
40
[rad/s]
[rad/s]
40
30
30
20
20
10
10
0
0
Zm Z2
0
5
10
15 [s]
20
25
30
Figure 12: Simulation considering PI regulator, windup filter using non linear model, motor speed and gearbox input.
0
1
2
3
4 [s]
5
6
7
8
Figure 13: Simulation considering PI regulator, windup filter using non linear model, motor speed and gearbox input (detail of starting step).
Z model
Z model
20
20
Zu2 15
Zu2
10
10
5
5
[rad/s]
[rad/s]
15
0 -5
-5
-10
-10
-15
-15
-20
0
5
10
15 [s]
20
25
30
Figure 14: Simulation considering PI regulator, windup filter using non linear model, upper spindle and lower spindle.
Zu2
0
-20
Zu2
0
1
2
3
4 [s]
5
6
7
8
Figure 15: Simulation considering PI regulator, windup filter using non linear model, upper spindle and lower spindle (detail of starting step).
1204 In Figure 18 and Figure 19 the results the estimates of upper and lower spindles speed (red and black) are also reported. Z model
Z model
60
60
Zm Z2
50
50
40
[rad/s]
[rad/s]
40
30
30
20
20
10
10
0
0
Zm Z2
0
5
10
15 [s]
20
25
30
Figure 16: Simulation considering PI regulator, windup filter and TVSS using non linear model, motor speed and gearbox input.
0
1
2
3
Z model
15
15
10
10 5
Zu1 Zu2
0
[rad/s]
[rad/s]
5
Zoss ,u1 Zoss ,u2
-5
-15
-15
15 [s]
8
20
25
Zu2 Zoss ,u1 Zoss ,u2
-5 -10
10
7
Zu1
0
-10
5
6
Z model 20
0
5
Figure 17: Simulation considering PI regulator, windup filter and TVSS using non linear model, motor speed and gearbox input (detail of starting step).
20
-20
4 [s]
30
Figure 18: Simulation considering PI regulator, windup filter and TVSS using non linear model and estimates of observer, upper spindle and lower spindle.
-20
0
1
2
3
4 [s]
5
6
7
8
Figure 19: Simulation considering PI regulator, windup filter and TVSS using non linear model and estimates of observer, upper spindle and lower spindle (detail of starting step).
CONCLUSIONS A control system aimed at regulating the angular speed of the electrical motor powering a rolling mill is presented in this paper. It is composed by 2 contributions: the first one has the target to obtain the defined angular speed; the second one to reduce the torsional oscillations on spindles. To follow the desired speed profile, a PI regulator defined on the difference between motor angular speed and reference motor speed has been implemented; moreover a torque dependent on the difference between motor and upper spindle angular speed has been added to damp torsional vibrations. Since in large plant the spindles speed is unknown, an observer is needed to estimate the spindles speed. The state observer improved PI regulator is deemed to be within reach of commissioning and service personnel, and therefore immediately applicable to industrial drives. The control system performances has been evaluated with a torsional lumped parameters model of the rolling mill taking into account non linear phenomena like backlash in gears. Simulation results confirmed the theoretical studies and showed a strong improvement of the dynamic performances with an almost total reduction of oscillations.
1205
REFERENCES [1] F. L. Mapelli, M. Matuonto: SUPPRESSION OF TORSIONAL VIBRATIONS IN INDUSTRIAL INVERTER DRIVES USING A STATE OBSERVER, Proc. 7th International Power Electronics & Motion Control Conference Sett.1996 – Budapest [2] D. G. Luenberger: AN INTRODUCTION TO OBSERVERS, IEEE Trans. Automatic Control, Vol. AC-16, NO. 6, DEC 1971; [3] Zhanghai Wang, Dejun Wang: DYNAMIC CHARACTERISTICS OF A ROLLING MILLDRIVE SYSTEM WITH BACKLASH IN ROLLING SLIPPAGE, Journal of materials processing technology, 20 May 1998; [4] Adam Bar, Andrzej Swiatoniowsky: INTERDIPENDENCE BETWEEN THE ROLLING SPEED AND NONLINEAR VIBRATIONS OF THE MILL SYSTEM, Journal of materials processing technology, 2004; [5] Katsuhiko Ogata: MODERN CONTROL ENGINEERING, 5-th edition, 2009; [6] David H. E. Butler, Yoshiharu Ande: COMPENSATION OF A DIGITALLY CONTROLLED STATIC POWER CONVERTER FOR THE DAMPING OF ROLLING MILL TORSIONAL VIBRATION, IEEE Trans. Automatic Control, 1992.
BookID 214574_ChapID FM_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Vibrations Control in Cruise Ships Using Magnetostrictive Actuators Francesco Braghin, Simone Cinquemani, Ferruccio Resta Mechanical Engineering Department, Politecnico di Milano, Campus Bovisa Sud, via La Masa 1, 20156, Milano, Italy
ABSTRACT A big problem in cruise ships is related to noise and vibrations generated by engines and exhaust stacks. Reduction or control of ship noise has traditionally been implemented by passive means, such as by the use of vibration isolation mounts, flexible pipe-work, and interior acoustic absorbing materials. However, these passive noise control techniques are effective mostly for attenuating high-frequency noise, while they are generally ineffective for controlling the low-frequency one. This paper presents an active vibration control of ship bulkheads based on independent modal control technique using magnetostrictive actuators. In the first part of the research, a mock up of the vibrating bulkhead is reproduced in laboratory and a mechanical model of both the system and actuators has been realized. The modal control has then been simulated focusing on actuators and sensors position and number to improve the system’s controllability and observability properties and hence allow to obtain optimal performances in terms of vibration reduction. The influence of boundary conditions has also been taken into account in order to be able to predict the control logic performances in the various possible scenarios. 1. INTRODUCTION There are many sources of noise within a ship structure. Among these are the propulsion systems, exhaust stacks, and various onboard equipment. The principal noise source is the engine system. Figure 1 shows a section of a cruise ship highlighting the engines room, the smokestacks, the chimney and the decks. Generally, there are five possible energy transmission paths associated to vibration diffusion, including the mounting system (consisting of the engine cradle, isolation mounts, raft, and foundation); the exhaust stack; the fuel intake and cooling system; the drive shaft; and the airborne radiation of the engine [1]. Through these paths the energy exchanged causes the propagation of vibrations throughout the ship structure. In general, passive and active control methods can be used to reduce acoustic noise and radiation. Passive noise control essentially reduces unwanted noise by utilizing the absorption property of materials. In this approach, sound absorbent materials are mounted on or around the primary source of noise or along the acoustic paths between the source and the receivers of noise. At low frequencies, however, passive control techniques are not effective because the long acoustic wavelength of the noise requires large volumes of the passive absorbers [2]. Active vibration control involves the use of active systems to reduce the transmission of vibration (e.g., transmission of periodic vibration from the ship engine to its hull). Such an active system is used in practice to complement passive isolation. Actually, the active control of vibrations in ship structures is related mainly to: active vibration isolation (mounting system) [3]; active control of noise in ducts and pipes (exhaust stack; fuel intake and cooling system) [4,5]; active control of vibration propagation in beam-type structures (drive shaft) [6,7] and active control of enclosed sound fields (airborne radiation of the engine) [8,9,10]. The paper fits into this context, deepening the problems relating to control of vibrations on the ship structure, especially in that places where noise is annoying to the crew. In particular the study is focused on vibration control of the bulkheads forming the structure of housings and corridors of a cruise ship.
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_106, © The Society for Experimental Mechanics, Inc. 2011
1207
1208
Chimney
Decks
Smokestacks Engines room
Figure 1 – Cross section of a cruise ship
Figure 2 – Firsts 8 vibration modes of a plate
2. THE SYSTEM 2.1 The Bulkhead Bulkheads subjected to vibratory phenomena are thin structures that extend along two dimensions. For this reason they can be considered as plates. According to a modal approach, typical of mechanics, a vibrating system can be represented, in a certain range of frequencies, by its vibration modes. Following this approach, in figure 2 the first n = 8 modes of vibration of a plate, calculated using the finite element method, are reported. Naturally, such a representation is a simplification of reality, where each system is identified by an infinite number of vibration modes. The equation of motion governing the dynamic of the system stems from the consideration that the response of a mechanical system to a disturbance is the sum of the independent responses of its vibration modes [11]. The dynamics of this system is governed by the equation of motion:
>M @q >R @q >K @q
> @
I T ] D FD I T ] C F C
(1)
where: x q, q , q are (n x 1) vectors containing the modal coordinates of the system (respectively displacements, x
velocities and accelerations); >M @, >R@, >K @ are the matrices of modal mass, modal damping and modal stiffness of the real system (n x n), respectively calculated as:
>M @ x
diag (m1 , m 2 , m3 , m 4 , m5 , m6 , m7 , m8 ) ;
>K @
calculated using finite element technique. FD is a scalar describing the disturbance of the system,
diag (k1 , k 2 , k 3 , k 4 , k5 , k 6 , k 7 , k8 ) ;
>R @
0.01>M @
(2)
1209
x x
FC is a (nC x 1) column vector describing the system control forces exerted by the nC magnetostrictive actuators, I T ] D is a (n x 1) column vector extracted from the modal matrix at coordinate ]D, representing the point of
x
application of the disturbance FD; IT ] is a (n x nC) matrix, obtained putting side by side nC column vectors, each extracted from the modal C
> @
matrix at coordinates ]C representing the point of application of the nC control forces FC; 2.2 The sensors The relationship between physical and modal coordinates is described by: z
>I ] @q T
> @ is a (n x n ) matrix, obtained putting side by side n
where I T ]
M
(3)
M
M
M
column vectors, each extracted from the
modal matrix at coordinates ]M representing the point where the physical coordinates are measured by m sensors.. To calculate the modal coordinates, known m physical coordinates, it is necessary to invert the equation (3). This operation requires the system to be observable, that is the matrix I T ] has rank equal to the number of
> @ M
modes used to describe the system. In other words, a number of m of sensors is needed, each suitably positioned, at least equal to the number of modes n used to represent the physical system. Since this condition is often onerous in terms of number of sensors required, it is possible to reduce the complexity of the problem: in a certain range of frequencies 'f in fact, the real system can be described by a reduced system characterized by nR modes of vibration, where nR
> @ is a (m x n ) ~ ~
where I ]
M
R
> @
matrix, obtained putting side by side nR column vectors, each extracted from the ~
modal matrix of the reduced system at coordinates ]
M
, and q~ is a (nR x 1) vector with the modal coordinates
of the reduced system. To invert equation (4) two cases must be distinguished depending on the number of sensors used: x m = nR, the system is determined q~
x
>I~ ]~ @
1
M
z
(5)
m > nR, the system is over-determined and inversion can be done using pseudoinverse matrix: q~
1 ª ~ˆ ~ º «¬I ] M »¼ z
(6)
Note that, to ensure the observability of the system, it is not possible to reconstruct the modal coordinates of the reduced system with a number of measures m
1210
magnetostrictive actuator supply current (I) with the force transmitted to the structure on which it is bound (FT) is calculated as: G (s)
FT I
C
s2
(7)
s 2 r / m s k / m
The gain C is: C
ndA
(8)
s H GL
where: A, GL are respectively the cross section area and the length of the magnetostrictive bar, n is the number of coils and m is the inertial mass of the actuator, d is the piezomagnetic coefficient and sH is the material compliance. Bar stiffness k can be calculated as: A k H (9) s L while the damping coefficient r is experimentally obtained. Figure 4 shows the transfer function G(s) between the supply current and the transmitted force for the used magnetostrictive actuator (Fig. 5). Permanent Magnets Actuator
Terfenol-D bar
Spring
Coil
Inertial mass
Figure 3 – Scheme of a magnetostrictive actuators
Figure 4 – Transfer function of the magnetostrictive actuator
Figure 5 – The magnetostrictive actuator used
2.3 The controller The term modal control is used to describe a wide variety of control techniques which find their origin in a description of the system through its own modes of vibration. This approach stems from the consideration that the response of a mechanical system to a disturbance is the sum of the independent responses of its vibration modes. This motivates the desire to design a controller that doesn’t alter these mode forms, but allows to change independently the natural frequency and the damping associated [16, 17, 18]. The active control logic is designed to generate a control force that is proportional to the speed of the mode to be controlled and synchronous with it. In this way the damping associated with the controlled mode increases, reducing the amplitude of vibration in the range of frequencies near the resonance. Using an actuator, and then controlling independently just a mode, the term FC can be written as: FC
~ ~ ~ I T ] C r q
(10) where: ~ ~ x I T ] C is a scalar calculated as the inverse of the element extracted from the modal matrix at coordinate ]C limited to the controlled mode, x r is a scalar representing the additional damping introduced by the control force on the controller mode, ~ x q is the modal velocity of the controller mode.
1211
Note that, to guarantee the force control is effectively synchronous with the modal velocity of the controlled mode, the dynamic of the actuator has to maintain the phase of the output. This condition is verified only for frequencies greater than the actuator natural one. Using a modal approach is natural to verify the controllability of the system by observing the lagrangian components of the control with respect to the modal coordinates. That simply means the actuators can’t be placed in a node of the controlled mode of the system. 3. NUMERICAL AND EXPERIMENTAL COMPARISON The system to be controlled is a thin square steel plate (550mm x 550mm x 15mm) hanged by two elastic ropes used to reduce the boundary conditions (Fig. 7). The system is forced by a shaker, connected to the plate by a spring, able to provide the desired excitation, while a current-controlled magnetostrictive actuator is used to control the vibration of the plate. To know the number of sensors needed, numerical simulation are performed In the range of frequencies of interest. Tests are conducted forcing the system with a sweep sine excitation in a range of frequencies between 500Hz and 800Hz. Figure 6 shows the comparison between modal velocities of the reduced system (on the right) and the corresponding modal velocities of the real system (on the left). Since the coordinates for each modes are very similar, the real system can be properly described by a reduced system with nR=3 modes and, therefore, as many sensors are required to calculate the modal coordinates from the available measurements.
Figure 6 – Comparison between modal velocities of the reduced system (on the right) and the corresponding modal velocities of the real system (on the left) The controller is designed to reduce the effects of vibration related to the 8th mode, while it doesn’t modify the remainders. Figure 8 shows the location of excitation, sensors and magnetostrictive actuator on the plate, as function of the shape of the mode to be controlled. Numerical simulations are conducted evaluating the behaviour of the system with and without the control. Estimated modal velocities are depicted in figures 9, 11, 13, in case of uncontrolled (up) and controlled system (down). The modal control effectively acts on the 8th mode, while the others remain unchanged. The damping introduced by the control reduces the amplitude of vibration in the resonance up to 65%. Experimental trials are carried out on the mock-up under the same functioning conditions. Figures 10, 12, 14 shows the modal velocities measured. The control acts only on the 8th mode increasing the damping. Amplitudes are reduced by about 60%, perfectly consistent with the numerical results. When the control is active, the velocity of the 8th mode has its peak of resonance shifted of about 6Hz with respect to the uncontrolled one. As known, this phenomenon can be charged to the sweep sine excitation that moves the resonance frequencies towards greater values, so much greater with the damping system.
1212
M1 FC M3 M2
FD Figure 7 – Position of FD, FC and sensors with respect to the modal shape of the controller mode (8th)
Figure 8 – The mock-up of the bulkhead with shaker, magnetostrictive actuator and accelerometers
Figure 9 – Modal velocity q 6 [m/s 10-3] calculated for the excited system (up) and the controlled one (down)
Figure 10 - Modal velocity q 6 [m/s 10-3 measured for the excited system (up) and the controlled one (down)
Figure 11 - Modal velocity q 7 [m/s 10-3] calculated for the excited system (up) and the controlled one (down)
Figure 12 - Modal velocity q 7 [m/s 10-3] measured for the excited system (up) and the controlled one (down)
1213
Figure 13 - Modal velocity q 8 [m/s 10-3] calculated for the excited system (up) and the controlled one (down)
Figure 14 - Modal velocity q 8 [m/s 10-3] measured for the excited system (up) and the controlled one (down)
The graphs shows that the experimental modal velocities instead of having only one peak at the natural frequency of the mode considered, have also some small peaks at the frequencies associated with other modes. This is quite evident on the 7th modal velocity. This phenomenon is easily explained by observing that the introduction of the magnetostrictive actuator inevitably leads to a change in the modes of vibration of the plate. This change is reflected on the modal matrix that may be different from the one actually used. Consequently, vectors I ] C and I ] D and the matrix I ] are inaccurate. Correct values can be reached performing a new
> @ M
modal identification of the system. 4. CONCLUSIONS Active vibration control of a plate, representing a flexible wall, has been investigated through a modal approach in order to analyze n single dof systems instead of a n-dof system. The control is based on independent modal control technique using magnetostrictive actuators to reduce the amplitude of modal vibrations. Firstly the mechanical models of both the bulkhead and actuator have been realized; the modal control has been simulated focusing on actuators and sensors position and number to improve the system’s controllability and observability properties and to obtain optimal performances in terms of vibration reduction. Secondly, the control technique has been implemented on a mock up of the vibrating bulkhead reproduced in laboratory. The comparison between numerical and experimental results confirm the effectiveness of the control. REFERENCES [1] [2] [3] [4] [5] [6]
U.O. Akpan, “Active noise and vibration control literature survey: Controller technologies”, Defence Research Establishment Atlantic, Dartmouth, (Nova Scotia). 1999 C.R. Fuller, A.H. von Flotow, “Active Control of Sound and Vibration”, IEEE Cont. Sys. Mag. 15(6):9–19 (1995). C.R. Fuller, S.J. Elliott, and P.A. Nelson, “Active Control of Vibration”. Academic Press, New York, 1996. S.C. Douglas and J.A. Olkin, ``Multiple-input, Multiple-output, Multiple-error Adaptive Feedforward Control Using the Filtered-X Normalized LMS Algorithm,'' Proc. Second Conference on Recent Advances in Active Control of Sound and Vibration, Blacksburg, VA, pp. 743-754, April 1993. L'Esperance A.; Bouchard M.; Paillard B.; Guigou C.; Boudreau A., “Active noise control in large circular duct using an error sensors plane”, Applied Acoustic, Vol.57(4), pp. 357-374 X. Pan, C. H. Hansen, “The Effect of Error Sensor Location and Type on the Active Control of Beam Vibration”, Journal of Sound and Vibration, Vol. 165(3), pp. 497-510
1214
[7]
S. J. Elliott, L. Billet, “Adaptive Control of Flexural Waves Propagating in a Beam”, ”, Journal of Sound and Vibration, Vol. 163(2), pp.295-310 [8] T. J. Sutton, S. J. Elliott, A. M. McDonald, "Active control of road noise inside vehicles", Noise Contr. Eng. J., vol. 42(4), pp. 137-147, 1994 [9] S.J. Elliot, P.A. Nelson, I.M. Stothers, C.C. Boucher, “ In-flight experiments on the active control of propeller-induced cabin noise”, Journal of Sound and Vibration, Vol. 140(2), pp. 219-238 [10] I.U. Borchers,U. Emborg, A. Sollo, E.H.Waterman, J. Paillard, P.N. Larsen, G. Venet, P. Goransson, and V. Martin. In Proc. 4th NASA/SAE DLR Aircraft Interior Noise Workshop 1992. [11] J. Inman, “ Active Modal Control for Smart Structures”, Philosophical Transactions: Mathematical, Physical and Engineering Sciences, Vol. 359, No. 1778, Experimental Modal Analysis, (Jan. 15, 2001), pp. 205-219 Published by: The Royal Society [12] S.J. Moon, C.W. Lim, B.H. Kim, Y Park, “Structural Vibration Control Using Linear Magnetostrictive actuators”, Journal of Sound and Vibration, Vol.302, 2007, pp.875-891. [13] T. Zhang, C. Jang, H. Zhang, H. Xu, “Giant Magnetostrictive Actuators for Active Vibration Control”, Smart Material and Structures, Vol.13, 2004, pp.473-477. [14] P.A.Bartlet, S.J. Eaton, J. Gore, W.J. Metheringham, A.G. Jenner, “High-power, low frequency magnetostrictive actuation for anti-vibration application”, Sensors and Actuators A, Vol.91, 2001, pp.133136 [15] A. Lundgren, H. Tiberg, L. Kvarnsj6, A. Bergqvist, “”A Magnetostrictive Electric Generator” JEEE Transaction on magnetics, Vol.29(6), 1993 [16] Y.H. lin, C.L. Chu, “A new Design for Independent Modal Space Control of General Dynamic Systems”, Journal of Sound and Vibration, Vol.180(2), 1995, pp.351-361 [17] S.P. Singh, Harpreet Singh Pruthi, V.P. Agarwal, “Efficient modal control strategies for active control of vibrations”, Journal of Sound and Vibration, Vol. 262, 2003, pp.563-75 [18] S. Hurlebaus, U. Stobener, L. Gaul, “Vibration reduction of curved panels by active modal control”, Computers and Structures, Vol.86, 2008, pp.251–257
BookID 214574_ChapID 107_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
A pedagogical image processing tool to understand structural dynamics Joseph Morlier Université de Toulouse, ISAE DMSM, Campus SUPAERO,10 av. Edouard Belin BP54032 - 31055 Toulouse Cedex 4 – France [email protected] Abstract This paper presents a framework and one pedagogical application of motion tracking algorithms applied to structural dynamics. The aim of this work is to show the ability of high speed camera to study the dynamic characteristics of simple mechanical systems using a marker less and simultaneous Single Input Multiple Output (SIMO) broadband analysis. KLT (Kanade-Lucas-Tomasi) trackers are used as virtual sensors on mechanical systems video. First we introduce the paradigm of virtual sensors in the field of modal analysis using video processing. Then we present a pedagogical example of flexible beam (Fishing rod) video. From KLT tracking we extracted displacements data (virtual sensors) which are then enhanced using filtering and smoothing and then we can identify natural frequency and damping ratio from classical modal analysis. The experimental results (mode shapes) are compared to an analytical flexible beam model showing high correlation but also showing the limitation of linear analysis. The main interest of this paper is that displacements are simply measured using only video at FPS (Frame Per Second) that respects the Nyquist frequency. There is no target needed on the structure only few critical pixels that are good features to track and which become virtual sensors. 1. Introduction In order to achieve the right combination of material properties and service performance the dynamic behavior is one of the main points to be considered. To better understand the dynamic behavior of the structure we need to characterize the resonances of the structure. A common way of doing this is to define its modal parameters i.e. natural frequency, damping ratio and mode shape [1]. The goal of our work is to develop a method to replace classical contact accelerometers based instrumentation with an optical camera working with an intelligent software in order to continuously assess the dynamic parameters of the structure. Previous works [2-5] obtained modal parameters by introducing real targets on the structure or by studying simple structure in ideal conditions. Real time displacement measurement have been done using different approach of digital image processing techniques (texture recognition algorithm) on a flexible bridge [6] or using LED targets (colour filtering) [7]. In our previous work [8] displacement of a bridge under harmonic excitation was reconstructed by using video openCV framework. Such bridges have low natural frequencies within 5 Hz and maximum displacement of several centimetres. In a previous paper [9] we continue in studying the linear dynamic response of a helicopter blade using a broadband excitation. Here the size of the structure is smaller: so the frequencies increase and the experimental setup changes (need of high speed camera to verify the Nyquist criteria) and the high displacement of the fishing rod permit to study more modes. This method can be used in structural dynamics according to three important hypothesis; firstly the number of Frame Per Second (FPS) verifies the Nyquist frequency criteria, secondly the camera axis is perpendicular to the studied 2D structure (to avoid angular errors) and finally the global displacement (in pixels) must be superior to the pixel resolution if not the mode would not appear in the Frequency Response Function (FRF). According to these hypotheses the main drawback of our works is that industrial applications are limited to large structures because they have lower frequencies and larger displacements. But however it remains interesting for educational purposes; it offers a simple measurement tool to understand and visualize structural dynamics engineering problems. From the application point of view, video camera and motion tracking algorithm replaces accelerometers for bending displacement measurements under broadband excitation. One advantage of this non contact measurement method (versus Laser Doppler Vibrometer) is that we can measure in one test several simultaneous outputs for SIMO modal analysis. A way to detect moving objects is by investigating the optical flow which is an approximation of two dimensional flow field from the image intensities. It is computed by extracting a dense velocity field from an image sequence.
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_107, © The Society for Experimental Mechanics, Inc. 2011
1215
BookID 214574_ChapID 107_Proof# 1 - 23/04/2011
1216 The optical flow field in the image is calculated on basis of the two assumptions that the intensity of any object point is constant over time and that nearby points in the image plane move in a similar way [10]. Additionally, the easiest method of finding image displacements with optical flow is the feature-based optical flow approach that finds features (for example, image edges, corners, and other structures well localized in two dimensions) and tracks their displacements from frame to frame. The LK [11] tracker is based upon the principle of optical flow and motion fields [12-14] that allows to recover motion without assuming a model of motion. For practical purposes we use this algorithm on the flexible beam example in order to track the motion of the target pixels and reconstruct displacement signals. It offers various advantages like stable and accurate motion results in non optimal environment. The paper is arranged in three parts, the first introduces the theoretical background of motion tracking and discusses of the optical flow algorithm in the domain of vibration measurement. Then we present the experiment of vibrating fishing rod. Finally the structural characterisation of a flexible beam is studied using virtual sensors data processing. The dynamic parameters are extracted from FRF reconstruction and experimental results are compared with analytical solutions. 2. KLT motion tracking Kanade-Lucas-Tomasi (KLT) features may be used to describe general motions within video images. The KLT algorithm finds several thousand features in each frame of video. It then attempts to find a correspondence between the features in one frame with the features in the next. The origins of the Kanade-Lucas-Tomasi Tracker go back to the work of Lucas and Kanade [11]. They introduced a way to select features that is explicitly based on the tracking equation. Their intention is to select those features that make the tracker work best. They also proposed using an affine model of image motion to monitor feature dissimilarity between the first and the current frame. Most of the time, it is impossible to determine the location of a single pixel in the subsequent frame based only on local information. Due to this, small windows of pixels are used as features. The goal of tracking is to determine the displacement d of a feature window from one frame to the next (Figure 1).
I(x,y)
J(x,y)
Figure 1: Optical flow principle. Pixel motion from image I to image J is estimated solving the pixel correspondence problem: given a pixel in I, look for nearby pixels of the same color in J. Two key assumptions are needed: color constancy (a point in I looks the same in J) and small motion (points do not move very far).
The displacement is chosen as to minimize the dissimilarity between two feature windows, one in image I and one in image J:
ε = ³³ [J(x + d) - I(x)] 2 w(x)dx W
(1) T
T
where W is the given feature window, x = [x, y] are coordinates in the image and d = [dx, dy] is the displacement. The weighting function w(x) is usually set to the constant 1. The aim is to find the displacement d that minimizes the dissimilarity. For this, we differentiate Equation (1) with respect to d and equate it to zero.
∂ε ∂J(x + d) = 2 ³³ [J(x + d) - I(x)] w(x)dx = 0 ∂d ∂d W
(2)
BookID 214574_ChapID 107_Proof# 1 - 23/04/2011
1217 Using the Taylor series expansion of J about x, truncated to the linear term, we obtain:
J(x + d) ≈ J(x) + d x
∂ ∂ J(x) + d y J(x) ∂x ∂y
Putting this into Equation (2) yields
∂ε = 2 ³³ [J(x) - I(x) + g(x)T d] g(x)w(x)dx = 0 ∂d W Where
ª∂ º « ∂x J » g(x) = « ∂ » « J» ¬« ∂y ¼» Rearranging terms yields a linear 2 × 2 system: Zd = e Where Z is the 2 × 2 matrix:
(3)
Z = ³³ g (x)g(x) T w(x)dx and e is e = ³³ [I(x) - J(x)]g(x)w(x)dx W
W
Equation (3) is only approximately satisfied, because of the linearization of Equation (2). However, the correct displacement can be found by minimizing Equation (3) using a Newton-Raphson algorithm. Several interest operators have been proposed based on intuitive ideas of what good features should look like. Shi and Tomasi [15] propose a more principled criterion that is optimal by construction: “A good feature is one that can be tracked well”. Thus our virtual sensors should be located on good feature for correct displacement measurements. When dealing with dynamic systems, it is clear that the temporal sampling frequency (or frame rate) fs must be greater than 2Bt (Equation 4), in order to avoid aliasing in the temporal direction (Nyquist criteria). If global motion is assumed with constant velocities vx and vy (in pixels per standard-speed frame) and spatially band limited image with Bx and By as the horizontal and vertical spatial bandwidths (in cycles per pixel), then the minimum temporal sampling frequency fs (in cycles per speed frame) to avoid motion aliasing is given by
fs = 2Bt = 2Bx × vx + 2By × vy .
(4)
The assumptions of optical ideal conditions and ideal blur filter have been done here. Typical high speed camera uses a state-of-the-art CMOS sensor that records images at 1000 FPS at 1024x768 pixel resolution (or more).
BookID 214574_ChapID 107_Proof# 1 - 23/04/2011
1218 3. Experiment : Vibrating fishing rod In this experiment a pixel represents 2.12 mm using a video at a resolution of 1024x256pixels ( sf = 0.1m / 47 Pixels ), so only the first three bending modes (higher displacements than 2.12 mm) can be measured. After 30 Hz the others modes (displacements inferior to the resolution) induce only noise. Figure 2 presents the cantilever flexible beam experiment. The LK optical flow is used to follow 9 targets (green arrows) in bending along Y displacement. These targets become virtual displacement sensors which allow doing a SIMO analysis in only one test. The main experimental hypothesis is to constrain the flexible beam in Y direction (using blocks). The dynamic behaviour of the fishing rod is more complex (chaotic) as the beam has a tapered geometry, the material is composite and the behaviour remains nonlinear due to large deflections and/or rotations.
1 Y X
9
Figure 2: Cantilever flexible beam example: KLT trackers are used to follow 9 targets in bending (Y displacement). The targets are numbered from 1 to 9.
The main problem occurs in signal reconstruction (displacement). In fact target pixels (which move around x axis) create partial modal data, so displacement signals are irregular data. Thus the small linear displacement hypothesis is used (stability diagram of the Figure 3) to enhance the resolution of the motion and to compensate the missing data,. If the absolute value of x (relative displacement on abscissa) is less than 3 pixels the data is used, otherwise it is not implemented. 2.5
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Figure 3: Using this stability diagram the linearity hypothesis is checked: Each target has does not vary in X direction.
BookID 214574_ChapID 107_Proof# 1 - 23/04/2011
1219 Some pre-processing like windowing (Hanning) and filtering (low pass) have been done. The transfer functions have been estimated using tfestimate in Matlab. Figure 4 illustrates the effect of a running moving average (size of the window is 5) on the temporal signals. All these signal processing tools aim at obtaining smoother FRFs for more precise analysis.
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Figure 4: Effect of the moving average function on the temporal signal. This pre-processing aims at obtaining smooth FRFs.
The type of excitation (pulling down) offers a correct bandwidth at low frequency which allows the identification of the first three modes. The excitation level is the static-force that creates the deflection (1.96N) which induces the free decay. The results of the FRFs reconstruction permit to do a classical modal analysis extracting the three dynamic parameters (frequencies, damping ratios and mode shapes). These parameters which characterize the dynamic behavior of the beam are estimated from FRFs using classical SDOF frequency method called Rational Fraction Polynomial (RFP, [18]) around resonances f i ± 1Hz . Results are listed in Table 1 and can be compared with very good correlation to previous results from classical accelerometers sensors.
E( f )
σ( f )
E( ξ )
σ(ξ )
3.32 (Hz) 9.78 (Hz) 21.69 (Hz)
6E-4 8E-2 3E-2
0.93 (%) 0.96 (%) 0.73 (%)
3E-4 3E-3 3E-3
Table 1: Estimated mean E and standard deviation ı for frequencies f and damping ratios ȟ for three first modes extracted using RFP. The figure 5 shows filtered FRFs (9 measurement points) and the result of the identification of the first resonance at 3.32 Hz for each transfer functions. Thus the FRF correlation between experimental data and identified data is very good for each virtual sensor.
BookID 214574_ChapID 107_Proof# 1 - 23/04/2011
FRF m/N (dB)
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Frequency (Hz) Figure 5: Filtered FRFs form 9 measurement points and identification of the first resonance at 3.32 Hz using SDOF RFP method.
4. Data Analysis: Fitting theoretical mode shapes In order to validate our experiment results our first approach was to use the theory of the tapered beam [19]. The aim was to compare experimental frequency ratio with theoretical to identify the pseudo frequencies β i of the beam. According to the results of table 2, a good correlation of experimental data is found with the tapered beam theory. Taking into account that for uniform and tapered beam the second frequency has almost the same value, it is easy to deduce from these ratios the estimation of β i : β1 = 1.05, β 2 = 1.81, β 3 = 1.60 (Table 2). Frequency ratio
ω2 § β 2 · =¨ ¸ ω1 ¨© β1 ¸¹
2
ω3 § β 3 · =¨ ¸ ω2 ¨© β 2 ¸¹
2
Tapered theory
Experimental
Error
2.61
2.945
12%
1.81
2.21
22%
Table 2: Estimated frequency ratios (tapered theory) compared with experimental results.
We can also notice that higher error occurs for the third parameter identification due to the nonlinear behaviour of this mode. The mode shape describes the structure’s motion when it is vibrating at a particular frequency. Classically, the equation used to represent a uniform (regular section) beam under free vibration can be written as follows:
ρS
∂2 ∂4 v ( x , t ) + EI v ( x, t ) = 0 ∂t 2 ∂x 4
where x is the longitudinal coordinate, v is the transversal displacement of the beam in Y direction (which is perpendicular to X), t is time, E is the Young’s modulus, S is the cross section area, I is the planar moment of inertia of the cross section, L is the length and ρ is the density of the beam. Assuming that bending stiffness is
BookID 214574_ChapID 107_Proof# 1 - 23/04/2011
1221 independent of time and that the steady state vibration has a harmonic form, we get:
d 4Y ( x ) − λ4 Y ( x ) = 0 dx 4 Using separated variables and solving the differential equation, we can express the mode shape as:
Yi ( x ) = A1i cosh( λi x ) + A2 i sinh( λi x ) + A3i cos( λi x ) + A4 i sin( λi x ) th
Where the spatial frequency of the i mode is defined as
λi 2 = ω i
(5)
m with i = 1...n . EI
Using Equation 5 experimental data can be fitted with the analytical equation of the dynamic motion of a beam 2
using least square method with very good correlation ( R 0.98 ) for the first two modes (figure 9). All the mode shapes highlight local singularities due to local change in the beam materials properties (beam having 15 variable sections). It can be noticed that the coefficient estimated from the analytical fit are typical of the Fixed-Free bending mode behaviour (opposite coefficients A1i / A3i and A2i / A4i [20]) which correlate well the shape of the mode (Figures 6).
1
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Figure 6 : First mode shape extracted using experimental data: The global content of this mode is close to the first mode of an uniform beam. Some important singularities appear at the discontinuities of the fishing rod. For the second mode, the experimental data correlate well with the general model of mode shape. This mode is very close from second mode shape of Fixed-Free beam, expected for several zones that highlight the discontinuities of the fishing rod.
BookID 214574_ChapID 107_Proof# 1 - 23/04/2011
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General model: f(x) = a.cos(b.x)+c.sin(b.x)+d.cosh(b.x)+e.sinh(b.x) Mode 1
Mode 2
Coefficients a = -0.232 b = 1.225 c = 0.116 d = 0.231 e = -0.180 a =-0.093 b =1.8 c =-0.099 d = 0.071 e =0.067
R²
RMSE
0.992
0.040
0.986
0.043
Table 3: Estimated parameters of the fit of experimental mode shapes with analytical formula (Eq 5). The R² and RMSE coefficients show very good correlation with the general model.
The motion of each part of the flexible beam has been assumed to be harmonic as explained before. The spatial th th part for each section is given by Eq 6 where indice i is the i mode and j is the j section [21]:
Yij ( x) = A1ij cosh(λij x) + A2ij sinh(λij x) + A3ij cos(λij x) + A4ij sin(λij x)
(6)
If the number of sections is n then the number of unknown parameters in the final solution is 4n. Matching the boundary conditions at each of the n-1 interfaces, as described in Eq. (6), insures continuity between the sections and equilibrium of the total bending moments and internal shear forces (Figure 10).
Figure 7: The boundary conditions are matched at each of the (n-1) interfaces between the elements. For each interface there are 4 equations in order to insure continuity and equilibrium for the shear force and bending moment. At the two extremities, the conditions are the usual ones for a beam with a clamped and a free end.
The poor global third mode shape regression can be explained by the fact that Eq 5 is used instead of Eq 6. Then the main influence is the beam geometry (the fishing rod has 15 section) which should be fitted using a piecewise multiple fit tool to adjust the experimental data with enhance model of stepped beam (Eq 6). In addition normalization of this mode shape (Figure 8) increases the nonlinear behaviour (the real amplitude is very small compared to the first mode).
BookID 214574_ChapID 107_Proof# 1 - 23/04/2011
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1 0.9
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0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
0.5
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1.5
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2
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Figure 8 : Third mode shape extracted using experimental data highlights a complex behaviour: the 15 zones of the stepped beam are describes by red dotted stem. Each part of the beam should be identified with a different parameter (Eq 6).
In the experimental results the spatial sampling is not regular (good features to track are zones that have good contrast i.e. stepped zones of the fishing rod), so the spatial resolution is very poor but easy to interpolate. One of other limitations is that the influence of the camera viewpoint and calibration has not being taken into account in this study. The other disadvantage is that in some applications the small linear displacement hypothesis is not verified. It induces some instabilities of the targets which lead to several partial displacement measurements instead of one fixed (x direction) virtual sensor displacement (in bending; y displacement). Finally the vertical sensitivity will be low due to the size of the deflection compared to the length of the beam. It allows us to identify with only the two first modes with accuracy; the others have complex behavior or displacement inferior or close to the pixel, so that the noise influences more on the mode shape estimation process. Nevertheless the main interest of this method is that no targets need to be placed on the structure and also no time consuming computation is needed (for real time applications) even if several virtual sensors are to be tracked. Conclusion The main goal of this paper is to show a pedagogical image processing tool to understand structural dynamics. OpenCV framework can easily be used for displacemet measurement on a video of a vibrating system according the speed of camera respect the Nyquist criteria. KLT trackers are simply used as virtual sensors to measure displacement on video choosing good features to track on the image. We succeed to estimate the first three main modes of a flexible beam (cantilever composites fishing rod) under broad band excitation. For educational purposes, this simple application can also be used with the help of less expensive tools than high speed camera, e.g. with a classical camera (frequency max is 12.5Hz at image resolution of 1024*768 pixels). Finally it will be interesting to develop a 3D framework with several synchronised camera to continuously monitor an important structure like a bridge. Acknowledgments nd
The author want to thanks Guilhem Michon for his technical help and the 2 year students Paul Sebellin and Emmanuel Godard of ISAE campus-Supaero for being part of this pedagogical research project.
BookID 214574_ChapID 107_Proof# 1 - 23/04/2011
1224 References [1] D. J. Ewins, Model Testing: Theory and Practice, Research Studies Press, 1984. [2] P. Olaszek, Investigation of the dynamic characteristic of bridge structures using a computer vision method. Measurement 25 (1999) 227–36. [3] S. Patsias and W.J. Staszewski, Damage detection using optical measurements and wavelets, Vol 1, Structural Health Monitoring, 1 (2002) 5–22. [4] U.P. Poudel, G. Fu and J. Ye, Structural damage detection using digital video imaging technique and wavelet transformation, Journal of Sound and Vibration 286 (2005) 869–895. [5] J.J. Lee, M. Shinozuka, A vision-based system for remote sensing of bridge displacement, NDT and E International 39 (5) (2006) 425-431. [6] J.J. Lee, M. Shinozuka, Real-time displacement measurement of a flexible bridge using digital image processing techniques, Experimental Mechanics 46 (1) (2006) 105-114. [7] A.M. Wahbeh, J.P. Caffrey, S.F. Masri, A vision-based approach for the direct measurement of displacements in vibrating systems, Smart Materials and Structures, Vol; 12 (5) (2003) 785-794. [8] J. Morlier, P. Salom, F. Bos, New image processing tools for structural dynamic monitoring, Key Engineering Materials Vol. 347 (2007) 239-244. [9] J. Morlier, G. Michon, Virtual vibration measurement using KLT motion tracking algorithm, Journal of Dynamic Systems, Measurement and Control (In Press 2009) [10] J.K. Aggarwal, and N. Nandhakumar, On the computation of motion from sequences of images - A review, in: Proceedings of the IEEE, Vol. 76(8), 1988 , pp. 917-935. [11] B.D. Lucas and T. Kanade, 1981, An iterative image registration technique with an application to stereo vision, in: Proceedings of Imaging understanding workshop, pp 121-130. [12] J.L. Barron, D.J. Fleet, and S.S. Beauchemin, Performance of Optical Flow Techniques, International Journal of Computer Vision, 12 (1994) 43–77. [13] S. Lim, J.G. Apostolopoulos, A.E. Gamal, Optical flow estimation using temporally oversampled video, IEEE Transactions on Image Processing14 (2005) 1074- 1087. [14] J.Y. Bouguet, Pyramidal Implementation of the Lucas Kanade Feature Tracker, openCV documentation. [15] J. Shi and C. Tomasi. Good features to track, In: Proceedings IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 1994, pp. 593-600. [16] Information on http://opencv.willowgarage.com/wiki/ [17] C.C. Chang, Y.F. Ji, Flexible videogrammetric technique for three-dimensional structural vibration measurement, 2007 Journal of Engineering Mechanics 133 (6), pp. 656-664. [18] M. H. Richardson and D. L. Formenti, Parameter Estimation from Frequency Response Measurements using Rational Fraction Polynomials, in: Proceedings of the International Modal Analysis Conference,1982, pp.167-181. [19] H. Wang and W.J. Worley, Tables of natural frequencies and nodes for transverse vibration of tapered beams, NASA CR 443, 1966. [20] S.S. Rao, Mechanical vibrations, Prentice hall, 2003. [21] S. Timoshenko, Vibration Problems in Engineering, Constable & Company, London, 1928.
BookID 214574_ChapID 108_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Modal-Based Camera Correction for Large Pitch Stereo Imaging
Prather Lanier, Graduate Research Assistant Nathan Short, Graduate Research Assistant Dr. Kevin Kochersberger, Research Associate Professor, Mechanical Engineering Dr. Lynn Abbott, Associate Professor, Electrical and Computer Engineering Unmanned System Laboratory Virginia Tech Blacksburg, VA 24061
NOMENCLATURE ܣൌ ݏݏݎܥെ ܵ݁ܿ݉ܽ݁ܤݎ݁ݒ݈݁݅ݐ݊ܽܥ݂ܽ݁ݎܣ݈ܽ݊݅ݐ ܤൌ ݈݁݊݅݁ݏܽܤǡ ݏܽݎ݁݉ܽܥݐ݄ܴ݃݅݀݊ܽݐ݂݁ܮ݊݁݁ݓݐ݁ܤ݁ܿ݊ܽݐݏ݅ܦ ܿ ൌ ݏݐ݊ܽݐݏ݊ܥ݊݅ݐܽݎ݃ݎ݁ݐ݊ܫ݈ܽ݀ܯ ݀ ൌ ݀݅ݕݐ݅ݎܽݏǡ ݁݃ܽ݉݅ݐ݄݃݅ݎ݄݁ݐ݊݅݊݅ݐ݈ܽܿݏݐ݅ݐݐܿ݁ݏ݁ݎ݄ݐ݅ݓ݁݃ܽ݉݅ݐ݂݈݈݁݊݅݁ݔ݂݅ݐ݁ݏ݂݂ ܧൌ ݏݑ݈ݑ݀ܯܿ݅ݐݏ݈ܽܧ ൌ
ܫൌ ݏݏݎܥെ ܵ݁ܿܽ݅ݐݎ݈݁݊݅ܽ݊݅ݐ ݈ ൌ ܾ݂݄݉ܽ݁ݐ݃݊݁ܮ ݉ ൌ ܽݎ݁݉ܽܥ݊݅ݏ݅ݒ݁ݎ݁ݐ݂ܵݏݏܽܯ ܻ ൌ ݊݅ݐ݈݂ܿ݁݁ܦ݉ܽ݁ܤ݈ܽܿ݅ݐݕ݈ܽ݊ܣ ܻௗ௦ ൌ ݊݅ݐ݈݂ܿ݁݁ܦ݉ܽ݁ܤ݈ܽݑݐܿܣ ܻ ሺݔሻ ൌ ݄݁ܽݏ݁݀ܯܾ݉ܽ݁ݎ݁ݒ݈݁݅ݐ݊ܽܥ ܼ ൌ ܽݎ݁݉ܽܥݐݐ݆ܾܱܿ݁݉ݎܨ݁ܿ݊ܽݐݏ݅ܦ ܻሷ ൌ ݏ݁ݑ݈ܸܽ݊݅ݐܽݎ݈݁݁ܿܿܣݎ݁ݐ݁݉ݎ݈݁݁ܿܿܣ ݔ ൌ ܽݎ݁݉ܽܥݐ݂݁ܮ݊݅ݎ݁ݐ݊݁ܥܽݎ݁݉ܽܥݐ݈݁ݔ݅ܲ݉ݎ݂݁ܿ݊ܽݐݏ݅ܦ ݔ ൌ ܽݎ݁݉ܽܥݐ݄ܴ݃݅݊݅ݎ݁ݐ݊݁ܥܽݎ݁݉ܽܥݐ݈݁ݔ݅ܲ݉ݎ݂݁ܿ݊ܽݐݏ݅ܦ ߚ݈ ൌ ܹ݄݁݅݃ݕܿ݊݁ݑݍ݁ݎܨ݉ܽ݁ܤ݀݁ݐ ߩ ൌ ݕݐ݅ݏ݊݁ܦ݉ܽ݁ܤ ȣ୶ ǡ ȣ୷ ǡ ȣ ൌ ܴ݈݁݊ܽܲ݁݃ܽ݉ܫݎ݂ݏݎݐܸܿ݁݊݅ݐܽݐ π ൌ ܰݎݐܸܿ݁݊݅ݐܽݐܴ݀݁ݖ݈݅ܽ݉ݎ ߱ ൌ ݉ܽ݁ܤݎ݁ݒ݈݁݅ݐ݊ܽܥ݂ݕܿ݊݁ݑݍ݁ݎܨ݈ܽ݀ܯ
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_108, © The Society for Experimental Mechanics, Inc. 2011
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BookID 214574_ChapID 108_Proof# 1 - 23/04/2011
1226 ABSTRACT Stereo imaging is typically performed using two cameras that have been calibrated to account for lens-induced distortion and pointing errors, resulting in rectified images that are processed to obtain distance information. The accuracy of a 3-D map obtained from stereopsis is closely tied to the calibration data, and so relative motion between the cameras must be kept small. In order to reduce errors from a stereo image caused by motion, the structural connection between the cameras can be stiffened, but this comes with a weight and size penalty. For cameras that have a large baseline (pitch) distance, it may be impossible to have enough stiffness in the structure to obtain reasonable error bounds. An alternative approach is to model the camera motion using a modal technique and account for this motion during imaging. This paper outlines a procedure for stereo camera correction using measured accelerations to optimally trigger the camera. Results of the technique are shown for a simple beam that is center-mounted to a shaker to induce symmetric bending.
INTRODUCTION Stereopsis is the technique of using two images, obtained from neighboring cameras that are triggered simultaneously, to generate a three-dimensional point cloud representing the physical boundaries of objects in the images. Stereo vision is experienced first-hand by an observer who can judge distances simply by looking at an object; a more accurate estimate of distance is obtained for closer objects since the disparity between the left and right images is increased. Likewise, stereo cameras function on the basis of determining a correlation between the left and right images, and knowing the separation between the cameras, the distance to objects in the scene can be determined. In a stereo vision system, the cameras must be calibrated relative to each other so that when the left and right images are captured, there is a known function relating the shift (disparity) of the scenes to the distance from the cameras. This calibration cannot change during imaging. If the cameras move relative to each other after the initial calibration, then the distance calculation will be in error. Most stereo vision systems are built with a relatively small pitch (distance between the cameras) so that the likelihood of relative camera motion is reduced. Commercially available systems are rarely found with a pitch exceeding 0.6 m, which limits the range of the system. A larger pitch system may be desirable if stereo imaging is to be performed from a greater distance, such as imaging from an aircraft. Again, the problem with larger systems is maintaining relative camera position since the resolution of distance measurements is dependent on camera orientation accuracy. A novel solution to the flexible camera problem is found in using a modal analysis to compensate for relative camera motion. Consider a stereo camera pair mounted on a flexible beam, as shown in Figure 1. Depending on the system configuration and the nature of disturbances to the system, the camera motion can be accurately described in the modal domain. Using analytical or numerical techniques, rotations of the cameras can be extracted from measured accelerations at the cameras, described in the frequency domain. Once the camera rotations are determined, a transformation can be applied to correct the camera images back to the orientation used in the original calibration, providing an accurate stereo pair for processing.
STEREO VISION THEORY A stereo vision system is a set of two or more cameras that are positioned to capture images of the same 3-D scene from different points of view. This camera placement causes slight differences in appearance in the images, and these differences (known as stereo disparities) can be used to extract 3-D information from the 2-D projections. A common imaging arrangement is
BookID 214574_ChapID 108_Proof# 1 - 23/04/2011
1227 motivated by biological vision, w with cameras that are horizontally aligned and separa ated by a small pitch (baseline) distance. Figure 1 shows an example stereo system with a 10 inch pitch.
Figure 1: Stereo vision syste em with 10 inch pitch. The two lenses are visible near the ends of the metal bar. The cameras s and cables are on the other side of the bar. Stereo ranging is illustrated with h the simple arrangement that is shown in Figure 2. In this ideal system, the optical axes of the ttwo cameras are perfectly parallel, both image plane es are coplanar, and no lens distortion is present. Scene point P projects onto both image planes, and we would like to recover its 3-D coordinates. In this case, the distance Z (also called d the range or depth) can be found using the fo ollowing equation:
Z=
Bf d
(1)
In this equation, f is the focal le ength, B is the baseline, and d is the disparity betwee en two corresponding points, which is given by d = xl – xr. [1] (In the figure, the value for xr is negative because it is on the left of the optical axis.) Although stereo ranging is simple in prin nciple, the identification of point correspond dences between the two images is a difficult problem m.
Figure 2: Simple geometry fo or stereo ranging. The usual goal is to find the range Z from the cameras to a point P in the scene. [2] In an actual stereo system, the optical axes are not perfectly parallel. A calibration procedure is used to determine the relative 3 3-D orientation between the two cameras, as well as intrinsic parameters such as the focal lengths and image center locations. With this informattion it is
BookID 214574_ChapID 108_Proof# 1 - 23/04/2011
1228 possible to rectify the two images with 2-D transformations that cause all corresponding points to be aligned horizontally. Rectification causes the images to resemble the ideal case shown in Figure 2, and it increases the efficiency of the search for corresponding points. A rotation matrix can describe the orientation of the left camera with respect to the right, using the camera-centered coordinate system shown in Figure 3. The origin is located at the point of projection, and the z axis coincides with the optical axis. The x and y axes are parallel to the image plane. Rotation about the x, y, and z axes will be represented as ߆௫ R (pitch), ߆௬ R(yaw), and ߆௭ (roll) of the camera, respectively.
y
x
z Figure 3: Camera-centered coordinate system. This shows a Watec 660D G3.8 mono board camera, which is the model used for this project. The composite rotation vector RV for one camera can be expressed as the row vector ܴܸ ൌ ሾ߆௫ ǡ ߆௬ ǡ ߆௭ ሿ
(2)
If RV represents a fixed axis through the origin, then the angle of rotation about this axis is given by the vector norm ߠ ൌ ԡܴܸԡ (3) and the corresponding unit vector is π ൌ
ܴܸ ߠ
(4)
with components π ൌ ൣπ௫ ǡ π௬ǡ π௭ ൧. If we now define the antisymmetric matrix Ͳ π π௩ ൌ ௭ െπ௬
െπ௭ Ͳ π௫
π௬ െπ௫ Ͳ
(5)
then the rotation matrix is given by ܴ ൌ ܫ π௩ ߠ π௩ π௩ ሺͳ െ
ߠሻ
(6)
where I is the 3×3 identity matrix. This is known as Rodrigues' rotation formula [3], and it can be rewritten as
ߠ π௫ ଶ ሺͳ െ
ߠሻ ܴ ൌ ൦ π௭ ߠ π௫ π௬ ሺͳ െ
ߠሻ െπ௬ ߠ π௫ π௭ ሺͳ െ
ߠሻ
െπ௭ ߠ π௫ π௬ ሺͳ െ
ߠሻ
π௬ ߠ π௫ π௭ ሺͳ െ
ߠሻ
ଶ
ߠ π௬ ሺͳ െ
ߠሻ
െπ௫ ߠ π௬ π௭ ሺͳ െ
ߠሻ൪
π௫ ߠ π௬ π௭ ሺͳ െ
ߠሻ
ߠ π௭ ଶ ሺͳ െ
ߠሻ
(7)
BookID 214574_ChapID 108_Proof# 1 - 23/04/2011
1229 For both cameras, the combined rotation matrix is ܴ ൌ ܴ ܴ ்
(8)
where Rr and Rl are the rotation matrices for the right and left cameras, respectively. In a typical stereo ranging system, it is assumed that the cameras will remain stationary relative to one another. After completing the calibration procedure, any change in relative pose between the two cameras will affect the stereo disparity values of imaged points, and will therefore reduce the accuracy of range estimates. If the system is subjected to vibrations that can affect the relative camera orientations, however, then it is expected that accuracy will suffer. Vibration can also introduce motion blur into the images, which affects the ability to localize feature points in the images and can further reduce ranging accuracy. This paper considers situations in which two cameras are separated by a relatively large baseline distance, using a structural support that can flex when the system is subjected to vibration. The primary concern is rotation by the cameras in opposite directions about their respective y axes. Vibration will tend to cause the cameras to converge (rotate toward each other) and diverge (rotate away from each other) repeatedly. At maximum deflection during convergence, the disparity values for corresponding points will be smaller than for the calibrated (stationary) system. Based on the analysis given above, the estimated range values would therefore be smaller than the true values. Conversely, at maximum deflection during divergence, the disparity values will be larger, and the estimated range values would be larger than the calibrated values. Ironically, maximum deflection corresponds to minimum camera motion, which is the best time to capture images in an effort to reduce the effects of motion blur. Accelerometers can be attached to the cameras for estimating deflection angles, and therefore to correct for small rotational changes of the cameras. Assuming periodic motion, accelerometers can also be used to predict the instants of maximum deflection, which is when image actuation should be triggered.
MODAL MODELING OF A CENTRALLY-SUPPORTED BEAM In order to develop a model by which we can approximate the deflection and rotation of our cameras, it is necessary to make several assumptions. The first of these is to assume that the cameras are centrally supported. For example, our cameras could be placed along the wings of an airplane with the fuselage acting as a mounting point. Next, since the cameras are centrally supported, we can assume that this central support creates a fixed end condition on either side with the beam on which the cameras are attached. The beam, or hypothetical wing, behaves as an Euler-Bernoulli beam with tip mass. Additionally, the cameras will be mounted on the elastic axis of the beam to eliminate torsional moments from transverse deflections. Under these constraints, the system will behave as cantilever beam in transverse vibration. The governing equation of vibration for the Euler-Bernoulli beam is, [4]: ߲ ସ ܻሺݔǡ ݐሻ ܫܧ ߲ ଶ ܻሺݔǡ ݐሻ ଶ ܿ ቆ ቇ ൌ Ͳܿ ൌ ඨ ߲ ݐଶ ߲ ݔସ ߩܣ
(9)
The solution of the spatial equation has the form of ݔߪ݁ܣ, and the general solution of the spatial equation is: ܻ ሺݔሻ ൌ ܿଵ ሺߚ ݔሻ ܿଶ
ሺߚ ݔሻ ܿଷ ሺߚ ݔሻ ܿସ
ሺߚ ݔሻ
(10)
BookID 214574_ChapID 108_Proof# 1 - 23/04/2011
1230 Where ܿ values are the modal constants that will be determined by the boundary conditions of a cantilever beam with a tip mass, which are: ݏ݊݅ݐ݅݀݊ܿݕݎܽ݀݊ܤ݀݊ܧ݀݁ݔ݅ܨሺ ݔൌ Ͳሻ ܻ݀ ܻ ൌ Ͳ ൌͲ ݀ݔ
(11)
ݏ݊݅ݐ݅݀݊ܿݕݎܽ݀݊ܤ݀݊ܧ݁݁ݎܨሺ ݔൌ ݈ሻ ߲ ଶܻ ߲ ߲ଶܻ ܫܧቆ ଶ ቇ ൌ Ͳ ቆ ܫܧଶ ቇ ൌ െ݉߱ሷ ሺݔǡ ݐሻ ߲ݔ ߲ݔ ߲ݔ
(12)
These boundary conditions assume that the tip mass (camera) has a very small rotary inertia compared to the beam. Differentiating equation 17 and applying the appropriate boundary conditions results in: Ͳ ͳ െ ሺߚ݈ሻ ଷ െ Ⱦ
ሺߚ݈ሻ ݉߱ଶ ሺߚ݈ሻ
൦
ͳ Ͳ െ
ሺߚ݈ሻ ଷ Ⱦ ሺߚ݈ሻ ݉߱ଶ
ሺߚ݈ሻ
ܿଵ Ͳ ͳ Ͳ ܿଶ ͳ Ͳ Ͳ ൪ ൦ ൪ ൌ ൦ ൪ ܿଷ ሺߚ݈ሻ
ሺߚ݈ሻ Ͳ ଷ ଶ ଷ ଶ Ⱦ
ሺߚ݈ሻ ݉߱ ሺߚ݈ሻ Ⱦ ሺߚ݈ሻ ݉߱
ሺߚ݈ሻ ܿସ Ͳ
(13)
For a non-zero solution, the leading matrix determinant is set to zero which provides the roots of the system. These roots can be used to determine the natural frequencies of our test camera boom. Using this approach, the fundamental bending frequencies of the system are found. The results are listed below. Table 1: Physical properties of beam. Beam Properties Young's Modulus,(E) Ͳ Ͳͳ כଽ ܽܲܩ ʹͲͲ݇݃Ȁ݉ଷ Density,(ȡ) ͶǤʹʹͷ ିͲͳ כଵଶ ݉ସ Inertia,(I) Cross sectional Area,(A) ͷǤͲ ିͲͳ כହ ݉ଶ Ǥ Ͳ͵ͷ݃ Mass of Tip Mass(m) Length of Beam (݈ ) Ǥ ͳͳͳͷ݉
Table 2: Natural frequencies of first three modes of cantilever beam system.
Mode
ࢼ Values
Frequency (Rad/s)
Frequency (Hz)
1
1.047
126.305
20.1028
2
3.974
1865.8
296.955
3
7.098
5952.3
947.34
After obtaining the natural frequencies of our stereo boom, the modes of vibration are determined by substituting the ࢼ values. Figure 3 shows the first mode shape (displacement) along with the derivative of this shape to obtain the modal slopes. It will be the slope plot that is used to correct the camera pose so that correct distances are obtained from the stereo image pair.
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1231
Normalized First Mode Shape of Cantilever Beam
Displacement(meters)
0
-0.5
-1
-1.5
-2
0
0.1
0.2
0.3
0.4 0.5 0.6 Length of Beam (m)
0.7
0.8
0.9
1
0.9
1
Normalized Slope Plot of the First Mode Shape of Cantilever Beam
Slope of Beam (radians)
0
-10
-20
-30
-40
0
0.1
0.2
0.3
0.4 0.5 0.6 Length of Beam (m)
0.7
0.8
Figure 3: Normalized First Mode Shape of Stereo boom (Top). Normalized Slope along the length of stereo boom (Bottom).
RESULTS FOR A STEREO VISION SYSTEM SUBJECT TO HARMONIC VIBRATION The stereo system was built using two Watec 660D cameras, as shown in Figure 4 and described in Table 3, networked with an Axis 241Q Video Server, shown in Figure 4 and described in Table 4. The cameras were mounted on a beam at a baseline of 10 inches, which was placed on a 50 lb. shaker. The cameras were connected to the video server through two coax video cables and powered by 9 volts through a bench top power supply.
Figure 4: Stereo System with two Watec 660D cameras and Axis 241Q Video Server.
BookID 214574_ChapID 108_Proof# 1 - 23/04/2011
1232 Table 3: Camera Specifications for Left and Right Cameras. [5] Watec 660D G3.8 mono board Sensor Type
¼’’ interline transfer CCD
Unit Cell Size
7.15 μm x 5.55 μm
(physical pixel size on sensor) Resolution
704 x 480
Minimum illumination
0.06lx.
Focal Length
3.8mm
Table 4: Video Encoder Specifications for stereo system. [6] AXIS 241Q Video Server Connection Type
Network Cameras
Resolution
Transmits simultaneous streams up to 704 x 576
Video Frame Rate
Up to 30 frames per second
Number of Channels
4
Video Format
Motion JPEG and MPEG-4
The data acquisition system was a Spectral Dynamics model 20-24 digital signal processing (DSP) unit running the SigLab analysis software. Figure 5 shows the test components.
Figure 5: Shaker is a Vibration System Model VG10054. Accel is a PCB 35C65 and the analyzer is a Spectral Dynamics 20-24.
The 3-Dimensional scene that was analyzed was a set of objects placed on the ceiling above the shaker, on which the stereo system was mounted. This scene is shown in Figure 6, newspaper was used as a backdrop to provide a textured background for correlation purposes in the stereo matching routine.
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1233
Figure 6: View of scene used for measuring distances with stereo system. Using the SigLab virtual network analyzer function, a continuous sinusoidal input signal was used to excite the stereo vision system at its first fundamental frequency of 20 Hz. Acceleration data was then sampled at a rate of 512 Hz at 20 second intervals. These acceleration values can then be converted into beam displacement using the following relationship ܻௗ௦ ൌ
ܻሷ ߱ଶ
(14)
where ܻሷ is the acceleration seen by the accelerometer and ࣓ is the frequency in radians per second. Figure 7 shows the results of one such sample run. Acceleration Data (Seconds)
Acceleration (m/s)
30
X: 0.3809 Y: 14.19
20 10 0 -10 -20 -30 0
0.1
0.2
0.3
-3
Displacement(meters)
3
0.4
0.5 Time
0.6
0.7
0.8
0.9
1
0.7
0.8
0.9
1
Accelerometer Displacement
x 10
2 1 0 -1 X: 0.3809 Y: -0.0009956
-2 -3
0
0.1
0.2
0.3
0.4 0.5 0.6 Time (Seconds)
Figure 7: Accelerometer Data in m/s (Top). Accelerometer Displacement (Bottom).
BookID 214574_ChapID 108_Proof# 1 - 23/04/2011
1234 Examination of the figure above shows a maximum displacement of about .995 mm at the location of our accelerometer. Figure 8 displays a plot of our calculated mode shape which can be used to determine camera displacement/rotation based on the locations of the accelerometer and the camera. The accelerometer is located at a distance of 87 mm from the fixed end of our stereo vision boom and the camera is located at 97 mm. Using the accelerometer values and the scaled mode shape, the camera displacement can be obtained. -3
0
Normalized First Mode Shape of Cantilever Beam
x 10
-0.2
-0.4
Displacement(meters)
-0.6
-0.8 X: 0.087 Y: -0.00104
-1
-1.2
-1.4
-1.6
-1.8
0
0.02
0.04
0.06 Length of Beam (m)
0.08
0.1
0.12
Figure 8: Scaled mode shape plot based on measured acceleration at 87 mm. డ
st
Additionally, the 1 mode beam slope, , can be extracted from the mode shape and used to డ௫ obtain camera rotation angles, and will be used for the camera corrections. In the experimental case, the measured accelerations were used to determine a peak camera rotation angle of.0265 radians, shown in Figure 9. Normalized Slope Plot of the First Mode Shape of Cantilever Beam 0
-0.005
Slope of Beam (radians)
-0.01
-0.015
-0.02
X: 0.097 Y: -0.02675
-0.025
-0.03
-0.035
0
0.02
0.04
0.06 Length of Beam (m)
0.08
0.1
0.12
Figure 9: Scaled beam slope plot based on distance to center of camera lens at 97 mm.
BookID 214574_ChapID 108_Proof# 1 - 23/04/2011
1235 Before the experiment was conducted, a set of images were taken while the cameras were motionless in order to compare the corrected results to the ideal results from the system. The disparity map for this scenario is shown in Figure 10. This figure shows the relative distance of the objects from the camera. Closer objects are represented as lighter intensities and objects farther away are shown as darker intensities. Table 5 shows the results of distance calculations before (top two rows) the shaker is turned on, and after (bottom two rows) the shaker is turned on where images are obtained at the peak acceleration (peak displacement and rotation). Note that distance errors of 19% represent the maximum error in the oscillatory cycle for an uncorrected set of images, but they also represent the highest quality images because the cameras are momentarily motionless at the peaks of the cycle. Table 5: Results from stereo matching: static (top) and dynamic (bottom). Measurements are distances to the real world points Pixel
Measured
Actual
% Error
{263, 345} White
1344.382 mm
1371.6 mm
1.98%
{222, 367} Black
1163.814 mm
1168.4 mm
0.39%
{274, 350} White
1109.814 mm
1371.6 mm
19.08%
{220, 341} Black
954.366 mm
1168.4 mm
18.31%
Figure 10: Disparity map from correlation without vibration (left) and with vibration (right). By placing accelerometers on or near the cameras, measured vibration can be used to calculate the deflection angle of the cameras which in turn is used to correct the images using the Rodrigues rotation.
RESULTS OF CAMERA CORRECTIONS In the experiment, the camera angle at peak motion was calculated to be 0.0261 rad., resulting in a șy of 0.0522 rad. which is used in Eqn. 11 to obtain the corrected camera pose. The disparity map in Figure 11 and the calculated error in Table 6 show the improvements in calculating depth using the angle calculated from the accelerometer data.
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1236 Table 6: Error in distance measurement using camera corrections. Pixel
Measured
Actual
% Error
{265, 334} White
1309.125 mm
1371.6 mm
4.55 %
{230, 322} Black
1093.952 mm
1168.4 mm
6.37 %
Figure 11: Disparity map using camera corrections. Comparing the results of Table 6 to Table 5, we see a significant improvement in distance measurements when the angle corrections are applied to the stereo rectification. The error in measured distance dropped 14.5% for the black marker and 11.94% for the white marker, providing an average reduction in error of 70.68%.
MODAL CORRECTIONS FOR SYSTEMS SUBJECT TO RANDOM NOISE The flexible beam stereo vision system used in this experiment represents a system with inherent flexibility which must be characterized to obtain accurate stereo images. For this experiment, the beam and camera system represent a system with very small distributed mass along the beam and a large concentrated mass and inertia at the end of the beam. One of the reasons for this set-up was to create a low-frequency first mode for a small-pitch camera system; it would normally be considered bad practice to build a stereo vision system with such a flimsy backbone structure. For this experiment however, the system provided the low frequencies necessary for the data acquisition system to sync the camera images with the measured accelerations. In practice, a properly designed system would have significant mass in the beam to keep the natural frequency high and the deflection low. One such example is in aircraft, where cameras mounted on the wingtips represent a wide baseline imaging system subject to dynamics that will require corrections for accurate stereo imaging. In this case, a modal analysis of the vehicle would be experimentally performed to determine how translational acceleration correlates to camera rotation, so camera corrections can be performed using accelerometer measurements. For slender beam systems such as aircraft wings, gust conditions are expected to excite predominately the first mode wing bending. Lee and Lee [7], and Eslimy-Isfahany and Banerjee [8] have shown that second wing bending mode frequencies for aircraft are typically five times higher than the first mode frequency, and Balakrishnan [9] has shown that turbulence energy for
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1237 subsonic flight is concentrated at very low frequencies. Bennett and Yntema [10] derived a method of calculating wing excitation from turbulence that only considers plunging rigid body motion and the first wing bending mode, reinforcing the concept that only the first flexible bending mode is typically excited in turbulence. A simple demonstration was performed to show how the first wing bending mode is expected to dominate the flexible response of an aircraft in turbulence. Using the current set-up, random noise from 0 – 200 Hz was input to the shaker, and the response at the beam tip was measured for 100 averages. Figure 12 shows the random noise input (top) and the beam response (bottom), and Figure 13 shows the results in the frequency domain with a clear indication that the first mode is dominant. In practice, camera trigger would occur at the peak response, and first mode bending behavior would be used to correct the camera pose. Plot of Input Signal to Shaker
Transducer Data(Volts)
0.06 0.04 0.02 0 -0.02 -0.04 -0.06
0
0.1
0.2
0.3
0.4 0.5 0.6 Time (Seconds)
0.7
0.8
0.9
1
0.8
0.9
1
Plot of Raw Accelerometer Data Accelerometer Data(Volts)
0.2
0.1
0
-0.1
-0.2
0
0.1
0.2
0.3
0.4 0.5 0.6 Time (Seconds)
0.7
Figure 12: Input to Shaker (Top).Raw Data from Accelerometer (Bottom). Coherence Plot 1
Coherence
0.8 0.6 0.4 0.2 0
0
10
20
30
40 50 60 Frequency (Hz)
70
80
90
100
80
90
100
Siglab Transfer Function(Volts/Volts)
2
10
1
Amplitude
10
0
10
-1
10
-2
10
0
10
20
30
40 50 60 Frequency (Hz)
70
Figure 13: Coherence Plot (Top). Transfer Function (Bottom).
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1238 CONCLUSIONS This paper has described a novel approach to stereo image acquisition in the presence of vibration. We found that modeling the stereo vision boom as a cantilever beam allowed for a reasonable prediction of the deflection angle of a stereo vision camera. When used in the rectification of stereo images collected during steady-state vibration, the predicted deflection angle reduced the error in distance measurements by more than 70% when compared to their non-corrected counterparts. This represents a significant improvement in stereo ranging accuracy, and has potential for broad application in the field of unmanned systems.
ACKNOWLEDGEMENTS The authors wish to acknowledge the Defense Threat Reduction Agency for supporting this work.
REFERENCES 1) Shapiro, L., and Stockman, G. (2001). Computer Vision. Upper Saddle River: Prentice Hall. 2) Bradski, G., and Adrian, K. (2008). Learning OpenCV: Computer Vision with the OpenCV Library. Sebastopol: O'Reilly Media Inc.
ϯͿ Wolfram Research, Inc. http://mathworld.wolfram.com/RodriguesRotationFormula.html (1999-2009). 4) Inman, Daniel J. 2001. Engineering Vibration. Upper Saddle River, New Jersey: PrenticeHall,Inc. 5) Watec. (2008). Watec Cameras. Retrieved September 28, 2009, from http://www.wateccameras.com 6) Axis Communications. (2009). Axis Communications. Retrieved September 28, 2009, from http://www.axis.com/products/cam_241q/index.htm. 7) Lee, I. and Lee, J. J., “Vibration Analysis of Composite Wing with Tip Mass Using Finite Elements,” Computers and Structures, Vol. 47, No. 3, p. 495 – 504, 1993. 8) Eslimy-Isfahany, S. H. R., and Banerjee, J. R., “Dynamic Response of Composite Beams with Application to Aircraft Wings,” Journal of Aircraft, Vol. 34, No. 6, p.785 – 791, Nov – Dec 1997. 9) Balakrishnan, A. V., “Modeling Response of Flexible High-Aspect-Ratio Wings to Wind Turbulence,” Journal of Aerospace Engineering, Vol. 19, No. 2, p. 121 – 132, 2006. 10) Bennett,F. V., and Yntema, R. T., “The Evaluation of Several Approximate Methods for Calculating Symmetrical Bending-Moment Response of Flexible Airplanes to Isotropic Atmospheric Turbulence,” NASA TN 2-18-59L, March 1959.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Structural Damage Identification Based on Multi-objective Optimization
Sungmoon Jung Assistant Professor, Dept. of Civil and Environmental Eng., FAMU-FSU College of Engineering, 2525 Pottsdamer Street, Tallahassee, FL 32310, USA Seung-Yong Ok BK Research Professor, Safe and Sustainable Infrastructure Research Group, Seoul National University, 599 Gwanak-ro, Gwanak-gu, Seoul 151-744, Korea Junho Song Assistant Professor, Dept. of Civil and Environmental Eng., University of Illinois at UrbanaChampaign, 205 North Mathews Avenue, Urbana, IL 61801, USA
NOMENCLATURE
f F nl nm N S Uc Um w
μ σ
= an objective function = multiple objective functions = number of load cases = number of measurements within a load case = number of Pareto-optimal solutions = condition of the structure (the solution from the optimization) = computed response = measured response = weighting factor = sample mean = unbiased sample standard deviation
ABSTRACT Structural damage identification is an inverse problem, which often is formulated as an optimization problem. The design variables are degrees of damage in computational model, and the objective is the discrepancies between the computed responses and the measured responses. Conventional single-objective optimization approach defines the objective function by combining multiple error terms into a single one, which leads to a weaker constraint in solving the identification problem. An alternative approach explained in this paper is a multi-objective approach that simultaneously minimizes multiple error terms. A stronger constraint from multiple objectives promotes the solutions to converge to the correct solution. Numerical examples based on static testing are provided to illustrate the multi-objective approach. Also, conceptual explanations are given to extend the approach to include both the static testing and the dynamic testing. Expected challenges will also be explained.
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_109, © The Society for Experimental Mechanics, Inc. 2011
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1240 INTRODUCTION Structural damage identification enables us to quantify the damage occurring in civil, mechanical, and aerospace structures. Extensive study has been conducted for this topic, primarily based on vibration signals (see [1]). However, due to the nature of civil structures, static load testing is also commonly performed. Static testing may be a simpler alternative than dynamic testing, or it may complement the dynamic testing effectively. Many algorithms have been proposed for the damage identification of bridge-like structures using static loading [2-6]. In this paper, we will first summarize previous findings on improving the robustness of the detection against the sensor noise, using static formulations and examples that used static tests. We employed multi-objective approach to provide stronger constraint to the problem that enabled lower false-negative detection rate [7]. Next, we will explain conceptual framework to integrate static testing and dynamic testing, to further improve robustness of the detection against the sensor noise. Challenges in integrating many tests will be also explained.
STRUCTURAL DAMAGE IDENTIFICATION USING OPTIMIZATION FORMULATION: STATIC LOADING Structural damage identification is often formulated as an optimization problem. The design variables of the optimization are the degrees of the damage in various members of the structure. For example, effective crosssectional areas of a truss bridge correspond to the design variables. The objective is to minimize the difference between the response predicted by a computer model and the response measured from an experiment. If the difference is minimized, it is concluded that the solution has been found, i.e. the degree of the damage in the computer model is assumed to be a good approximation of the damage in the real structure. Formally stated, nl
min F (S) = min ¦ wi f i (S)
(1)
i =1
§ U ijc − U ijm · ¸ f i (S) = ¦ ¨ ¨ Um ¸ j =1 © ij ¹ nm
2
(2) th
th
in which the subscript i represents the i load case, the subscript j represents the j measurement with the load case, nl is the number of load cases, nm is the number of measurements, w is the weighting factor, f is the c m objective function, S is the condition of the structure, U is the computed response, and U is the measured response. The optimization problem shown in equations (1) and (2) is an inverse problem, which is known to be highly sensitive to the noise in the measured response. A small variation in the response U due to the noise in the measurement tends to affect the solution S significantly. Since the measured response usually contains the noise, the sensitivity of the solution due to the biased response is one of the main challenges of the solving this inverse problem. We have proposed to use a multi-objective approach to address this issue.
MULTI-OBJECTIVE GENETIC ALGORITHM Algorithms for the multi-objective optimization (MOO) try to find a variety of Pareto optimal solutions in an efficient manner. Accordingly, they aim to guide the search towards the global Pareto optimal region and to maintain the population diversity in the current non-dominated front. Many computational implementations of MOO adopt genetic algorithm (GA) [8] due to its population based approach. Since GA inherently works with a population of solutions, multiple Pareto optimal solutions can be captured efficiently. This aspect makes GA well-suited for MOO problems. This multi-objective genetic algorithm (MOGA) needs to be able to find as diverse solutions as possible. If only a small fraction of the true Pareto optimal front is found, many interesting solutions with large trade-offs among the objectives may be neglected. Of particular interest among many MOGA implementations is a fast elitist non-dominated sorting genetic algorithm, often called NSGA-II [9], which is used in this paper.
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1241 The NSGA-II first generates a population consisting of “individuals” which are randomly distributed in the solution space. Each individual corresponds to a set of the design variables and is expressed in a form of “chromosome.” In this study, damage indices to represent the locations and degrees of damages are encoded in sequence into one chromosome. Based on the objective function values, NSGA-II ranks the individuals in the population based on the “non-domination” concept, as shown in Figure 1.
f2 ( x)
nf
f2 ( x)
=¦ i =1
max 1
d d1i
d
i 2
f1 ( x )
dik dimax
d 2max
f1 ( x )
Figure 1. Non-dominated sorting
Figure 2. Crowding distance
In order to maintain diversity in the current non-dominated front, NSGA-II introduces a density-estimation metric, called the crowding distance, which is defined as the sum of the normalized distances between two adjacent individuals along the multi-objective space. As shown in Figure 2, the crowding distance is calculated for each individual as the sum of the multi-dimensional distances between two adjacent individuals along the multiobjective space. Once the individuals are ranked and their crowding distances are assigned, NSGA-II finds nondominated (using rank), diverse (using crowding distance) individuals for the next generation. By repeating generations, the solutions evolve, and the Pareto optimal solutions are obtained.
DAMAGE IDENTIFICATION BASED ON MULTI-OBJECTIVE APPROACH The multi-objective based approach [7] seeks to improve the robustness of the damage identification. The method is composed of three steps. First, obtain multiple sets of measured responses for constructing multiple objectives. Second, obtain Pareto-optimal solutions by performing multi-objective optimization. The following equation represents the first two steps.
min F(S) = min[ f1 (S)
f 2 (S)
f n (S)]
(3)
th
in with the i objective function is defined in equation (2). Simultaneous minimization of multiple objective functions is performed using the NSGA-II algorithm explained in the previous section. The third step is to post-process the Pareto-optimal solutions to better identify the damage. Not all Pareto-optimal solutions will be close to the true solution, and therefore some false-positive solutions are expected. To quantify the level of confidence, the following statistical measures are employed.
μ=
σ=
1 N ¦ Si N i =1
1 N ¦ (Si − μ ) 2 N − 1 i =1
(4)
(5)
in which Si is each Pareto-optimal solution, N is the number of all Pareto-optimal solutions, μ is the sample mean, and σ is the unbiased sample standard deviation. Further use of these measures can be found in [7].
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1242 The obvious benefit of the multi-objective approach is that it will identify multiple candidates of solutions, which may better describe the true solution. If conventional single objective optimization is used, many analyses need to be repeated to obtain the same number of solutions, which will have higher computational cost. Another benefit as shown in [7] is stronger constraint provided by the multi-objective approach. When the measured response contains the noise, the stronger constraint better prevents the solution from converging to an incorrect solution, compared to the conventional single-objective optimization. The following example compares the performance of the multi-objective approach and the conventional singleobjective approach. Only one representative case has been selected out of various cases presented in [7]. To test the performance of the method without modeling errors or other uncertainties, “numerical” experiments are performed as shown in Figure 3. First, cross-sectional area of members 2, 5, and 11 of the numerical model are 2 reduced by 70%, 50%, and 30% respectively. Cross-sectional areas of undamaged members are 30,000 mm for 2 2 2 elements 1-4; 20,000 mm for elements 5-7; 22,000 mm for elements 8-13; and 25,000 mm for elements 14-15. The elastic modulus of each member is 200 GPa. Second, for each load case shown in Figure 3, displacements are measured at the “sensor” locations. Vertical displacements are measured for nodes 2, 3, 4, 6, 7, and 8, and horizontal displacement is measured for node 5. The focus of the proposed method is to improve the robustness against the sensor noise. To simulate the sensor noise, error is randomly selected from the Gaussian distribution 2 with zero mean and variance (5%) , and added to the displacement.
P
The goal of the damage detection is to inversely identify the location and the degree of damage using the “measured” displacements. Three objective functions are constructed from the three sets of displacements from the three load cases, which approximately describe loading from trucks. For multi-objective optimization, NSGA-II is used with 200 chromosomes and 2000 generations. For single-objective optimization, the three objectives are combined into single objective with uniform weighting factors, and then used as the objective function for the genetic algorithm with 200 chromosomes and 2000 generations. Since the single-objective genetic algorithm may be affected by the inherent randomness, 10 analyses are repeated to obtain mean and the standard deviation of the solutions.
Figure 3. Damage detection of a truss: damage locations, load cases, and sensor locations Figure 4 compares the performance of the multi-objective approach (MOGA) and the single-objective approach (SOGA). Although both approaches identify the most likely candidate of damaged members, SOGA shows significantly higher rate of false-positives. The likely reason is because in SOGA, once the solution converges to an incorrect solution, it cannot easily come back to the correct solution. On the other hand, MOGA’s simultaneous minimization of multiple objectives better prevents a solution from converging to an incorrect solution.
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Condition index
1 0.8 0.6 0.4 0.2 True solution
SOGA
0 1
2
3
4
5
6
7
8
9 10 11 12 13 14 15
Member index
Condition index
1 0.8 0.6 0.4 0.2 True solution
MOGA
0 1
2
3
4
5
6
7
8
9 10 11 12 13 14 15
Member index
Figure 4. Damage identification under 5% noise in the measurement
EXTENSION TO THE DYNAMIC LOADING AND THE INTEGRATION OF HETEROGENEOUS EXPERIMENTS Responses from dynamic testing such as vibration frequencies and modes are also commonly used in damage identification. Although static testing presented above has its merits, combination of static testing and dynamic testing may better detect the damage in the structure. Some authors have employed multi-objective optimization to improve the robustness of damage identification using dynamic testing [10-14]. Proposed forms of objective functions include: to minimize the difference between measured frequency and computed frequency, to minimize the difference between measured mode and computed mode, and to minimize the difference of a processed quantity (such as modal strain energy) from measurement and computation. Also, depending on the combination of these, and how multiple modes are considered, variations of these forms are also possible. We are currently working on the application of the multi-objective approach to integrate the response from the static testing and the dynamic testing. Damage identification using both static and dynamic testing may provide us more accurate result. The first objective is to minimize the static response, and the second objective is to minimize the dynamic response. Depending on how multiple load cases and modes are treated, the number of objective functions may increase to a large number. Currently, we limit the maximum number of objective functions to three because optimization of four or more objectives will be computationally too expensive. However, the development in the algorithms of multi-objective optimization, such as the dimensionality reduction [15-16], may enable us to use a general framework without the limit.
SUMMARY In this paper, we first highlighted the previous findings on the damage detection using multi-objective approach, focusing on the static loading. Multiple objectives are constructed to minimize measured displacements and computed displacements. Simultaneous minimization of the objectives improves the robustness against the noise
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1244 because of the stronger constraint provided by the multi-objective optimization. An example is presented, which compared the performance between the multi-objective approach and the conventional single-objective approach. The paper also explains how the framework may integrate heterogeneous experiments such as static testing and dynamic testing. One or more objective functions can be constructed from a type of test. Simultaneous minimization of the objective functions from various types of experiments may provide more accurate result. In order to make the framework general without the limit in the number of objective functions, development in the algorithms of multi-objective optimization is necessary due to the computational cost.
REFERENCES 1. Doebling SW, Farrar CR, Prime MB, Shevitz DW. Damage identification and health monitoring of structural and mechanical systems from changes in their vibration characteristics: a literature review, Los Alamos National Laboratory Report, LA-13070-MS, 1996. 2. Banan MR, Hjelmstad KD. Parameter estimation of structures from static response. I. Computational aspects. Journal of Structural Engineering 1994; 120(11):3243–3258. 3. Hjelmstad KD, Shin S. Damage detection and assessment of structures from static response. Journal of Engineering Mechanics 1997; 123(6):568–576. 4. Chou JH, Ghaboussi J. Genetic algorithm in structural damage detection. Computers and Structures 2001; 79(14):1335–1353. 5. Yeo I, Shin S, Lee HS, Chang SP. Statistical damage assessment of framed structures from static responses. Journal of Engineering Mechanics 2000; 126(4):414–421. 6. Kouchmeshky B, Aquino W, Bongard JC, Lipson H. Co-evolutionary algorithm for structural damage identification using minimal physical testing. International Journal for Numerical Methods in Engineering 2007; 69(5): 1085–1107. 7. Jung S, Ok SY, Song J. Robust structural damage identification based on multi-objective optimization. International Journal for Numerical Methods in Engineering; accepted for publication. 8. Goldberg DE. Genetic Algorithms for Search, Optimization, and Machine Learning, Reading. MA: AddisonWesley 1989. 9. Deb K, Agrawal S, Pratap A, Meyarivan T. A fast elitist non-dominated sorting genetic algorithm for multiobjective optimization: NSGA-II. Proceedings of the Parallel Problem Solving from Nature VI Conference, Paris, France, 16-20 September 2000; 849–858. 10. Kim GH, Park YS. An improved updating parameter selection method and finite element model update using multiobjective optimisation technique. Mechanical Systems and Signal Processing 2004; 18:59–78. 11. Haralampidis Y, Papadimitriou C, Pavlidou M. Multi-objective framework for structural model identification. Earthquake Engineering and Structural Dynamics 2005; 34(6):665–685. 12. Christodoulou K, Papadimitriou C. Structural identification based on optimally weighted modal residuals. Mechanical Systems and Signal Processing 2007; 21:4–23. 13. Jaishi B, Ren WX. Finite element model updating based on eigenvalue and strain energy residuals using multiobjective optimisation technique. Mechanical Systems and Signal Processing 2007; 21:2295–2317. 14. Perera R, Ruiz A, Manzano C. An evolutionary multiobjective framework for structural damage localization and quantification. Engineering Structures 2007; 29:2540–2550. 15. Brockhoff D, Zitzler E. Are all objectives necessary? On dimensionality reduction in evolutionary multiobjective optimization. In PPSN IX, Springer LNCS 2006; 533–542. 16. Deb K, Saxena DK. Searching for pareto-optimal solutions through dimensionality reduction for certain largedimensional multi-objective optimization problems. In Proceedings of the World Congress on Computational Intelligence 2006; 3352–3360.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Effects of Shaker Test Set Up on Measured Natural Frequencies and Mode Shapes
Chris Warren, Peter Avitabile Structural Dynamics and Acoustic Systems Laboratory University of Massachusetts Lowell One University Avenue Lowell, Massachusetts 01854
ABSTRACT The shaker set up used for modal testing can have an effect on the measured characteristics obtained. Several test setup configurations are presented to show the effects on the measured response of the system due to single shaker vs. multiple shakers as well as stinger types. For reference, impact testing is also performed to remove any effects from the shaker stinger/setup that might affect the results. Modal assurance criteria between several configurations are presented to illustrate the dramatic differences that can be obtained due to the test set up used. INTRODUCTION The most commonly used excitation techniques for modal testing are impact and shaker excitations. While both techniques have advantages and disadvantages, shaker testing tends to lead to higher quality frequency response functions (FRF) over greater bandwidths. Using shaker excitations, there generally is much better control on the frequency ranges excited as well as the level of force applied to the structure. While the measurements obtained with shaker excitation tend to be of higher quality and more consistent, greater caution must be take during the setup of a shaker test to obtain these pristine measurements. Various elements of the test setup can contaminate the FRFs, primarily due to the type of shaker attachment on the structure [1]. The most common way of attaching a shaker to a test structure is through a stinger. Stingers, also called quills, are typically made of drill or threaded rod. This type of geometry can provide high axial stiffness while attempting to keep bending stiffness to a minimum. While the main purpose of the stinger is to dynamically decouple the shaker from the test structure, this is impossible to fully achieve. Force transducers between the stinger and structure can only decouple the structure in the axial direction of the stinger. Some of the common problems that are typically encountered have been investigated in previous work; it has been shown that the location, alignment, length, and type of the stinger(s) used can dramatically affect measurement results [1]. In this paper, several test setup configurations are presented to show the effects on the measured response of the system due to single shaker vs. multiple shakers as well as stinger types. An impact test was also performed to remove any effects of the shaker/stinger assembly on the test article. There are many ways to set up for a shaker excitation test and no one way is always perfect. The following cases studied in this paper present common issues the test engineer may run into while performing shaker tests. Variations and inconsistencies in the measured FRFs and extracted mode shapes may or may not be avoided. Nevertheless, one must be conscious of the effect of his test setup on the measured system.
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_110, © The Society for Experimental Mechanics, Inc. 2011
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CASES STUDIED Modal testing was being performed on a small Southwest Windpower turbine as part of another study. Forced normal mode (FNM) testing was planned; 3 shakers were to be attached to the roots of the 3 blades of the turbine to ensure an even energy distribution throughout the structure. A preliminary impact test was performed to provide a reference without stingers in lieu of a finite element model. Two tests using a single shaker were then performed as an intermediate step to observe the degree to which the shaker setup may affect the test article. Nylon stingers typically come equipped with aluminum sleeves, so single-shaker tests were run with and without a sleeve. Finally, a 3-shaker MIMO test was performed in preparation for the FNM testing. These tests are all compared to the impact test; the 4 cases studied and presented in this paper are: Case 1 – 3 Reference Impact Case 2 – 3 Reference Impact vs. Single Shaker, sleeved nylon stinger Case 3 – 3 Reference Impact vs. Single Shaker, sleeveless nylon stinger Case 4 – 3 Reference Impact vs. 3 Shaker MIMO, sleeveless nylon stingers Structure Description & General Testing Performed The test article used throughout these cases was a 46 inch (1.17m) diameter Southwest Windpower Airbreeze wind turbine, which was mounted to the shaft of a commercial electromechanical fan motor. Figure 1 depicts the test setup from 4 angles. The numbers in Figure 1a correspond to the blade numbers used throughout the paper. The stingers were 10-32 nylon threaded rod, approximately 12 inches in length. For the sleeved case, the threaded rods were sleeved in a thin aluminum tube secured by jam nuts which were tightened to the point where no apparent rattling was present. Furthermore, small pieces of balsa wood were mounted to the turbine blades to align the axes of the shaker, stinger and force gage while accounting for the curvature of the blades. Also, the shaft of the motor was secured to prevent rotary motion of the turbine.
a
b
1
3 2
c
d
Figure 1. Turbine and shaker test setup. When the impact test was performed, the 3 force gages were mounted to the structure to maintain as much consistency with the shaker tests as possible. The excitation technique used in each of the shaker cases was a random signal with a Hanning window. All shaker measurements were taken over a 0-128 Hz bandwidth with 4096 lines of resolution. The impact test was over the same band having 2048 lines. For the single shaker tests, only the shaker attached to blade 1 was used. When multiple shakers were used, principle component analyses were performed to ensure that uncorrelated forces were input to the
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turbine. For each case, a laser Doppler vibrometer was used to measure the response at roughly 56 points, indicated by green squares in Figure 1a. LMS Test.Lab was used to capture all data and extract modal parameters with PolyMAX. RESULTS & CORRELATION The mode shapes obtained in each test appeared to be very similar and slight differences between the results were difficult to see. Modal assurance criteria (MAC) values are calculated and used for comparison of the tests, but shapes obtained from the individual tests will not be displayed. Figure 2 displays shapes that typify the response of the turbine in the frequency range of interest. (The modes are intentionally labeled with alphabetic characters to facilitate later referencing.)
Figure 2. Typical mode shapes of the wind turbine. Case 1: Impact Results To provide a benchmark that was free from the effects of any stingers, a three-reference impact test was performed. Input and measurement points were consistent with those used in the shaker tests. The first 9 flexible modes, which occur in the 060 Hz bandwidth, will be compared. Table 1 lists the frequencies of these modes and the auto-MAC values for the impact test. High off-diagonal terms are indicators of high correlation between modes due to spatial aliasing resulting from unmeasured portions of the structure. The intent of this paper is not to identify the differences between those modes but to compare the results of the similar modes of the different tests.
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16.46 Hz 16.53 Hz 17.90Hz 25.24 Hz 27.89 Hz 28.96 Hz 31.04 Hz 53.46 Hz 54.60 Hz
16.46 Hz 100.0 0.9 9.9 72.1 1.1 1.6 41.7 4.1 0.3
Table 1. Impact auto-MAC values. Impact 16.53 17.90 25.24 27.89 28.96 Hz Hz Hz Hz Hz 0.9 9.9 1.1 1.6 72.1 3.7 1.0 57.2 51.2 100.0 3.7 100.0 5.9 2.3 2.6 1.0 5.9 1.4 3.2 100.0 57.2 2.3 1.4 100.0 98.8 51.2 2.6 3.2 98.8 100.0 4.1 2.7 1.3 0.3 84.7 0.1 1.7 36.2 1.2 1.2 1.3 0.3 5.2 34.7 39.9
31.04 Hz 41.7 4.1 2.7 84.7 1.3 0.3 100.0 61.4 10.8
53.46 Hz 4.1 0.1 1.7 36.2 1.2 1.2 61.4 100.0 6.6
54.60 Hz 0.3 1.3 0.3 5.2 34.7 39.9 10.8 6.6 100.0
Case 2: Impact vs. Single Shaker – Sleeved The first step taken after the impact test was to excite the wind turbine with a single shaker and a sleeved nylon stinger. A sleeve around the nylon quill is typically used to prevent buckling in the middle while allowing flexible ends. Table 2 lists the mode shape pairs between the impact and single shaker test. When comparing the frequencies, the average of the absolute difference between the two tests is 3.56% while the maximum difference was 8.60%. Also, adding a stinger/shaker assembly to the turbine increased the frequency of most of the modes. Most of the MAC values indicated a reasonable amount of correlation. The average was 71.6%, but some modes correlate very poorly. The fifth mode shape pair had a MAC value of 12.4. This degradation clearly indicates that the addition of the stinger/shaker has a dramatic affect on the extracted mode shapes even though the frequencies are relatively unaffected. While average MAC values are not usually considered, they are provided her as a metric to compare how the various tests faired relative to the others. Table 2. Mode shape pair table comparing the Impact and Single Shaker – Sleeved tests. Mode Pair # 3 Ref. Impact Frequency Single Shaker Frequency Diff. (%) 1 1 16.46 Hz 1 16.80 Hz 2.02% 2 16.53 Hz 2 17.08 Hz 3.36% 2 3 17.90 Hz 3 18.07 Hz 1.02% 3 4 25.24 Hz 4 24.78 Hz -1.83% 4 5 5 27.89 Hz 5 25.58 Hz -8.26% 6 28.96 Hz 7 29.77 Hz 8.60% 6 7 7 31.04 Hz 6 31.45 Hz -4.08% 8 53.46 Hz 8 54.23 Hz 1.45% 8 9 54.60 Hz 9 55.35 Hz 1.38% 9
MAC 88.2 78.4 58.6 91.4 12.4 55.8 82.6 94.7 82.7
Case 3: Impact vs. Sleeveless Single Shaker The results of the single shaker with a sleeved stinger were drastically different from the impact test, so the sleeve was removed to investigate its effects on the frequencies and mode shapes. Table 3 lists the mode shape pairs comparing the impact and sleeveless stinger, single shaker test. The average frequency difference was 3.30%, so the type of stinger used does not appear to affect the frequency greatly. Only 8 modes were found with this test, but the MAC values improved significantly; the average value increased from 71.6% to 77.2%. Furthermore, the number of mode shape pairs with MAC values above 90% doubled. Both the sleeved and sleeveless tests yielded poor results compared to modes 3, 5, and 6 of the impact test. Mode 3, which is similar to mode C in Figure 2, has all three blades moving in phase. When a shaker is attached, the blade to which the attachment was made has a clear phase difference relative to the other two blades. This phenomenon also occurred in modes 5 and 6 – both are similar to mode E. The trend appears to be that whenever the attachment blade has a significant amount of modal displacement, the MAC value degrades. In the case of the sleeveless test, no mode was found at all for mode 6. Relative to the impact test, neither single shaker excitation yielded adequate results even though there were only small variations in the natural frequencies.
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Table 3. Mode shape pair table comparing the Impact and Single Shaker – Sleeveless tests. Mode Pair # 3 Ref. Impact Frequency Single Shaker Frequency Diff. (%) 1 16.46 Hz 1 16.59 Hz 0.79% 1 2 16.53 Hz 2 17.02 Hz 2.95% 2 3 17.90 Hz 3 18.08 Hz 1.03% 3 4 4 25.24 Hz 5 24.78 Hz -1.80% 5 27.89 Hz 4 23.81 Hz -14.62% 5 6 6 28.96 Hz ---7 31.04 Hz 6 30.08 Hz -3.09% 7 8 53.46 Hz 7 53.94 Hz 0.91% 8 9 9 54.60 Hz 8 55.25 Hz 1.19%
MAC 97.8 97.7 45.6 92.0 18.5 -93.9 84.9 87.3
Case 4: Impact vs. 3 Shaker MIMO As mentioned before, these tests were done in preparation for a forced normal mode test, so additional shakers were mounted near the roots of the other two blades. Table 4 lists the mode shape pairs between the impact and MIMO tests. Table 5 lists the full MAC matrix. Immediate improvements were observed: all 9 modes were found with an average frequency difference of 2.48% and average MAC of 90.0%. The phase-lag of the first attachment blade observed before in modes 3, 5, and 6 disappeared, so their MAC values improved dramatically. Each MAC value for these three modes was at least 88%, compared to a maximum of 58.6% in the single shaker tests. Also note that the high off-diagonal terms in the MAC matrix are consistent with the impact auto-MAC and are not of concern for this study. In the impact test, the first two modes are separated by a 0.40% difference. Due to the pseudo-repeated root for modes 1 and 2, a coordinate transformation had to be performed to obtained the MAC values shown in Tables 4 and 5. Without this extra transformation, the MAC values were much lower. The transformation was required to show the true correlation that does exist.
3 Reference Impact
Table 4. Mode shape pair table comparing the Impact and 3-Shaker MIMO tests. Mode Pair # 3 Ref. Impact Frequency MIMO Frequency Diff. (%) 1 1 16.46 Hz 2 17.16 Hz 4.26% 2 16.53 Hz 1 16.88 Hz 2.11% 2 3 3 17.90 Hz 3 18.10 Hz 1.19% 4 25.24 Hz 4 25.77 Hz 2.11% 4 5 27.89 Hz 5 28.68 Hz 2.84% 5 6 28.96 Hz 6 30.16 Hz 4.14% 6 7 7 31.04 Hz 7 31.75 Hz 2.29% 8 53.46 Hz 8 54.41 Hz 1.77% 8 9 9 54.60 Hz 9 55.50 Hz 1.66% * - MIMO rotated 120 degrees relative to the impact test
16.46 Hz 16.53 Hz 17.90 Hz 25.24 Hz 27.89 Hz 28.96 Hz 31.04 Hz 53.46 Hz 54.60 Hz
Table 5. Full MAC comparison for the Impact and MIMO tests. 3 Shaker MIMO 17.16 16.88 18.10 25.77 28.68 30.16 31.75 Hz Hz Hz Hz Hz Hz Hz 11.5 0.2 7.5 38.9 42.2 16.5 94.8 2.9 3.7 68.3 4.2 4.4 39.7 89.8 0.2 2.5 88.3 6.6 1.2 11.3 4.2 1.0 65.0 1.8 1.2 1.2 88.5 89.4 58.6 2.8 0.3 2.0 3.1 87.8 91.8 13.8 37.1 0.4 0.7 87.8 91.5 4.9 1.5 38.9 0.7 1.7 2.1 83.7 88.9 0.1 1.1 0.6 32.9 0.4 0.5 59.4 2.1 0.3 0.5 4.0 41.3 43.2 5.5
54.41 Hz 3.7 13.1 2.1 43.1 0.4 0.3 68.5 98.1 10.8
MAC 89.8* 94.8* 88.3 88.5 87.8 91.5 88.9 98.1 82.7
55.50 Hz 9.0 0.3 1.8 3.3 45.9 51.6 0.6 2.7 82.7
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Summary of Results The results of the single shaker tests using both sleeved and sleeveless nylon quills were poor representations for the mode shapes based on the impact reference. Even though an average frequency difference of about 3.5% between the impact and both single shaker tests for 9 modes over a 60Hz bandwidth was obtained, average MAC values were poor (~75%). The MIMO test provided a set of mode shapes and frequencies which were most consistent with the impact test. The average frequency difference was 2.5% and the MAC average was 90.0%. Clearly, significant differences in the mode shapes are the result of the changes in shaker/stinger setup. CONCLUSION A study was conducted to determine the effects of additional stingers on the natural frequencies and mode shapes of a small, 3-blade wind turbine. A 3-reference impact test was used as the benchmark to which single- and 3-shaker tests were compared. Changes to the test setup as subtle as removing the sleeve of the nylon stinger had a clear impact on the results. Overall, dramatic differences between the mode shapes obtained in the various tests were seen even with minimal changes in natural frequencies, demonstrating that great care must be taken when using shaker excitation in modal testing. REFERENCES 1. 2. 3.
Cloutier, David, Peter Avitabile, Rick Bono & Marco Peres. “Shaker/Stinger Effects on Measured Frequency Response Functions.” Proceedings of the IMAC-XXVII. February 9-12, 2009 Orlando, Florida USA. LMS Test.Lab – Leuven Measurement Systems, Leuven, Belgium Polytec Scanning Laser Doppler Vibrometer, Polytec Optical Measurement Systems
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Experimental modal analysis of non-self adjoint systems: inverse problem regularization
Morvan Ouisse, Emmanuel Foltête FEMTO-ST LMARC, Université de Franche-Comté - 24 rue de l’épitaphe - 25000 BESANCON – France [email protected], [email protected]
ABSTRACT When dealing with experimental modal analysis of non-self adjoint problems (rotordynamics, vibroacoustics, active control...), the nonsymmetry of the system induces specificities that must be considered for proper use of identification techniques. In this paper, the particularities of this kind of problem are addressed in order to be able to efficiently identify the dynamic behaviour. The first matter which is detailed is related to the ability of the technique to identify both right and left eigenvectors. The second point is associated to the regularization of inverse problem for matrices identification using the complex eigenvectors. The inverse procedure, which is one of the ways allowing the damping matrix identification, is very sensitive to noise. The technique of properness enforcement, already available in the context of symmetric systems, has been extended to nonself adjoint in order to regularize the problem. A numerical test-case has been performed on a rotordynamics application, and some experimental results are presented on a structural active control application.
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_111, © The Society for Experimental Mechanics, Inc. 2011
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1252 1. INTRODUCTION Experimental modal analysis is a very common tool in structural dynamics. It allows people to represent, using only experimental data, the dynamic behavior of a structure in a frequency range of interest using identified parameters (eigen frequencies, eigen shapes, modal damping ratios and modal masses). Using the modal basis, one can evaluate the response of the structure at measured points to an arbitrary excitation. A more evolved objective can be the determination of a matrix-based model, which can be seen as an experimental reduced model. For some specific applications, one can be interested to ask this model to be physical, in other words to have some matrices topology which are the same as those of a physical system : for example, on a large structure exhibiting a small component that vibrates mainly on its firsts components modes, one can be interested to have only a few degrees of freedom for it in the experimental model, and one would like the associated identified mass (resp. stiffness or damping) terms to be related to the physical mass (resp. stiffness or damping) of the component. This can also be of first interest in the context of experimental identification of damping matrices, which can not always be easily found through modelization. This paper will focus on this particular point. It has been shown [1] that for structural dynamics with symmetric matrices, the existence of an equivalent physical experimental reduced model is equivalent to the properness condition of complex vectors. In the same paper, an efficient methodology has been proposed to enforce that property when identified modes do not verify that condition. Some particular problems lead to second-order non symmetric formulations, like vibroacoustics [2] or rotordynamics [3]. For those systems, the quadratic eigenvalue problem [4] must be solved to obtain a coherent modal description of the problem, using complex modes. The first part of the paper proposes to extend the properness condition to non-self adjoint problems, before showing a methodology to enforce the properness condition on complex modes that do not verify the required property. The third part will be dedicated to a numerical example, on which the original model is available, before an experimental validation on an active control application. 2. PROBLEM DESCRIPTION AND MODAL DECOMPOSITION 2.1 Second-order typical problem The typical second-order problem which is considered in this paper is: (1) The notations used here are in accordance with those proposed in [5]. To this time-domain equation are associated the following direct quadratic eigenvalue problem: (2) and the corresponding ajoint eigenvalue problem: (3) in which: o
is the vector of unknown discretized field,
o
is the mass matrix, supposed to be nonsingular (all eigenvalues are then finite),
o
is the stiffness matrix,
o
is the damping matrix,
o
is the force vector,
o
is the j-th eigenvalue associated to the j-th right eigenvector
and j-th left eigenvector
All matrices are also supposed to be real, but not necessary symmetric, so the eigenvalues of the problem are real or come in conjugate pairs. A very complete review of this kind of problem is addressed in [4].
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1253 2.2 Modal decomposition of the permanent harmonic response The eigenmodes of the system can be efficiently used for the modal decomposition of the permanent harmonic response. This can be done considering the state-space representation of the system: (4) in which:
(5)
The associated first-order eigenvalue problem can then be solved to find the right eigenvectors left ones diagonal matrix
and the
, while the orthogonality relationships can be written using 2n arbitrary values to build the : (6)
The modal decomposition of the permanent harmonic response at frequency
is finally: (7)
A classical way to use these modes is to perform the calculation of the harmonic response according to previous equations, using a limited number of modes, depending on the maximum frequency value that should be obtained. In the field of experimental modal analysis, an experimental reduced model is built from complex modes, which are identified using FRFs (Frequency Response Functions) using techniques like the one described in [6] for symmetric problems. In the following, for practical reasons, without loss of generality, one will assume that the eigenshapes are normalized such as . 3. PROPERNESS CONDITION The properness condition for complex modes is very well detailed in reference [1]. Starting from a given set of 2n identified complex modes, this condition is related to the fact that the system can be exactly represented in the frequency range of interest, by a n degrees of freedom equivalent physical model, built from identified modes. If a set of vectors does not verify the properness condition, it means either that the system can not be represented by these sole modes, or that the experimental identification has introduced some errors on the eigenshapes (the properness condition is very sensitive to noise). In this case, Balmès has proposed a methodology to enforce the properness condition in structural dynamics [1]. In the same paper, it has also been shown that the properness condition is equivalent to the completeness of the basis, which is discussed for example in references [7] and [8]. In the present work we extend these notions to the case of non symmetric systems. The properness condition is associated to the inverse problem. The orthogonality relationships (6) can be inverted: (8) or:
(9)
and (10)
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(11)
It is then clear that the properness condition for nonsymmetric second order systems can be written as: (12) Once this relationship is verified, the matrices can be found using the inverse relations:
(13)
In the context of symmetric systems, this is one of the most popular ways to identify damping matrices from experimental measurements. An important remark is that these relationships require the knowledge of n modes to build a n-degrees of freedom physical equivalent system. 4. ENFORCEMENT OF THE PROPERNESS CONDITION One supposes in this part that a modal identification has been done, allowing one to obtain the eigenvalues, the left and right eigenvectors. The objective is to enforce the properness condition (12) on the eigenvectors. In a practical point of view, the experimental eigenshapes identification is generally quite sensitive to measurement errors, and a small shift on vectors can induce some large changes not only in the properness relation but also on the values of matrices after the inversion process. One then want to find the matrices and that verify the properness relation (12) and that minimize the norm of can be interpreted as a regularization of the inverse procedure.
and
. This
An analytical derivation of this minimization problem can be done using a Hamilton function that includes a Lagrange multipliers matrix: (14) and after a few operations, the solution that correspond to minimization of Hamilton function is: (15) with: (16) The Riccati equation (16) that has to be solved is not the one which is classicaly used in active control theory, since both matrices in factor of unknown matrix are not transposed one from another. Nevertheless, a quite efficient method can be used to find a solution of this equation using a Newton technique. 5. ROTORDYNAMICS APPLICATION A wide class of non self-adjoint problems is constituted of rotordynamics applications. A whirling beam example, which has been described in [10], is considered here to illustrate the application of the method in this context. This high speed gyroscopic system includes a lumped mass at the center of the beam. The system is discretized using 10 degrees of freedom, and the corresponding matrices (including gyroscopic and circulatory terms) are:
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(17) including 5 x 5 matrices such as:
(18)
where
is the Kronecker symbol, and the system properties are given in reference [10].
The complex eigenvalues and left and right eigenvectors are evaluated using the state-space form, and an artificial noise is added to the calculated values (random noise on real and imaginary parts with a relative magnitude of 1% for eigenvalues and 3% for eigenvectors). The matrices reconstruction is first performed directly from the complex vectors, and in a second way with properness enforcement; the FRFs are finally evaluated from the reconstructed matrices. One can observe in figure 1 that the properness enforcement allows a correct reconstruction of the FRFs, while the direct inverse technique leads to significant errors.
Figure 1: Frequency Response Functions between dofs 5 and 7 - Reference, reference with noise, direct reconstruction and reconstruction with properness enforcement One can clearly observe that the low level of noise which has been introduced in the simulation leads to very large errors on the identified matrices. This is due to the poor conditioning of the problem, which induces large errors in particular on damping values reconstruction. The figure 2 shows the efficiency of the properness enforcement using a graphical representation of system matrices, since the direct reconstruction introduces extra terms in matrices that lead to large errors in FRFs. In particular, one can observe that the damping matrix is the one which is the most affected by the reconstruction errors.
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Figure 2: Graphical representation of matrices - Original system, direct reconstruction and reconstruction with properness enforcement (white = zero value, black = maximum amplitude) 6. EXPERIMENTAL ILLUSTRATION 6.1 Presentation of the experimental set-up In this section an experimental illustration of the methodology is presented. The figure 3 shows the experimental set-up which has been used. It is constituted with two bending beams which are coupled through their bases by a common “clamping” device. The frequency range of interest concerns the two firsts modes of the coupled system, which could be represented by a 2-degrees of freedom equivalent model, using points 1 and 2 indicated in figure 3 as reference points. These points are equipped with accelerometers and some contactless force transducers are used to excite the structure, with force sensors. An electrical intensity probe has also been used to check the value of the force sensors and to verify that the moving masses do not perturb the measured information. The system itself is characterized by symmetric matrices, the unsymmetric parts are introduced using an active device.
Figure 3: Experimental set-up with analogical loop
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1257 The active part includes an analogic signal processing circuit, which allows one to introduce a force at point 3 depending on the value of displacement or acceleration of point 4. The results which are presented here correspond to a force feedback which is directly proportional to the displacement: it can be seen as a nonsymmetric term in the stiffness matrix. The gain of the feedback loop was chosen such that the system remains in the stable domain. All measurements have been performed using a sine-stepping approach to avoid undesirable effects due to broadband signal in feedback loop. 6.2 Left and right modes identification The classical LSCF method [11] can be easily adapted to identify the left and right complex modes of the structure. As indicated in [12], the full identification of the left and right vectors requires the use of sensors and excitations at every point of interest (i.e. every degree of freedom of the model). Fortunately, as soon as relationships between left and right eigenvectors are available, this condition is no longer necessary and only a limited number of excitation points can be used. In the present case, in which there is no link between left and right vectors, there is no alternative to the excitation of all points of interest. The identification procedure is based on complex fitting of Frequency Response Functions (FRFs) matrix which can be written using equation (7): (19) The first step of the method is the identification of the complex poles, this can be done exactly in the same way as if the system would be symmetric, since the eigenvalues are common to the left and right eigenvectors. Once the poles have been identified, one has to identify the eigenvectors. The excitation of degree of freedom number e allows identification of residue matrix (20) in which can be seen as the modal participation matrix, which is diagonal and whose terms are components number e of each left vector (which are unknow). This indicates that each column of is proportional to a right eigenvector. Since this matrix can be evaluated for each excitation point, an efficient way to find the orientation of right eigenvectors is to normalize the columns of (which defines a modified matrix ), and to use the mean of all measured values to build the direction of the right eigenvectors: (21) The matrix
includes right eigenvectors which are not normalized, this point will be discussed later.
The left eigenvectors can be found using an equivalent procedure based on dual residue matrices are constituted of lines j of all residue matrices :
which
(22)
This matrix is the equivalent of matrix
for the left eigenvectors. Its columns are then normalized to obtain
and the left eigenvectors are then proportional to: (23) The final step is to change the norm of vectors such as that
, which can be done by finding coefficients such
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and in order that
(24)
.
6.3 Results and discussion The figure 4 shows two FRFs measured with open loop. The measured curve is black dotted, from this curve the LSCF techniques allows identification of complex modes and modal synthesis (blue curve), and the modes are used to identify the matrices using equations (13). Calculation of FRFs from these matrices leads to green curve, while the red one corresponds to the synthetized FRFs obtained from complex vectors after properness enforcement. In this case the system is symmetric, and one can observe that, even in that case, the direct reconstruction of the matrices fails because of the perturbation due to experimental noise.
Figure 4: examples of FRFs with open loop The right and left eigenvectors which have been identified from the experimental FRFs are: (25) The properness enforcement leads to small changes in these vectors: (26) The small modifications in complex vectors correspond to very large changes in the damping matrix. The matrices which are deduced from original vectors are: (27) while those obtained after properness enforcement are: (28) One can clearly observe that the mass and stiffness matrices are not very sensitive to the changes in the complex vectors, while the damping values are very affected by the properness enforcement. The modified vectors clearly lead to more physical damping matrices than the original ones, this corresponds to the observations done in figure 4. These tests have been done using the open loop circuit, which means that the system matrices should be symmetric. They are clearly not symmetric, even if the extradiagonal terms have the same signs and same order of magnitude, which is not the case of the ones obtained from the original vectors. In the same way, the left and right complex vectors should be identical, which is not the case even if the differences are quite small.
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1259 A better procedure would be to apply the methodology dedicated to symmetric systems which has been presented in [1]. A similar analysis can be performed with the closed loop. In this case the system is clearly not symmetric. The results presented here correspond to a gain value of 6 in the feedback loop, corresponding to a strong feedback that remains in the stable domain. The figure 5 shows two measured FRFs and the corresponding synthetized ones using the identified complex modes, the matrices obtained by direct inversion and those corresponding to properness enforcement. It is then clear that once again the properness enforcement gives very good results compared with the direct procedure. Another point which can be observed is the impact of the feedback loop which induces shifts of the resonance pics compared with open loop (fig. 4).
Figure 5: examples of FRFs with closed loop The modified LSCF technique presented in section 6.2 leads to the identification of right and left eigenvectors: (29) The non symmetry of the system is clear. Once again, the properness enforcement induces small shifts in the identified complex modes: (30) The changes in complex shapes is illustrated in figure 6, in which one can observe that the changes mainly occur on the phase values, while the amplitudes of the vectors remain almost unchanged.
Figure 6: Right and left eigenmodes: original shapes (dashed lines), modified shapes (solid lines); mode 1 (blue), mode 2 (red)
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1260 The direct inversion of the problem to estimate the values of the matrices is clearly not the good way to obtain a physical value of the damping matrix: (31) The reconstruction after properness enforcement is much better: (32) The values of these matrices are clearly in accordance with the experimental conditions: the mass matrix is almost unchanged, the damping matrix is slightly changed, according to the phase delays in the feedback loop, while the extradiagonal term is very affected by the force feedback. 7. CONCLUSION In this paper, the properness condition has been extended to non self adjoint problems, a methodology has been proposed to enforce this condition on complex mode shapes which have been obtained experimentally, in order to physically regularize the inverse procedure to identify the system matrices. A numerical application on a rotordynamics test-case has been shown and an experimental validation has been done on a structure with a feedback loop to illustrate the methodology. References [1] E. Balmes, New results on the identification of normal modes from experimental complex ones, Mechanical Systems and Signal Processing 11 (2) (1997) 229-243. [2] K. Wyckaert, F. Augusztinovicz, P. Sas, Vibro-acoustical modal analysis: Reciprocity, model symmetry, and model validity, The Journal of the Acoustical Society of America 100 (1996) 3172. [3] Q. Zhang, G. Lallement, R. Fillod, Relations between the right and left eigenvectors of non-symmetric structural models, applications to rotors, Mechanical Systems and Signal Processing 2 (1) (1988) 97-103. [4] F. Tisseur, K. Meerbergen, The quadratic eigenvalue problem, SIAM Review 43 (2) (2001) 235-286. [5] N. Lieven, D. Ewins, Call for comments: a proposal for standard notation and terminology in modal analysis, The International Journal of Analytical and Experimental Modal Analysis 7 (2) (1992) 151-156. [6] R. Fillod, J. Piranda, Research method of the eigenmodes and generalized elements of a linear mechanical structure, The Shock and Vibration Bulletin 48 (1978) 3. [7] A. Sestieri, S. Ibrahim, Analysis of errors and approximations in the use of modal co-ordinates, Journal of Sound and Vibration 177 (2) (1994) 145-157. [8] Q. Zhang, G. Lallement, Comparison of normal eigenmodes calculation methods based on identified complex eigenmodes, Journal of Spacecraft and Rockets 24 (1) (1987) 69-73. [9] A. Jameson, Solution of the equation ax+xb = c by inversion of an (m_m) or (n_n) matrix, SIAM Journal on Applied Mathematics 16 (5) (1968) 1020-1023. [10] L. Meirovitch, G. Ryland, A perturbation technique for gyroscopic systems with small internal and external damping, Journal of Sound and Vibration 100 (3) (1985) 393 - 408. [11] H. Van der Auweraer, P. Guillaume, P. Verboven, S. Vanlanduit, Application of a fast-stabilizing frequency domain parameter estimation method, Journal of Dynamic Systems, Measurement, and Control 123 (2001) 651. [12] I. Bucher, D. Ewins, Modal analysis and testing of rotating structures, Philosophical Transactions: Mathematical, Physical and Engineering Sciences (2001) 61-96.
BookID 214574_ChapID 112_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Frequency Domain Tracking of Time-Varying Modes ir. J. Lataire and Prof. dr. ir. R. Pintelon Vrije Universiteit Brussel, Pleinlaan 2, 1050 Elsene, e-mail: [email protected]
Nomenclature Symbol A(k) Δn δ• δ(•) F RF F RFinst fs ϕk G• ke Kexc LTI LPV MIMO Nk Np O(•) OPAMP Sskirt SISO SIMO t, t∗ T u(t), y(t) U (s), Y (s) U (jω), Y (jω) ums , Ums Ums,T ωk x ˆ Y1 Y1,ke Y1,ke Yinst
Definition (repeated in the text) amplitude of the sine at the kth bin nth differencing operator Kronecker delta Dirac delta function Frequency Response Function instantaneous frequency response function sampling frequency phase of the sine at the kth harmonic linear time invariant (LTI) system excited frequency bin set of excited frequency bins Linear Time Invariant Linear Parametrically Varying Multiple Input, Multiple Output number of excited frequencies order of time variation order of magnitude OPerational AMPlifier spectrum of a windowed linearly increasing function of time Single Input, Single Output Single Input, Multiple Output continuous time and fixed time instant respectively length of the measured time record (in seconds) time domain input and output signals respectively Laplace domain input and output signals respectively Frequency domain input and output signals respectively multisine signal, respectively time domain and frequency domain representations spectrum of the windowed multisine (rectangular window) discretized angular frequency (ωk = 2πk ) T estimate of x contribution of linearly varying part of the system to output part of Y1 obtained from skirt centered around ke Y1 − Y1,ke output spectrum of the instantaneous system
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_112, © The Society for Experimental Mechanics, Inc. 2011
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ABSTRACT Many engineering applications involve systems whose dynamics vary with time. Consider, for example, the dynamics of the wings of a plane that is airborne. The resonance modes of the wings, and the associated damping and stiffness are functions of the flight speed and height. Therefore, the dynamics vary as the plane accelerates and/or its altitude increases. This paper provides a methodology for tracking the evolving dynamics of linear, slowly time-varying systems. The responses of the systems to multisine excitations are explained and shown to provide an insight into the timevarying behavior of the system. The acquired knowledge of the responses is used to estimate non-parametrically the instantaneous frequency response function from one single experiment. A simplified case, where the variations are linear with time, is discussed in this paper. A glance of a generalization procedure to a polynomial variation is given. Results are illustrated on simulations of a lumped time-varying system.
1
Introduction
Time-varying systems are found in a lot of engineering applications. As functions of the flight speed and height, the resonance frequency and damping of most vibrating parts of a plane (i.e. principally the wings) are time-varying. A robot arm is a nonlinear system that can be seen as a linearized system around continuously evolving (and thus time-varying) set points. The dynamics of a bridge vary as a train is moving over it,... In general, vibrations of mechanical structures are described by partial differential equations. In practice, however, matrices of transfer functions, describing the relation between applied forces at some points to the vibration at others are used as a preliminary step to modal analysis. As each element of the matrix corresponds to a transfer function of a SISO (Single Input Single Output) system, this well known and simplifying framework of ordinary differential equations applies to the considered systems [1]. Time-varying structures can be described – or at least be approximated well – by, either, (matrices of) ordinary differential equations whose coefficients are functions of time, or (matrices of) interpolated linear time invariant (LTI) systems. In this paper, the latter model will be considered. For slow variations an approximate equivalence between both can be shown [7]. If the variation of the dynamics is slow with respect to the time constants of the system, the concept of an instantaneous system makes sense. The latter is defined as the system that one obtains by freezing the dynamics at some time instant. An estimate of the evolution of the instantaneous dynamics provides valuable insight into the system. Informally, it reveals which poles and zeroes are varying and which are not. This paper provides a simple methodology for extracting a non-parametric estimate of the evolving instantaneous FRF. The method is discussed for SISO systems, but is straightforwardly generalized to SIMO (Single Input, Multiple Output) systems, such as structures with a single excitation point. MIMO systems are not considered in this paper. Non-parametric time-frequency methods, providing a time dependent spectral representation of non-stationary signals are thoroughly discussed in the literature. Popular methods are, among others, the short-time Fourier Transform, wavelet transforms, Cohen’s class of distributions, an overview of which are found in [3]. The application of these methods to vibrational signals for the characterization of time-varying mechanical systems is found for instance in [2]. However, neither an accurate estimate of the instantaneous FRF nor uncertainty nor bias bounds are obtained. The current study fills this gap, assuming that the device under test can be excited by a user-defined signal, in this case a multisine. This paper shows how the instantaneous FRF is obtained for a simplified case and explains conceptually how it can be generalized. The use of multisine excitations has been shown to provide some non-parametric frequency domain information on dynamic systems using easy algorithms. For LTI systems for instance, the frequency response function (FRF) is calculated non-parametrically as a simple division of input and output spectra at the excited frequencies. For nonlinear time invariant systems, the level of the nonlinear distortions is readily calculated by using the information at the unexcited frequencies (i.e. the detection lines) of the multisine [9], and the best linear linear approximation is readily provided.
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In this paper, it is shown how the response of time-varying systems to multisines allows one to immediately distinguish the time invariant part from the time-varying part by inspecting the output spectrum. Also, the response to the multisine is used to estimate the evolution of the instantaneous frequency response function (FRF) over the measured time window. This estimate is non-parametric in the frequency domain and parametric in the time domain. This means that, at each user-defined excited frequency (drawn from a discrete set), the evolution of the FRF is provided as a polynomial in time within the measured time range. This paper is restricted to first order polynomial (i.e. linear) variations. The generalization to variations of any order is given in [7]. However, note that any variation can be approximated by a piecewise linear variation, provided it is slow w.r.t. the typical time constants of the measured system, such that the current method can be applied to successive time-pieces of the signal. This method gives complementary results to the well known recursive identification methods [4], [8], which identify the instantaneous dynamics parametrically, but at discrete time instants. The current method is, however, implemented as an offline algorithm. It does not introduce delays in the identified time-varying models. A special case of time-varying systems are LPV (linear parametrically varying) systems, whose dynamics depend on one or more external scheduling parameters. If the dependence of the dynamics on the scheduling parameters is static, then the method described in this paper immediately provides the instantaneous FRF of the system under test as a function of the scheduling parameter. The evolution of the dynamics for the whole measured range of the scheduling parameter is then estimated at once. Parametric identification techniques exist, both in the time domain [10] and in the frequency domain [5]. The systems are usually (however not always) described by differential (or difference) equations with time dependent coefficients. The latter are identified as projections on arbitrary basis functions. The obvious advantage of these methods is parsimony. The disadvantages (w.r.t. the method described in this paper) are the complexity of the required identification algorithms (involving usually a nonlinear optimization) and the requirement of choosing a model order. The remainder of this paper is organized as follows. Section 2 describes the system model considered. Section 3 explains how the multisine excitation is constructed. Section 4 discusses the algorithm for estimating the instantaneous FRF in a simplified case, where the transfer function is evolving linearly in time. Results on simulations and measurements are given in Section 5 and Section 6 draws the conclusions. Notational conventions: Small letters x(t) and capital letters X(s) denote, respectively, time domain signals and their Laplace transforms.
2
System model U
+
G0 G1
t
G2 . . . GN p
t2
Y
t Np
Figure 1: Considered system model: combination of LTI systems, followed by monomially varying gains. The considered model is given in Fig. 1 and consists of a parallel connection of LTI systems, G0 , G1 , G2 , . . . followed by monomially varying gains: 1, t, t2 , . . . . The time domain input/output relation is given by
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phase (RAD)
amplitude (dB)
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60 80 0
π 0
−π
100 200 300 frequency (Hz)
0
100 200 300 frequency (Hz)
Figure 2: Frequency domain representation of the spectrum of a multisine.
y(t) =
Np
yn (t) =
n=0
Np
Gn {u(t)}tn
(1)
n=0
where u(t) and y(t) are the time domain input and output signals respectively. yn (t) is thus the output signal of the monomially varying gain of order n.
2.1
The instantaneous transfer function
The instantaneous transfer function at a particular time instant t∗ is defined as the transfer function one obtains by freezing the time-varying gains in Fig. 1 at t∗ , given by (in the Laplace domain):
F RFinst (s, t∗ ) =
Np
Gn (s)(t∗ )n
(2)
n=0
The corresponding block schematic is obtained by replacing all t’s by t∗ ’s in Fig. 1.
3
Applying multisine excitations to time-varying systems
A multisine signal is mathematically defined as follows: 1 ums (t) = √ A(k) cos(ωk t + ϕk ) Nk k∈K
(3)
exc
where k is an integer index drawn from the discrete set Kexc ⊂ [1, T2fs ] of excited frequency bins, ωk = 2πk is the T discretized angular frequency, T is the period of the multisine, and fs is the sampling frequency. The multisine thus consists of a sum of sines (or cosines) whose frequencies are all multiples of the same fundamental frequency f0 = T1 . Their phases ϕk are randomly distributed, s.t. E ejϕk = 0. The frequency domain representation of this multisine is given by discrete points, as illustrated in Fig. 2. Its Fourier spectrum is given by: 1 Ums (jω) = √ A(k)πδ(ω − ωk )ejϕk (4) Nk k∈±K exc
where δ(•) is the Dirac delta function, A(k) = A(−k), and ϕ−k = −ϕk . The Fourier spectrum of the windowed multisine (3) evaluated in ωk equals T Ums,T (jωk ) = √ A(k)ejϕk . 2 Nk
(5)
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LTI
LSTV
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Figure 3: Output spectrum of the (pseudo) steady-state response of an LTI system (left) and an LSTV system (right).
In this equation and in the remainder of this paper it will be assumed that, when windowed time domain signals are considered, a rectangular window of length T and height 1 is used. In the case of periodic signals with period T , leakage is then avoided. The amplitudes of the sines coincide with the function A(k), which can be chosen arbitrarily, depending on the application. A(k) is set to zero beyond a certain frequency index kmax such that the signal is band-limited, and aliasing is avoided when sampled at an appropriate sample frequency. In addition, some frequencies are not excited (i.e. A(kn.exc ) = 0). The constant Nk in (3) is equal to the number of actually excited frequencies. The division by its square root renders the RMS value of the signal independent of the chosen number of excited frequencies. The signal at these unexcited frequencies in the output spectrum contains valuable information about the timevariation of the system. Henceforth, it will be assumed that in between any two excited frequencies a considerable amount of unexcited frequencies are present. The precise ratio of excited to unexcited frequencies is a design parameter of the multisine excitation that determines the trade-off between the frequency resolution on the one hand, and, on the other hand, the noise variance on the estimate and the ability of tracking faster varying dynamics [7]. When a multisine is applied to an LTI system, the steady-state output is also a multisine, but with the phases and amplitudes shaped by the system’s transfer function. In the noiseless case, no signal is expected between any two excited frequencies. This is illustrated on the left figure of Fig. 3. For linear, slowly time-varying (LSTV) systems, a small alteration occurs. In this case, the most important contributions are also found at the excited frequencies. But, contrary to LTI systems, the unexcited frequencies also contain a deterministic part of the signal. These frequencies are given by the grey dots in Fig. 3, right. Skirt-like contributions are formed around each excited frequency. To illustrate this effect, many frequencies were left unexcited between each pair of excited frequencies. An insight into the origin of these skirts allows for the extraction of an accurate non-parametric estimate of the evolving instantaneous FRF, as explained in the next section. The concepts cited earlier are illustrated in Section 4 on the simple case where only linearly varying contributions are present (i.e. Gp = 0 for p > 1 in (1)).
4
Linearly varying contribution
The application of a multisine to an LSTV system is easily understood in the case where only linearly varying contributions are present, as shown in Fig. 4. Here, the only non-zero LTI systems from Fig. 1 are G0 and G1 .
U
G0 G1
+
t
Y +
Y1
Figure 4: Equivalent block schematic of the considered systems where the time variation appears as a time invariant dynamic part followed by a linearly increasing gain with time.
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When a multisine is applied to the system in Fig. 4, the output of both G0 and G1 are multisines. The multisine at the output of G1 is multiplied by a linearly increasing gain. This multiplication (in the time domain) corresponds to a convolution in the frequency domain of the multisine with a skirt-like signal, given in Fig. 5, left (the skirt being the spectrum of a windowed, linear ramp). Convolving the skirt with a series of Dirac delta functions (the spectrum of the multisine) means that the skirt is scaled, phase-shifted and copied around each excited line. Fig. 5, right, illustrates the result: the output spectrum of a linearly increasing gain, excited by a multisine. Similar results are observed when this multisine is applied to a varying gain with a higher order monomial of t, except that the skirts look a little different, as shortly described in Section 4.1. The conclusion of this discussion is that the time-varying part of the system – i.e. the signal branch through G1 and the time-varying gain in Fig. 4 – is responsible for a small contribution at the excited lines, and for the skirts at the unexcited frequencies in the output spectrum. The signal branch through G0 is the ‘time-invariant part’ of the system and is only responsible for (the major part of) the energy at the excited frequency lines. As a result of this clear distinction between the excited and the non-excited lines, it is possible to easily extract a non-parametric estimate of the evolving instantaneous FRF of the system. This is elaborated in the following paragraphs. As mentioned earlier, the spectrum of the multisine at the output of G1 is convolved with the windowed linear ramp’s spectrum. The latter, when evaluated at the DFT frequencies, gives: ⏐ T ⏐ ⏐ Sskirt (jωk ) = te−st dt⏐ (6) ⏐ 0 s=jωk ⎧ ⏐ 2 ⎨ − T e−sT + 1−e−sT ⏐ = jT (for k = 0) ⏐ s s2 2πk s=jωk = ⎩ T2 (for k = 0). 2
Elaborating the convolution of Sskirt with the output spectrum of G1 at the DFT frequencies gives:
Y1 (jωk ) = [G1 (jω)Ums (jω)] ∗ Sskirt (jω)|ω=ωk ∞ 1 = Sskirt (jωk − jω )G1 (jω )Ums (jω )dω 2π −∞ G1 (jωk )A(k ) jT 2 2 √ = δ + T δ ejϕk k,k k,k π(k − k ) 4 Nk
(7)
k∈±Kexc
80 60 40 20 0
amplitude (dB)
amplitude (dB)
(where δk,k is 1 iff k = k and 0 in all other cases; a bar denotes a negation). This is a sum of shifted hyperbolas, as seen on the right side of Fig. 5. Equation (7) shows that, in a small band around an excited frequency (i.e. for a certain ke + d s.t. A(ke ) = 0 and with d smaller than the minimum distance between two excited frequencies),
40 60 80
2 0
– 2 100 0 100 frequency (Hz)
0
phase (rad)
phase (rad)
phase (rad)
100 0 100 frequency (Hz)
100 200 frequency (Hz)
0
– 0
100 200 frequency (Hz)
Figure 5: Spectrum of a windowed ramp (left), multisine convolved with a ramp’s spectrum (right).
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Y1 (jωke +d ) is a sum of two contributions: Y1 (jωke +d ) =Y1,ke (jωke +d ) + Y1,ke (jωke +d ) jT 2 G1 (jωke )A(ke ) jϕke √ Y1,ke (jωke +d ) = T 2 δd + δd e πd 4 Nk G1 (jωk )A(k ) √ Y1,ke (jωke +d ) = ejϕk ) 4π N (k + d − k k e
(8) (9) (10)
k ∈±{Kexc \ke }
(where δ0 = 1 and 0 elsewhere, δ0 is its complement). Intuitively, Y1,ke is the contribution of the skirt centered around ke and is the main contribution to Y1 in that small frequency band. Y1,ke contains the contributions from all the other neighboring skirts. From (9) and (10), one learns that, at the first left and right bins of the excited frequency ke , the following holds: |Y (jωke ±1 )| 1,ke = O(πΔke ), Y1,ke (jωke ±1 )
(11)
where Δke is the (nominal) distance between two excited frequencies, expressed in bins. It follows that |Y1,ke (jωke ±1 )| Y1,ke (jωke ±1 ) for sufficiently separated excited frequencies, and thus Y1 (jωke ±1 ) ≈ Y1,ke (jωke ±1 )
(12)
Furthermore, it can be shown [6] that the first term evaluated at the excited frequency, Y1,ke (jωke ), corresponds to the windowed signal’s spectrum that would be obtained at that frequency if the time-varying gain in the system in Fig. 4 was replaced by the constant gain with value T2 . In other words, if Y1,ke was not present, the response of the instantaneous transfer function at the middle of a time record (i.e. at t∗ = T2 ) would simply be obtained by discarding the signal at the unexcited frequency lines. Define Yinst (jωk , t∗ ) to be the spectrum of the windowed output signal of the instantaneous system at time instant t∗ . It is thus clear that: T Yinst jωke , = Ums,T (jωke )G0 (jωke ) + Y1,ke (jωke ) 2 Δ2 Y (jωke ) 1 =− +O 2 (Δke )3
(13)
where Δ2 X(k) ≡ −2X(k) + X(k − 1) + X(k + 1) is the second difference of X(k). The second equation is proven in [6] and is valid when the first left and right frequencies of each excited frequency are not excited, viz. ∀ke ∈ Kexc , Ums (jωke −1 ) = Ums (jωke +1 ) = 0.
(14)
The estimate of the instantaneous response at t∗ = 0 is found by setting the time-varying gain to 0 or, equivalently, by removing the lower branch in Fig. 4. It is thus found as: Yinst (jωke , 0) =Ums,T (jωke )G0 (jωke ) =Y (jωke ) − Y1 (jωke ) ≈−
(15) (16)
2
Δ Y (jωke ) − Y1,ke (jωke ) 2
(17)
The last approximation is deduced from (13). Yinst at all other times is now obtained from a linear interpolation as:
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2 Yinst (jωke , t∗ ) = Yinst (jωke , 0) + Y1,ke (jωke ) t∗ T Δ2 Y (jωke ) 2 ∗ ≈− + Y1,ke (jωke ) t −1 2 T
(18)
Note that evaluating (18) at t∗ = T2 gives (13), as expected. Y1,ke is approximately known at the unexcited frequencies in a small frequency band around ke . But, for the evaluation of (18), its value is required at the excited frequencies. An estimate of Y1,ke (jωke ) can be extracted from the spectrum at the unexcited frequencies consistent with the following reasoning. From (9), it is clear that Y1,ke (jωke ) −Y1,ke (jωke ) π = = πe−j 2 Y1,ke (jωke +1 ) Y1,ke (jωke −1 )
(19)
Combining (8), (12) and (19), the contribution of the time-varying branch at the excited frequency is thus well approximated by: π
Y1,ke (jωke ) ≈ ±Y (jωke ±1 )πe−j 2 ,
(20)
where the choice of the sign is a priori arbitrary. A more robust estimate is, however, found as the mean value of both the positive and the negative estimates of (20): Y (jωke +1 ) − Y (jωke −1 ) −j π Yˆ1,ke (jωke ) = πe 2 2 Y1,ke (jωke +1 ) − Y1,ke (jωke −1 ) −j π = Y1,ke (jωke ) + πe 2 2 1 = Y1,ke (jωke ) + O (Δke )2
(21) (22) (23)
The second term in (22) is a differenced smooth spectrum and is thus prone to be much smaller than Y1,ke (jωke ), s.t. Yˆ1,k (jωk ) ≈ Y1,k (jωk ). The order of magnitude of the induced error follows from (10). e
e
e
e
An estimate of the instantaneous FRF at all excited frequencies is obtained by substituting (21) into (18) and by dividing the result by the windowed input spectrum at the excited frequencies.
F RF inst (jωke , t ) = ∗
−
Δ2 Y (jωke ) 2
+
π Y (jωke +1 )−Y (jωke −1 ) πe−j 2 2
Ums,T (jωke )
2 T
t∗ − 1
(24)
Clearly, this estimate is computed very easily. When evaluated at t∗ ∈ [0, T ], it gives a very accurate estimate of the evolution of the instantaneous FRF inside the measured time window for systems described by the block schematic in Fig. 4. Two error terms, introduced in (13) and (23), affect this estimate. The error term O (Δke )−2 being dominant, the deterministic error on the estimate in (24) is inversely proportional to the density squared of the excited frequency grid. It is important to note that this error analysis only applies if the considered systems fulfill the model in Fig. 4.
4.1
Generalization to polynomial variations
Till now, the system (1) has been considered for Np = 1. For higher order of variations, conceptually the same methodology can be followed (see [7]). In short, the multisines at the outputs of G1 , . . . , GNp are all multiplied by
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monomials in t. These multiplications yield convolutions in the frequency domain with spectra consisting of sums of hyperbolas of increasing orders (i.e. they are proportional to ω1 , ω12 , . . . , ωN1 p ). Thus, the resulting output spectrum still consists of skirts. In this case, however, the skirts are sums of hyperbolas, centered around each excited frequency. The contribution from each individual branch is now extractable from the output spectrum using a simple least squares fitting algorithm. The results of this generalization are shortly described in Section 5. The use of this method also allows the derivation of a model error and the noise variance on the estimated instantaneous FRF (not discussed in this paper).
5
Results
5.1
On simulations
The described algorithm (essentially given by (24)) was applied to simulation data. The simulated system was described by an ordinary differential equation with linearly time-varying coefficients. The instantaneous pole-zero configuration, is given at the left side of Fig. 6. The zeroes were fixed (i.e. they were time invariant) at the black circles. At t = 0 and t = T , the poles of the system were given respectively by the grey and the black crosses. A multisine excitation with a period of 1 second was applied to the system. 64 harmonics, between 1Hz and 255Hz, were excited (the sample frequency was fs = 1024Hz, and the number of sampled points per period was 1024). Only odd harmonics were excited (i.e. only odd multiples of 1Hz), and 10 periods of the multisine were simulated. The first period was discarded to assure the LTI systems to attain a steady-state (the whole system thus attained a pseudo steady-state). The estimated instantaneous FRF, as computed by (24), at three linearly spaced time instants are given by the grey shaded dots. The corresponding actual instantaneous FRF’s are given by the grey shaded full lines. The difference between both is given by the black crosses and lies at least 10 dB below the estimate. A good agreement is thus obtained. The most important (relative) difference is found at the least damped resonance and suggests that higher order variations are expected. Fitting a 5th order time-varying model (i.e. Np = 5 in (1)) on the data by using the method described in [7] yields the error given by the black dashed lines, which lies at least 50dB below the estimate.
5.2
On measurements
The method discussed in [7] was applied to measurement results too, as illustrated in Fig. 7. The measurements were conducted on an electronic circuit, whose schematic is given in the middle figure. It is a second order bandpass
20
500 0
0 FRF (dB)
FRF (dB)
imaginary
1000
10 0
ï50
ï500
ï10
ï1000 ï150
ï100 ï50 real
0
0
50
100 150 200 frequency (Hz)
250
ï100
0
50
100 150 200 frequency (Hz)
250
Figure 6: Results on simulation data. Left plot: initial (grey) and end (black) configuration of instantaneous poles (crosses) and zeroes (circles). (The zeroes are fixed.) Middle and right plot: actual (full grey shaded lines) and estimated (grey shaded dots) instantaneous FRF’s. Black crosses: estimation error. Black dashed lines: estimation error when using a model for higher (5th) order variations. The right plot is a zoomed out version of the middle plot.
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ï10
10nF
Imag. Real
FRF (dB)
ï20
10k
input sched.
10k
10nF
ï40 ï50 ï60
output 470k
ï30
ï70 ï80
0
1
2 frequency (Hz)
3 4
x 10
Figure 7: Results on measurements, using a 7th order time-varying model. Left plot: sketch of the expected evolution of the instantaneous poles (crosses) and zeroes (circles) (one fixed zero at the origin). Middle plot: electronic schematic of measured circuit (second order bandpass filter with variable resonance frequency controlled by the scheduling input, ’sched.’). Right plot: the grey shaded dots give the estimated instantaneous FRF’s obtained using the discussed method. Each grey shaded line gives an estimate of the FRF from a measurement with constant value of the scheduling input. Black dashed lines: estimation error.
circuit in which one of the resistors was made variable (physically implemented as a transistor of type BF245B), thus allowing the instantaneous poles to vary, as sketched in the left figure. The OPAMP used was a CA741CE. The input signal was a multisine consisting of 50 excited frequencies chosen between 300Hz and 40kHz. Only odd multiples of the fundamental frequency, 300Hz, were excited. The sampling frequency was 625kHz. The signal applied to the scheduling input was a linearly increasing voltage with time. When using a 7th order time-varying system (i.e. Np = 7 in (1)), the results in the right plot are obtained. The estimated instantaneous FRF is given by the grey shaded dots and are compared with the grey shaded full lines. Each one of the latter is the estimate of the instantaneous FRF from an experiment where the value of the variable resistor was fixed, allowing the use of classical LTI identification techniques. Both estimates agree very well. The error, given by the black dashed line at the bottom of the figure, lies at least 30dB below the estimate.
6
Conclusions
The paper presents a method for extracting the instantaneous frequency response function of a slowly time-varying system which can be approximated by a parallel connection of two LTI systems, one of which is succeeded by a linearly increasing gain with time. The instantaneous transfer function is identified non-parametrically in the frequency domain. As such it provides valuable information on the evolution of the system’s dynamics with time. An intuitive explanation for a generalization to higher order variations is provided.
Acknowledgements This work was supported in part by the Fund for Scientific Research (FWO-Vlaanderen), by the Flemish Government (Methusalem Fund, METH1), and by the Belgian Federal Government (IUAP VI/4). J. Lataire’s work was supported by the Research Foundation – Flanders (FWO) under a Ph. D. fellowship.
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References [1] Avitabile P. Experimental modal analysis (a simple non-mathematical presentation). In proc. of 46th Annual Technical Meeting and Exposition of the Institute-of-Environmental-Sciences-and-Technology (ESTECH 2000), APR 30-MAY 04, 2000 [2] Conforto S. and D’Alessio. T. Spectral analysis for non-stationary signals from mechanical measurements: a parametric approach. Mechanical Systems and Signal Processing, 13(3):395–411, May 1999. [3] Hammond J. K. and White P. R. The analysis of non-stationary signals using time-frequency methods. Journal of Sound and Vibration, 190(3):419 – 447, 1996. ¨ ¨ T. Theory and Practice of Recursive Identification. MIT Press, Cambridge, 1983. [4] Ljung L. and Soderstr om [5] Lataire J. and Pintelon R. Frequency domain identification of linear, deterministically time-varying systems. In proc. of the 15th World Congres IFAC - International Federation of Automatic Control, pages 11474–11479, 2008. [6] Lataire J. and Pintelon R. Extracting a non-parametric instantaneous FRF of a linear, slowly time-varying system using a multisine excitation. In proc. of the 15th IFAC Symposium on System Identification, Saint-Malo, France, July 6-8, pages 617–622, 2009. [7] Lataire J. and Pintelon R. Non-parametric estimate of the instantaneous transfer function of a time-varying system. Internal note J. Lataire, [email protected], 2009 [8] Niedzwiecki M. Identification of Time-Varying Processes. J. Wiley & Sons, Chichester, 2000. [9] Schoukens J., Pintelon R., Dobrowiecki T., and Rolain Y. Identification of linear systems with nonlinear distortions. Automatica, 41(2):491–504, 2005. [10] Spiridonakos M.D. and Fassois S.D. Parametric identification of a time-varying structure based on vector vibration response measurements. Mechanical Systems and Signal Processing, 23(6):2029 – 2048, 2009. Special Issue: Inverse Problems.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Variability in Natural Frequencies of Railroad Freight Car Components
William C. Shust, P.E. Mechanical Engineer Objective Engineers, Inc. 1557 Aztec Circle Naperville, IL 60563 [email protected]
Darrell Iler Senior Car Engineer Canadian National Railway 1764 S. Ashland Ave. Homewood IL 60430 [email protected]
ABSTRACT For many years, the North American Railway industry has been interested in the shock and vibration environment of freight cars. Initially this interest was related to the potential damage to cargo. More recently the interest has expanded to enhanced car and component reliability. A recent important subtopic has been the installed resonance frequencies of pneumatic equipment for air brake control valves (CVs). Perhaps the most critical accessory feature of a railcar is the braking system -- a complicated assembly of pneumatic controls and air reservoirs. Millions of pneumatic brake control valves are in service every day in North America. These devices depend on spring-loaded parts moving within very small dimensional clearances to work properly. As pneumatic controls, they are challenged to receive and repeat subtle air difference signals down the length of any train they comprise. Further, they are expected to work in all weather and operating conditions, and for several years at a time without any maintenance or inspection. At this time, the North American rail industry does not have any standards for CV design or mounting that accounts for vibration. The CV builders themselves have put one proposal forward. The industry is currently discussing this proposed design limit -- it would require installed natural frequencies to be above a minimum value. The valves themselves have been studied at length. However an examination of the current variations in both acceptable and “suspect” railcars has not been available thus far. In this paper, variations in resonance for existing attachment methods will be presented, as well as common response modes. In addition, a preliminary method for predicting the severity of vibration levels for different freight cars will be presented. NOMENCLATURE SR L E Z V
Severity Rating of likely field vibration at Control Valve Location Factor Empty Weight Factor Railcar input vibration, assumed from vertical spectrum found in AECTP-400 Inertance FRF of the Control Valve, driving point vertical (g/lb, These are the traditional H1 calculations using no windowing on a 4096-sample records that completely contain the accelerometer response. Data was collected with a 2000 Hz sampling rate using a 48 msec pretrigger.)
Axes
In this and most railroad literature, longitudinal is the direction of train travel. Lateral is to the left in the direction of travel, and vertical is up. Roll, pitch and yaw motions are taken about the longitudinal, lateral, and vertical axes respectively.
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_113, © The Society for Experimental Mechanics, Inc. 2011
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INTRODUCTION Automatic air brakes were first introduced ion trains in the early part of the twentieth century. Prior to this time brakemen ran perilously atop railcars applying the brakes by hand. The idea and implementation to automate air brakes conceived by George Westinghouse was a major leap forward for the rail industry in productivity and safety. The initial design was a simple air valve on the locomotive that controlled brake cylinder pistons on each car to apply the brakes. The idea quickly exposed its limitations when trains were separated and the cars were without brakes, and cut-off from any control. Westinghouse quickly engineered the solution by developing the “triple valve” to control and store air on each railcar. With the triple valve, the engineer no longer would apply pressure to set the brakes. Instead, a line of compressed air was maintained throughout the train, and the engineer would reduce pressure (the pneumatic signal) to brake. Then each triple valve would respond by using the car’s stored air to apply the brakes. Brake control valves (CVs) operating today still employ this method. This has allowed trains to grow in length, size and tonnage. In 1910, a heavy car was 50,000 lb (22.3 kg) whereas today 290,000 lb (129.5 kg) is common. Also, as loaded capacity has increased, empty (tare) weights have decreased. Today’s 290,000 lb (130 kg) railcar only weighs 45,000 lbs (20.1 kg) when it is empty. For decades, freight car design has been driven by three prime goals: “longer asset life, lighter weight, and higher payload capacity.” [1] Regarding efficiency of the transportation mode, the results have been quite good. However, starting in the early 1990s some in the rail industry began to believe that lighter weight car designs were responsible for a harsher operating environment felt by the accessories mounted on cars [2, 3]. The aforementioned papers were results of extensive tests by the CV (control valve) makers. They outlined modifications of their products to lessen vibration-induced wear. This again restored the CV service life, but the need for optimized mechanical car designs, and greater railroad efficiency (more and faster miles) continues to challenge the hardware. As the CV manufacturers gathered more data and understanding of the problem, they began to focus on the localized supporting brackets used to attach the CV to the freight car. In particular, they have proposed that the supporting brackets and car structure be designed such that its resonance frequency is between 135 and 200 Hz. [2] Railroads aggressively push the limits of braking performance to accommodate today’s heavier trains, and increase productivity while reducing costs. Due to design longevity, many parts within today’s automatic airbrake valves were engineered with the freightcar of the 1970s in mind. Considering the light weight of today’s cars their st insensitivity to 21 -century vibrations is still successful. However, recent reliaibliity concerns show that some attention to the supporting structures may be merited. SOURCES OF RAILCAR VIBRATION Some amount of vibration will occur any time a train is moving. Some of the track-based energy is due to random irregularities, having no periodicity. Track designed for higher speeds is inherently smoother due to U.S. government regulations for track quality [4]. However, in general the random inputs get stronger as the train encounters them faster. This leads to a well-known increase in ride roughness with increased train speed. Note that if certain rigid body resonances (pitch, roll, bounce) or kinematic wheelset lateral hunting occur, the response will be an exception to this overall trend of higher speeds leading to rougher ride. In addition to random track roughness, other energy sources can be periodic and speed-dependent, as shown in Figure 1. This shows forcing frequencies versus train speed for three common sources: historical 39-foot (11.9 m) spacing of rail joints, wheel RPM, and sleeper (or crosstie) spacing. The three colored regions bound the fundamental through 3rd harmonic frequency for these common sources of periodic energy. The lowest shaded region in the figure is due to rail joints. If the track is made of jointed rail in North America, most commonly one will find 39.5-ft. pieces of rail. Since the bending strength of the rail is less near a joint, dips in the track can develop over time. Several railcar types tend to respond to these dips by showing rigid body roll (rotation about the X-axis) at 1.9-2.8 Hz, which corresponds to 12-18 mph train operations. This is sometimes called “rock and roll” and is the most common car dynamic issue relative to jointed rail. Car/payload combinations
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with higher centers-of-gravity (CGs) exacerbate this. For this reason, the A.A.R. limits the overall car CGs to 98“ (or less) above the top of rail [5]. 100 Sleeper Spacing
90
Frequencies (Hz)
.
80
3rd Harmonic
70
2nd Harmonic
60 50
Fundamental 40 30 20
3X/rev
10
Wheel Revolutions
2X/rev
Once/rev Rail Joints
0 0
5
10
15
20
25
30
35
40
45
50
Train Speed (mph)
Figure 1. Speed-dependent energy sources of railcar vibration. Colored regions bound the fundamental and 3rd harmonic frequency for three common sources of tonal energy. The next higher shaded region is related to wheel imperfections and RPM. These can be either flat spots (which are tolerable to about 1 sq. inch in area) on a portion of the tread, or out-of-round (which allows runout of up to 0.07” or 1.78mm) wheels. The top shaded region in the plot is due to sleeper (crosstie) spacing. Placement of wooden sleepers varies somewhat, but they are usually spaced along the track at 19.5-inch (0.48m) centers. When directly over a sleeper, the rail is effectively in vertical compression, resting directly on the sleeper below. When a wheel is between two sleepers, the rail is supporting the wheel in bending, as a simply supported beam with a different vertical stiffness. As the railcar rolls, it sees these periodic changes in stiffness as a varying force input. Given the wide variety of track and operating speeds, it is difficult to represent the overall vibration source for rail vehicles. Even so, various North American and EU specifications exist for designing components that will fulfill their service lives on rail vehicles (rolling stock). A long-standing illustration of the broad nature of the energy available can be found in U.S. and NATO military specifications MIL-STD-810F [6], and AECTP 400 [7]. The North American rail spectra are essentially equal in both documents. Regarding vibration, AECTP 400 incorporates much of MIL-STD-810F and is a bit more comprehensive. Figure 2 shows rail vibration profiles specified therein. These are intended to verify that a piece of equipment shipped by rail to a forward staging area will survive the transport. Relative to the CV design issue at hand, the useful features of these plots are the indication of vertical as the most severe axis, and the gradual reduction of source energy above 40 to 80 Hz (depending on the axis). This would indicate that the most robust orientation of design should be vertical, and the least robust orientation should be longitudinal. Also, any longitudinal resonances are subject to energy that begins to fall off above 40 Hz, while reduction in vertical energy is above 80 Hz. This shape of the vertical PSD (power spectral density) will be used as a representative vibration input for a suggested vibration severity rating (SR) later in this paper.
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Figure 2. Rail vibration test PSDs as specified for equipment to be transported by rail for NATO, from document “AECTP 400, Mechanical Environmental Tests.” RAILCAR RESPONSES TO VIBRATION The response amplitudes to vibration can be affected by whether the input energy is close to a natural frequency of the system. A strong source may have little effect on the system if it is far from a resonant frequency. As well, a lesser source may dominant a particular response if it matches some localized resonant condition. With regard to the lowest frequency responses of railcar bodies, the three modes that most commonly are brought to the attention of railroad dynamicist are the car body roll natural frequency, the bounce natural frequency, and hunting activity. The first two are rigid body resonances; the third is a kinematic effect of the tapered wheel profiles. Roll (0.5 to 0.8 Hz) can be excited while traveling 12-20 mph (19-32kph) especially if the rails have outof-phase vertical irregularities (also called crosslevel deviations). Bounce (1.6-2.2 Hz) can be excited at transient bumps sometimes found near highway (road, or grade) crossings if the car is traveling in the 45-55 mph (7289kph) range. Hunting (2 to 4 Hz) is a lateral instability of the axles alternatively moving between the left and right wheel flange gaps, and may occur at 45 to 60 mph (72-97kph). Generally above the rigid body modes is flexure of the car structure. These are affected by whether the car is heavily laden or empty. Figure 3 shows typical ranges of the three fundamental flexure modes of six car styles [compiled from ref. 8]. Vertical Torsional Lateral
Well car Gondola Coal car Cov. hopper Boxcar Tank car 0
20
40
60
80
100
120
140
160
Typical Range of Flexural Frequencies (Hz, Loaded-to-empty)
Figure 3. Typical range of flexural frequencies for various fundamental modes of various railcars. (The fully loaded conditions are shown at the left end of a band, the empty conditions at the right end.)
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The bands are representative, but will vary beyond the limits shown for atypical car configurations. Note that the top three car types (Intermodal well cars, gondolas, and coal cars) have open tops, which result in low torsional stiffness and frequencies. Also note that the tank car bodies are generally much stiffer than all other car designs. SURVEY OF FREIGHT CAR CONTROL VALVE MOUNTING Within this paper a control valve (CV) will be considered to be each of two pneumatic portions (containing various ports, pistons, and manifolds) known as the service portion and the emergency portion. Both of these portions are needed, and are mated to either side of a heavy casting known as the pipe bracket. In North America, vendors supply the valves and bracket in both steel/iron and aluminum. Any combination of these valve portions and brackets may be found on any particular railcar in the field due to long service lives and the likelihood of single portion repairs or replacements. The pipe bracket may come in a double-sided or single-sided design. “Double-sided” indicates a pipe bracket that is nominally a rectangular prism with valve portion mounted on opposite vertical faces of the bracket, as shown in Figure 4. (A single-sided bracket uses a much wider manifold with both portions mounted to the same vertical face, but is not the focus in this document.) The stout box-shaped casting of the double-sided pipe bracket causes the important modes of vibration to be rigid unto the assembly itself, but flexural relative to its supporting structure on the railcar. This supporting bracket is often a ½” (12.7mm) plate formed into an inverted “U”-section or a short piece of structural channel. Regardless of the vendor and material, the overall dimensions of the valve assembly are quite consistent (approx. 8 x 22 x 11 inches, or 20x56x28cm). The range of weights for the assembly is approximately 75 lb (34kg, all items aluminum) to 170 lb (77kg, all steel). Thus, all other things equal, an all-aluminum installation could have a 50% higher natural frequency (flexure of the as-mounted assembly) than a steel/iron installation. In practice, many more steel/iron valves than aluminum are found in the fleet. And again these will be the focus within this paper. All valve combinations interface with the overall train brake system via tubular steel air pipes, as shown in Figure 5. Certain unfortunate combinations of the supporting structure and the piping runs have been found to fatigue the pipe flange connections. The pipes are believed to be especially sensitive to low frequency vibrations (due to its attendant greater deflection for a given g-level). The authors have not studied this general view.
Figure 4. A CV (Control Valve) assembly. From left, the emergency portion, the rectangular-shaped pipe bracket, the service portion.
Figure 5. CV installed on center sill of freight railcar.
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COMMON CV SUPPORT HARDWARE AND LOCALIZED RESPONSE MODES This section will show a range of the most common responses found in a survey of about 149 freight cars. The natural frequencies and actual hardware designs were found to be quite variable even within this small population. Regardless, the authors hope to present an incomplete but orderly summary of the analytical and experimental results. For the analytical part of this study, a linear finite element model was created for a typical two-sided valve assembly and supporting “U”-section bracket. The vibrations of the service and emergency portions of the valve have been reported elsewhere by two North American valve manufacturers in an excellent paper [2]. The resonances of the portions themselves were 200-350 Hz (well above the 10-135 Hz band of interest herein). Therefore, this FEA model only represents the appropriate inertia of a two-sided pipe bracket and associated steel hardware. Thus, the CV is represented as the nominal envelope of a valve assembly, but without internal details (and with artificial lightening holes to mimic the suitable overall inertial properties). The focus of the models is not the CV itself, but the support brackets. The resulting generic mode shapes can help show whether natural frequency increases could be made with minor stiffeners, or whether the larger car frame structure is likely to control the response frequency. Further, due to the many small variations in railcar dimensions and larger variations in vehicle lengths and capacity, this model was not calibrated in detail, nor was it tuned using the experimental FRFs collected herein. Rather it was used to show the baseline modes of a CV assembly when fixed to ground, and then to compare the results to typical car installation techniques. The next subsections show the FEA results for the first few flexural modes for several CV boundary conditions assumptions. The results will show that (commonly) the car’s local flexibility tends to dominate the as-installed frequencies. The section of mode shape predictions is organized into four cases: x Case 1: Baseline, CV on rolled “U” channel and fixed to ground x Case 2: CV on pinned-pinned beam (shallow channel), often found hanging under Boxcars x Case 3: CV on rolled “U”, and then to larger plate (with or without stiffening flanges) usually as outrigger at corner of car, common on some Coal cars, Well cars, and Tank cars (but without vertical leg) x Case 4: CV again on “U” section but welded to more substantial member of car, shown here on the end of the car’s “backbone” known as the center sill. Case 1 Figure 6 shows a common mounting style, but fixed to ground. The model shows only 3 modes below 500 Hz, lateral translation, pitch, and vertical. The vertical mode incorporates a small amount of roll due to the slight asymmetry in the location of the CV on the “U”-section. As expected, the frequency of the first mode can be increased significantly with a gusset in the interior, or with additional legs. For example, simple Euler equations show that increasing the “U”-section plate thickness by 50% and adding a third leg will increase this frequency by 70-75%. Both the pitch and vertical modes would also be increased greatly. This suggestion was outlined at an industry conference in 1994 [3]. On more flexible freight cars, such a modification may have less effect however. Case 1: Baseline, CV on rolled “U” channel and fixed to ground
105 Hz Lateral Translation
256 Hz Pitch Figure 6. Modes for baseline FEA.
403 Hz Vertical (shown plus some Roll, when not centered on “U”)
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Although intended as a hypothetical extreme, effectively this baseline design was assumed to be found on some covered hopper cars where the “U”-section foot welds were made directly on a shear plate that was itself fixed to the car’s “backbone” or center sill. This conclusion is made because the response appeared to duplicate this Case 1 scenario. Lateral translation of the valve assembly occurred with the two legs of the c-shape lozenging (parallelogramming) together at 95-105 Hz. Case 2 Figure 7 is an initial step beyond ground fixation. Case 2 shows a simple and common design for actually supporting a CV on a rail car. Herein, the valve is attached to a longer but shorter portion of channel, and the ends of the channel attached to the railcar frame. This channel is either rolled plate or a structural “C”-shape. Weld patterns found in the field were judged to approximate of a pinned-pinned beam fixation. The first natural frequencies are roll, vertical bounce, and pitch. Case 2: CV on pinned-pinned beam (shallow channel), common when hanging under Box cars
64.5 Hz Roll
71.8 Hz Pitch 66.2 Hz Vertical (plus some roll, when non-perfect symmetry) Figure 7. Modes for pinned-pinned beam support.
Regarding experimental results on actual cars, at least seven subfamilies of this Case 2 style were seen. For example, Figure 8 shows three pinned-pinned span designs just inboard of the car’s side sill and between floor supports. Overall for the several boxcars, lowest experimental (pitch and roll frequencies) were measured between 39-58 Hz, with higher (vertical bounce) frequencies at 88-104 Hz. The higher bounce frequencies may indicate fixed-fixed behavior rather than pinned-pinned boundary conditions. When the channel is firmly connected to a significant element of the railcar frame, these frequencies are likely to vary with the dimensions of the channel. Therefore minor design changes are likely to be effective if one or more frequencies are deemed too low. However, in some car designs this hardware is further connected to “outrigger” sections built from structural angles in order to place the valve at the car’s extreme corner to allow for better service and maintenance access. Such is the situation for Case 3. Case 3 Figure 9 shows a merger of Cases 1 and 2, that is, the CV is directly connected to the “U”-section, which then is welded to a shallow wide plate or channel. In turn, this is supported on truss-like elements. The benefit is that the CV is placed out near the extreme corner of a railcar making assembly and service easier. The illustration below shows a vertical leg at the extreme rear (right) corner of the rail car, as found on many coal cars. A similar design but without the vertical leg is found on tank cars. In this case the CV is arranged as a cantilevered assembly with supporting structural angles directed forward and laterally to the more-substantial frame members of the car. In this Case 3, the first FEA mode is at 37.5 Hz with flexure of the underlying plate largely controlling the response. The second are third modes are nearly identical, with pitching motions about slightly different centers. In all three modes, changes to the “U”-section are unlikely to affect the response frequencies significantly. This design was measured on many coal hoppers. Depending on the width and flanges of the plate steel, the fundamental mode was found in a 32-80 Hz range and controlled by the flexural stiffness of the plate. In this case, the c-shape and valve assembly tended to roll together in nearly a rigid-body response.
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Further, for the tank car situation (not shown above), lowest cantilever modes have been measured at 12-14 Hz. These occurred when this “outrigger” design was not vertically triangulated as a truss; four such styles are shown in Figure 10. Secondary bending of the support angles resulted in a pitching mode in the 48-52 Hz range. Thus, the strengths of the structural angles are heavily involved in the frequencies. In this scenario, changes to either the “U”-section or the underlying plate are likely to have only secondary effects.
Figure 8. One of many forms of the Case 2 beam support tested for impact FRFs. Case 3: CV on rolled “U”, and then to larger plate (with or without stiffening flanges) usually as outrigger at corner of car, common on some Coal cars (as shown here), some Well cars, and Tank cars (but without vertical leg).
71.0 Hz Pitch 37.5 Hz Lateral 67.9 Hz Pitch (about forward edge of (about rear edge of “U”, controlled by (controlled by underlying plate “U”, controlled by underlying plate underlying plate flexure, not “U” section) flexure, not “U” section) flexure, not “U” section) Figure 9. Modes for railcar rear-corner “outrigger” support. This Case 3 design was measured on many coal hoppers. Depending on the width and flanges of the plate steel, the fundamental mode was found in a 32-80 Hz range and controlled by the flexural stiffness of the plate. In this case, the c-shape and valve assembly tended to roll in nearly a rigid-body response. Further, for the tank car situation (not shown above), lowest cantilever modes have been measured at 12-14 Hz. These occurred when this “outrigger” design was not triangulated as a truss; four such styles are shown in Figure 10. Secondary bending of the support angles resulted in a pitching mode in the 48-52 Hz range. Thus, the
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strengths of the structural angles are heavily involved in the frequencies. In this scenario, changes to neither the “U”-section nor the underlying plate are likely to have significant effects.
Figure 10. Four of the several styles of tank car outrigger supports tested for impact FRFs. Case 4 Finally Figure 11 is a less common method of mounting the CV to coal cars and covered hoppers (wherein it is mounted more closely to the car’s backbone, known as the center sill). Because of the increased size and mass of the center sill, its vertical and lateral bending modes dominate the lower two natural frequencies for this case, 21.5 Hz vertical bending and 27.2 Hz lateral bending. The third frequency is controlled by the torsional stiffness of the center sill, at 62.9 Hz. For this model, the boundary conditions for the center sill were only roughly estimated. However, actual tests of an operating loaded car were performed at the center sill with peaks in the PSDs at 22.4, 29.3, and 72.8 Hz, showing good agreement with the assumed boundary conditions. Regardless, with reasonable confidence it can be said that a stiffener in the “U”-section is unlikely to change the actual experimental frequencies, although it may beneficially reduce the overall CV deflection. This design will be shown later to have an overall advantage in terms of severity. Unfortunately, service and repair access to the CV is compromised in this location. Case 4: CV again on “U” section but welded to more substantial member of car, shown here as the end of the car’s “backbone” known as the center sill.
62.9 Hz Higher torsional mode of center sill, “U”-section participates 27.2 Hz Lateral bending mode 21.5 Hz Vertical bending mode but does not control. of center sill “backbone” of car of center sill “backbone” of car Figure 11. Modes for railcar “backbone” center sill support.
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Other Cases Although also investigated in the field, several other design styles were not included in the FEA analyses. Cantilevered bookend designs (for example, Figure 12), usually hanging from the car’s side sill are found on steel coil gondola cars and some boxcars. This style of support has the widest variation in experimental frequencies of the various hardware families, with fundamental frequencies ranging from 19-95 Hz. The variation is greatly affected by gussets when provided, and by the local stiffness of the car’s side sill. There is no indication that the rail industry is experiencing problems with any of these valves. 7 Valves Coil cars
FRF Mag(g/lb)
10 -1
10 -2
10 -3
10 -4
25
50
75 Freq (Hz)
100
125
Figure 12. One of many cantilevered bookend style supports tested (left), and wide variety of driving point inertance FRF magnitudes measured with impact tests (right). Another family of mounting styles includes built-up beams cantilevered off a vertical face of a car end, commonly found on intermodal well cars (for example, Figure 13). These have a wide array of first natural frequencies depending on the existence of additional gussets, and the vertical section size of the cantilevers. For thin or shallow sections (e.g. 6” and under) first frequencies of 34-69 Hz were found (left). For thicker sections, the frequencies increased to between 80-194 Hz (right). The first response mode is usually torsion of the cantilever, which often couples to lateral motion due to the valve assembly center-of-percussion being above the cantilever.
Shallow cantilever style support, 32.7 Hz Deep cantilever style support, 80.1 Hz (considered to be a design of concern) Figure 13. Two of a few dozen cantilevered built-up beam supports tested for impact FRFs. Finally hybrids between the design styles abound. Figure 14 shows a small sampling. None of these styles are designs suspected of vibration issues.
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Wide right-angle plate, 34.1 Hz
“U” section of stepped leg length, “U”-section with internal gusset along 66.4 Hz 50% of legs, 87.4 Hz. Figure 14. Three of many hybrid support designs and dominant lowest natural frequencies.
INHERENT VARIATIONS FOUND WITHIN ONE SERIES OF COAL CARS FRFs were measured on 67 cars from an empty coal train, of which 54 were the Case 3 design and same car series. (This small diversion is presented because of the author’s long-standing interest in how close an FEA model should be tuned to a single test specimen. Perhaps the demonstration is of utility to some readers.) Figure 15 shows these FRFs in a waterfall fashion, and ordered by increasing serial number of the car. The dominant response mode (highest mobility) was at an average frequency of 79.51 Hz. Given this sampling, and assuming a lognormal distribution of this natural frequency, the +/-3 sigma limits for this response are 73.71 to 85.77 Hz. Such a band is approximately +/-7.5% of the mean natural frequency, and would be expected to contain 99.73% of the coal cars of this particular design. This is at the upper extent of the 3-7% variation in natural frequencies found in a previous study which investigated assembly-to-assembly variation of such “nonprecision” mechanical assemblies [8].
0 -1 -2 -3 -4 -5 60 50 40 30 20 10 0
0
50
100
150
200
250
Railcar No.
Figure 15. Vertical driving point FRF magnitudes for control valves on 54 coal cars from one car series and support style (Case 3).
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PREDICTION OF RAILCAR VIBRATION SEVERITY This final section of the paper will propose a ranking algorithm suggested by the first author. The method is in an initial stage of hypothesis and does not reflect the opinion of the second author, nor the engineering approaches of CN Railway, nor any other members of the North American rail industry. General discussions on CV vibration and infrequent but accelerated wear of valves tend to focus on lightweight aluminum coal gondolas (Case 3, and Figure 13-left, above) and on certain intermodal well cars with shallow-section cantilevered CV supports. The ranking method described herein has heuristically evolved during the project. It has emerged based on the extensive FRF tests of these cars, as well as many trial and error examinations of hammer-to-valve driving point FRFs, hammer-to-car FRFs, and car-to-valve transmissibility’s. Various other methods of discriminating railcars were tried along the way, including frequency of resonance (low vs. high), amplitude of resonance, and effective mass lines. None of these showed results that matched informal industry feedback about frequency of repair or replacement of valves for various car types. The method combines several numerical quantities: INPUT VIBRATION, Zi The nominal available input vibration at a hypothetical fully loaded railcar is assumed to be as shown by the rail vibration test spectra as shown in Figure 2 (as specified “AECTP 400, Mechanical Environmental Tests” and MIL-STD-810F). The primary axis of consideration is assumed to be vertical due to its larger overall amplitude. The other axes are ignored for simplification. LOCATION FACTOR, L Some car styles mount the CV near the wheels and therefore closer to the track input energy, for example coal cars. Other cars support the CV at mid-span between front and read wheels under the car, such as boxcars. The second style cars receive a reduction in assumed input, via a simple fraction of the car length between the car and the wheels. A multiplier of 1.00 is used for the input spectrum for a CV directly over the wheels; a multiplier of 0.50 is used if the CV is halfway along the car length. This factor is applied across the input spectrum for the car under consideration. EMPTY WEIGHT FACTOR, E As any pickup truck driver knows, an empty truck rides harsher than a loaded truck. The situation is similar with freight cars. Thus, a second multiplier (applied to the assumed input spectrum) is defined by the ratio of sprung weight while fully loaded to the sprung weight while empty (tare). This multiplier can vary from about 4 (for a heavy steel-hauling gondola) to 11 (for a very lightweight aluminum coal gondola). The coal car has a 286,000 lb. gross rail load, and only a 41,400 lb. empty load (of which approximately 17,000 lb. are unsprung). INERTANCE OF THE CONTROL VALVE, Vi The final piece of the rating algorithm is the driving point vertical inertance frequency response function (FRF) of the control valve in (g/lb). To better focus on the ranges of natural frequencies found within the surveyed cars, and the frequencies of greatest interest to the valve manufacturers, this summation is made across a band from 30-135 Hz. SEVERITY RATING, SR The severity rating is calculated by multiplying the assumed available vibration on the car by the magnitude of the driving point FRF at the control valve (on a spectral line by spectral line basis). Then sum this product across the 30-135 Hz band. Finally multiply this scalar by the location factor and the empty weight factor. Taken together, the above steps are indicated by equation (1) below. The result is an estimated severity rating of field operating vibrations, as made from a single driving point FRF of a given car.
SR
135
L E ¦ Zi x V 30
(1)
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COMPARISON OF CAR SEVERITIES
DTTX680787 WELL SHALLOW
DTTX657048 WELL SHALLOW
DTTX646796 WELL DEEP
DTTX645422 WELL DEEP
DTTX741701 WELL CCHAN
DTTX729055 WELL CCHAN
UTLX640736 TANKCAR
GATX54809 TANK CAR
PLCX43761 COV.HOPPER
CIC2064 COV.HOPPER
CEFX151923 COV.HOPPER
TTPX804490 COIL CAR
IHB19362 COIL CAR
CN187192 COIL CAR STOUT
PNJX51215 COAL CAR
PNJX50724COAL CAR
CEFX43852 COAL CAR CSILL
CEFX43824 COAL CAR CSILL
IC53309 BOXCAR
5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0
AOK14166 BOXCAR
Ctrl. Valve Severity Rating
Figure 16 shows the results of the severity rating for a random selection of 20 railcars from the 149 tested. This rating system indeed shows the “suspect” car styles would have the largest severities. That is the 4 highest bars indicate the lightweight coal cars with Case 3 design, and shallow beam cantilever supports on intermodal well cars. The other CV support styles and car types may have shown natural frequencies below 135 Hz, and/or larger peak FRF amplitudes. However, with the employment of the various modification factors and frequency weighting from Equation (1), this method indicates that they do not require further attention by the industry.
Figure 16. Severity Ratings for 20 of the 149 surveyed cars, based on preliminary equation (1). CONCLUSION / RECOMMENDATION This paper has attempted to support the authors’ opinions that a proposed design criteria (blanketed across the North American railcar fleet) for 135 Hz as the minimum allowable first mode [2] is not advised. That is not to say that the concept is not a useful ideal. As indicated back in Figure 2, input energy falls away above 40-80 Hz (depending on the axis), and it is a widely accepted rule of thumb that designing brackets for higher natural frequencies is better than lower. Further, this is especially true in typical ground-vehicle random vibration environments. The field survey and FRF measurements for 147 of 149 rail cars do not pass the 135 Hz proposed rule. Many of these cars are accepted and long-standing design styles. Therefore, the authors have continued to seek a simple method to objectively rank designs in terms of likely CV vibration severity during field operating conditions. One method would be recording the vibrations for each car type and mounting style, throughout a several day train trip. This is not feasible due to the many combinations, and the costs involved. The simpler proposed severity rating method was shown to approximate concerns within the industry about certain car types and CV support structures. It yields a single value to help prioritize the railcar fleet from a simple 2-channel driving point FRF. If adopted or developed further, this severity rating could assist the industry with
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allocating resources toward the cars of highest interest. Note again, this approach is quite preliminary; therefore the author welcomes reader comments. ACKNOWLEDGEMENTS The authors would like to express gratitude to the AAR Brake Control Valve Task Force for ongoing interest and feedback, and to the Canadian National Railway and the BNSF Railway for providing access to the railcars. REFERENCES [1] Vantuono, W., “Next-generation freight cars take shape,” Railway Age, Sept. 2000. [2] Wright, E., and Troiani, V., “A Study of Vibration Response of DB-60 and ABDX Freight Brake Control Valves and Recommendations for Installation on Freight Cars,” The Air Brake Assn. Annual Technical Conf. Chicago, IL Sept. 26, 2004. [3] Hart, J., “Severe Vibration of ABD, ADBW and ABDX Control Valves,” The Air Brake Assn. Annual Technical Conf. Chicago, IL Sept. 20, 1994. [4] United States Codes of Federal Regulations 49 CFR 213.9, Part 213 Track Safety Standards [5] Association of American Railroads, Interchange Rule 89, Section B, 1, e. [6] MIL-STD-810F, Jan. 2000, Method 514-5 Vibration (Annex B, Engineering Information, paragraphs 2.1.4 Endurance Test, and 2.2 Fatigue Relationship), pg. 514.5B-2 and 514.5B-3. rd [7] “Mechanical Environmental Tests,” AECTP 400, NATO, 3 Edition, a subset of STANAG 4370, 2006.
[8] Przybylinski, P., and Anderson, G., “Engineering Data Characterizing the Fleet of U.S. Railway Rolling Stock, Vol. II: Methodology and Data,” Federal Railroad Admin. Report FRA/ORD-81/75.2, Nov. 1981. [9] Shust, W.C., Smith, K.B., "Bounding Natural Frequencies in Structures II: Local Geometry, Manufacturing and Preload Effects," IMAC-XXII: A Conference on Structural Dynamics, January 2004, Dearborn, Michigan USA, Society for Experimental Mechanics, Inc.
BookID 214574_ChapID 114_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Identification of Nonlinear Systems with a State-Dependent ARX Model Robert W. Clark, Jr. Technical Fellow, The Boeing Company, 13100 Space Center Blvd, Houston, TX 77059 David C. Zimmerman Professor, Mechanical Engineering Dept., 4800 Calhoun Rd., Houston, TX 77204
ABSTRACT The utility of linear stochastic models is well known. By developing a straight forward, time-domain method of determining nonlinear coefficients for stochastic models, this paper expands the utility for nonlinear systems. A single-input multipleoutput nonlinear identification algorithm is formulated and demonstrated for systems that exhibit both soft and hard nonlinearities. The identification method is based on an Auto Regressive Exogenous stochastic model. The nonlinear characteristics of the system being identified are represented with coefficients that are a function of the output states. These coefficients are formulated using linear combinations of orthogonal vectors chosen from basis sets. The effects of noise on the input and outputs are minimized by utilizing a Generalized Least Squares algorithm. The developed identification method is demonstrated on a nonlinear numerical example with simulated corrupted measurements. NOMENCLATURE a b c e eo M N n P Q R (S N ) u uf u~
ARX model output coefficients ARX model input coefficients Filter coefficients from residual AR model Residual error Residual error from previous iteration Matrix containing the products of the basis with the input and output arrays Number of data samples Vector containing the values of 1 to number of data samples (N) Number of past output measurements in ARX model Number of past input measurements in ARX model Number of coefficients in AR filter Signal-to-noise ratio Input to system Scaled system input filtered with residual AR filter
um x xf
Measured input to system Input to discrete pre-filter Output of discrete pre-filter
y
Output of system
yf ~ y
Scaled system output filtered with residual AR filter
f
f
ym
α β γk
System input filtered with residual AR filter
System output filtered with residual AR filter Measured output of system Input basis scaling factor Output basis scaling factor Coefficient expansion basis
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_114, © The Society for Experimental Mechanics, Inc. 2011
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ηu ηy
σ( σn σS Ψ
Input signal noise Output signal noise
)
Standard deviation of argument Standard deviation of measured noise Standard deviation of measured signal Array of basis weighting coefficients
INTRODUCTION
The effectiveness of system identification for health monitoring and damage detection has been documented in publications for numerous years [1,2]. Historically, this effort has focused on assuming the system being evaluated is linear. This assumption is adequate for many systems over the operational envelope of interest, an example being small amplitude oscillations of a welded structure. This paper explores a method of identification of nonlinear systems focused towards, but not limited to, structures and mechanisms. These nonlinearities can be soft, as in geometric stiffening, and hard [3], such as bilinear springs and stiffness dead-bands generated by cracks opening and closing. Furthermore, the method can be applied to a Single-Input/MultipleOutput (SI/MO) systems with both the input and outputs corrupted with noise. IDENTIFICATION METHOD FOR TIME-INVARIANT NONLINEAR SYSTEMS
The value and simplicity of system identification of a time-invariant linear system using the ARX approach is well known. Using solution tools such as least-squares contributes to the simplicity. Researchers have expanded this type of identification to time-variant systems by making the coefficients functions of time [4 - 7], while retaining the structure of the problem to use least-squares to find the coefficients. Using this type of methodology, except making the coefficients a function of the system’s state, as opposed to time, will be developed in the following. Nonlinear Single-Input/Single-Output
If one had insight into a system to the level of knowing how its characteristics varied with its states, the task of identifying it would be straightforward. For instance, if it were known that a Single-DOF (SDOF) system had a spring that behaved like a cubic stiffness, it would be included in the setup of the formulation. Unfortunately, one rarely has this luxury. So a method of forming a general nonlinear system is needed. Currently a popular way of approaching this is by utilizing basis functions [5]. In [5] the methodology focused on time-varying systems. What follows is an adaptation of the method developed in [5] for time varying systems to systems that have time invariant nonlinear characteristics. The ARX model modified to have coefficients as a function of the results from the previous time step is y (n ) =
P
Q
i =1
j =0
¦ a(i, y(n − 1))y(n − i ) + ¦ b( j, y(n − 1))u (n − j ) + e(n) .
(1)
Expanding the coefficient using a basis set results in
a(i, y(n − 1)) =
V
¦α (i, k )γ ( y(n − 1)) k
k =0
b( j, y(n − 1)) =
V
(2) and
¦ β ( j, k )γ ( y(n − 1)) , k
(3)
k =0
where α and β are scalar weighting coefficients, γ is a basis function, and e is error due to white noise. Figure 1 shows examples of basis sets for use in Eqs. (2) and (3). The first two basis sets (Walsh functions and Block pulse) are well suited for modeling systems with hard nonlinear characteristics, where as Chebyshev polynomials are more appropriate for smooth nonlinear systems. Chebyshev polynomials have also proved useful for systems with excessive noise corruption. Prior to use in Eqs. (2) and (3), the basis need to be mapped to the range of y (0 becomes the minimum and 1 the maximum value of y).
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Figure 1. First Eight Walsh Functions, Block Pulses and Chebyshev Polynomials, respectively
Substituting Eqs. (2) and (3) into Eq. (1) results in y (n ) =
P
Q
V
V
¦ ¦α (i, k )γ ( y(n − 1))y(n − i ) + ¦¦ β ( j, k )γ ( y(n − 1))u(n − j ) + e(n) . k
k
i =1 k = 0
(4)
j =0 k =0
Eq. (4) can be simplified by defining two variable definitions y k (n − i ) = γ k ( y (n − 1)) y (n − i ) and
(5)
u k (n − j ) = γ k ( y (n − 1))u (n − j ) . Using the above variables transforms Eq. (4) to y (n ) =
P
Q
V
V
¦ ¦α (i, k )y (n − i ) + ¦¦ β ( j, k )u (n − j ) + e(n ) . k
i =1 k = 0
k
(6)
j = 0 k =0
Writing out the product of the basis with the input and output arrays yields M = [ y 0 (n − 1) ! yV (n − 1) y 0 (n − 2) ! y 0 (n − P ) ! yV (n − P ) ! u 0 (n ) ! uV (n − 1) u 0 (n − 2) ! u 0 (n − Q ) ! uV (n − Q )],
where
(7)
y k (n − i ) = γ k ( y (n − 1)) y (n − i )
= [γ k ( y (0 )) y (1 − i ) γ k ( y (1)) y (2 − i ) !
γ k ( y ( N − 1)) y (N − i )] , T
u k (n − j ) = γ k ( y (n − 1))u (n − j )
= [γ k ( y (0))u (n − j ) γ k ( y (1)) y(2 − j ) !
γ k ( y ( N − 1)) y (N − i )]
T
and n = 1 to N, where N is the number of time steps. The coefficients in Eq. (6) can be arranged in a vector as Ψ = [α (1,0 ) ! α (1,V ) α (2,0 ) ! α (P,V )
β (0,0) ! β (0,V ) β (1,0) ! β (Q,V )] . T
(8)
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Using the definitions shown in Eqs. (7) and (8) with Eq. (6), one obtains, y = MΨ + e .
If
(9)
M T M is nonsingular, the coefficients can be found using, Ψ = (M T M )M T y .
(10) T
If each of the columns of M are not linear independent and therefore M M is singular, a method must be applied to remove the columns of redundant information. In [5] they compare column by column to check that each one is linearly independent. A more efficient method of separating the linear independent vectors is by using QR factorization with pivoting [8]. Once the coefficients are found for Eq. (9), it is possible to construct the state-dependent coefficients for the ARX model, Eq. (6). This can be accomplished by using Eqs. (2) and (3) for a and b, respectively. Due to the model only being valid for the range of y that was used in the identification process, it should not be used to predict behavior outside that range. In the event y does go slightly outside the limits, the coefficients should not be extrapolated, but held at the value at the limit. Using the above outlined method, an algorithm can be written to generate a state-dependent Nonlinear ARX (NL-ARX) model for SI/SO systems. The following outlines the steps for such an algorithm. 1. 2. 3. 4. 5. 6. 7. 8.
Choose model parameters: model order (P & Q), basis set(s) to be used, and number of vectors to be used in basis set Construct yk and uk vectors Find the linear independent vectors in yk and uk Construct M matrix from the linear independent yk and uk vectors Solve for the α and β coefficients Construct the state-dependent NL-ARX coefficients a and b Truncate the coefficients above and below range of identification Reconstruct y to verify model
Example Nonlinear Single-Input/Single-Output System A SDOF system with a hard nonlinearity is used to demonstrate this method (Figure 2). The parameters used for this system are listed in Table 1. The forcing function is a summation of two sine waves u = sin ( 10.1 ⋅ t)+ sin ( 3.2 ⋅ t) and is shown in Figure 3 along with the response of the mass. y k c
m
u(force on mass)
khs
Figure 2. SDOF Nonlinear System with a Hard Stop Table 1. SDOF Nonlinear System Parameters Parameter
Value
m
1
k
20
c
0.5
khs (y < 0)
2000
(11)
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Figure 3. Nonlinear SDOF Example: Applied Force and Response The NL-ARX method outlined above was applied to the data in Figure 3 (sampled with a delta-time of 0.01). The parameters used in the identification are listed in Table 2 and the resulting coefficients are plotted in Figure 4, as a function of y. Walsh functions were chosen for the basis set due to their ability to duplicate discontinuities, which is demonstrated in Figure 4 by the discontinuity in the coefficients at zero displacement resulting from the hard stop. The optimum number of Walsh functions was found by performing a series of identifications with a range of values of Walsh functions (e.g., 1 to 16). The Root-Sum-Square-Error (RSSE) of the synthesized output compared to the original output was calculated for each identified model. The results from the case with the lowest RSSE were then compared visually with the original output to confirm the fit. Table 2. NL-ARX Parameters used in SDOF Nonlinear System Identification Parameter
Value
P
2
Q 1 Number of Walsh Functions 14 Number of Points in Basis Vectors 1000
Figure 4. Nonlinear SDOF Example: NL-ARX Coefficients as a Function of Displacement
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To demonstrate the validity of the identified model, the response was synthesized. It can be seen in Figure 5 that the synthesized response matches the original response very well.
Figure 5. Nonlinear SDOF Example: Synthesized Response Compared to Original The original system and the model constructed from the data were driven by an input other than the one used for the identification process as shown in Figure 6. These two outputs were than compared to further demonstrate the accuracy of the identified model. Again the match, shown in Figure 7, is close.
Figure 6. Nonlinear SDOF Example: New Forcing Function and Response of Original System
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Figure 7. Nonlinear SDOF Example: Response of Original and Identified System to Different Forcing Function The accuracy of the identification process can be greatly improved if data at a higher sampling rate is used. This will be demonstrated by repeating the previous example with a delta-time of 0.001 (the previous example used 0.01). The identification results in an even closer fit, which is nearly indistinguishable from the original system, as shown in Figure 8.
Figure 8. Nonlinear SDOF Example: Identification of System with 0.001 Sampling Period The above exercise demonstrated how the accuracy of the identified model can be improved by using a smaller time-step. For the approximate eight and one-half cycles in Figure 8 smaller time step equates to nearly 600 data samples in each oscillation. Although this level is achievable with today’s data acquisition systems, it is excessive, so the remainder of the identification performed will be closer to 100 data samples in each oscillation. If increased accuracy is required when applying the methods addressed, a smaller time step is an option. Nonlinear Single-Input/Single-Output with Corrupted Data Up to this point only idealized input/output signals with no noise have been used, which is never the case in practice. Due to many different reasons, measurements are often corrupted with noise. The method developed up to this point does not perform well when uncorrelated noise is superimposed on the input and output signals. To illustrate this, simulated noise was used to corrupt the signals shown in Figure 3. Figure 9 schematically shows how this was performed. The noise is represented by η, and the subscript m represents the measured signal. The noise was constructed using a random signal with maximum and minimum values chosen to result in an 80 dB Signal-to-Noise Ratio (SNR), as calculated by [9]
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(S N ) = 10 log
10
(12)
§σ s2 · ¨ σ2¸ . n ¹ ©
Separate uncorrelated noise models were constructed for the input and output. um
ηu
ym
ηy
Σ
unknown system
u
Σ
y
Figure 9. Illustration of Measurement Noise The same parameters listed in Table 2 were used with the NL-ARX method developed and data corrupted as explained above. The identified parameters resulted in the synthesized output shown in Figure 10. Although the method successfully identified the higher stiffness for displacements less than zero, the general response does not match the data from the original system.
Figure 10. Nonlinear SDOF Example with Noise: Synthesized Response Compared to Original using NL-ARX To reduce this effect, the ability to accommodate data with noise was needed. Therefore, an AR model of the noise was added to the algorithm. The identified noise model was then used to filter the input and output signals. This approach is referred to as Generalized Least Squares (GLS) [10, 11]. After the coefficients are found for Eq. (9) using a least-squares method, the residuals can be found by solving e = y − MΨ .
(13)
An AR model of the error can be constructed with the residuals, e(n ) =
R
¦ c(l )e(n − l ) .
(14)
l =1
The coefficients found for Eq. (14) can now be used as a filter for the input and output signals as ~ y f (n ) =
R
¦ c(l )y(n − l ) and l =1
(15)
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u~ f (n ) =
R
¦ c(l )u(n − l ) . l =1
The filtering influence of Eq. (15) reduces the noise content; however, it also linearly scales the data. The application of GLS in [11] is to linear systems, so this weighting phenomenon does not adversely affect the identification of the system. This is not the case when applying it to nonlinear systems. Therefore, a scale factor has to be applied to the filtered data. The scale factor is calculated by dividing the standard deviation (σ) of the original output data by that of the filtered, scale =
σ (y) . σ (~y f )
(16)
Multiplying both the filtered input and output from Eq. (15) by this scalar value restores the data to its original range, y = scale * ~ y and f
f
(17)
u f = scale * u~ f . By recalculating the coefficients of Eq. (9) using the filtered results from Eq. (17), the accuracy of the fit is increased. Repeating the process until the solution converges further increases the accuracy of the fit. Convergence has been achieved when the results from the two most recent iterations generate the same residuals. Likeness of two residuals is calculated using test =
e T eo − eoT eo eoT eo
,
(18)
where the subscript “o” represents the residuals from the previous iteration step. In practice, the best results have been achieved by setting a tolerance on Eq. (18) of less than 1.0 × 10 −9 . In addition to the identified AR filter, it is also beneficial to apply a pre-filter to both signals. The simple discrete filter
x nf +1 = (x n+1 + x nf ) 2
(19)
has proven to be adequate in reducing the noise without masking the systems characteristics. The NL-ARX method outlined in earlier can be updated with both the pre-filter and GLS routine to generate a statedependent NL-ARX-GLS model for SI/SO systems with corrupted data. The following outlines the additional steps needed for a NL-ARX-GLS algorithm. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
Pre-filter the input and output signals with Eq. (19) Choose model parameters: model order (P, Q & R), basis set(s) to be used, and number of vectors to be used in basis Construct yk and uk vectors with the filtered data Find the linear independent vectors in yk and uk Construct M matrix from the linear independent yk and uk vectors Solve for the α and β coefficients Find the residuals using Eq. (13) Filter the input and output using GLS Using Eq. (18), check for convergence If the algorithm has not converged and number of iterations is less than the maximum allowed, go to Step 3 and use the filtered data from Step 8 to create the vector pool, otherwise go to Step 11 Construct the state-dependent NL-ARX coefficients a and b Truncate the coefficients above and below range of identification Reconstruct y to verify model
Example Nonlinear Single-Input/Single-Output with Corrupted Data
To demonstrate the ability of the NL-ARX-GLS method, it was applied to the corrupt data from the above example. The parameters used for the identification are listed in Table 3. Chebyshev Polynomials were used instead of Walsh functions because identifications performed revealed they often work well with noisy data.
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Table 3. NL-ARX-GLS Parameters used in SDOF Nonlinear System Identification with 80 dB Noise on Input and Output Parameter
Value
P
2
Q
1
R Number of Chebyshev Polynomials Number of Points in Basis Vectors
8
1000
GLS Tolerance
1.0E-09
Maximum Iterations
100
19
As with the previous example, to demonstrate the validity of the identified model, the response was synthesized. It can be seen in Figure 11 that the synthesized response does not match as close as the clean data (Figure 5), but much better than the attempt of matching corrupted data with the NL-ARX method (Figure 10).
Figure 11. Nonlinear SDOF Example with Noise: Synthesized Response Compared to Original using NL-ARX-GLS Nonlinear Single-Input/Multiple-Output with Corrupted Data
Unlike linear systems, some nonlinear systems dictate a level of spatial resolution of outputs to adequately perform system identification. For instance, if a multiple degree of freedom system contains a softening spring, a single input and output would not provide enough information to accurately identify the nonlinear characteristics; however, a measurement of both sides of the spring would. For this reason, a multiple output method more accurately captures the characteristic of nonlinear systems. The method developed for the identification of SI/SO nonlinear systems with input/output signals corrupted with noise (NLARX-GLS) was extended to SI/MO systems. This was accomplished by using Eq. (1) for each output and adding to it terms to represent nonlinear influences from other outputs. Coefficients, which are a function of the difference between outputs, are applied to the quantity resulting from differencing the output of interest with another output. It is recognized that this will not represent all possible nonlinear combinations, but since this is focused on mechanical systems, it should suffice. For a more general form, the influence of each output as well as products can be added to the model. With the above-mentioned conditions, the SI/SO model shown in Eq. (1) is transformed to
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y h (n ) = +
P
¦ i =1
mout −1
ª
Q
¦ b( j, y (n − 1))u (n − j )
(20)
h
j =0
º
P
¦ «¬¦ a(l * P + i, y (n − i ) − y (n − i ))( y (n − i ) − y (n − i ))»¼ + e(n) , h
l =1
where
a (i, y h (n − 1))y h (n − i ) +
r
h
r
i =1
if l < i i r=® . i + 1 if l ≥ i ¯
The best results have been achieved by performing the identification on each output individually using the other outputs measured data during the identification. The process is similar to the method used to find the optimum number of basis functions for SI/SO systems, with the number of basis functions for the output being identified, as well as for the coefficients being the variables. The number of basis functions for the input is kept the same as that of the output of interest.
Example of Nonlinear Single-Input/Multiple-Output System To demonstrate the SI/MO NL-ARX-GLS identification algorithm, a three-DOF system with two nonlinearities was created (Figure 12). The two nonlinearities consist of a cubic spring (k3) and a hard stop (khs); the values of the parameters are shown in Table 4. The system was excited with the input on mass 2 (21)
u = 100 * sin ( 10.1 ⋅ t)+100 * sin ( 3.2 ⋅ t) . y1
k2
k1 m1
y3
y2
m2
c1
c2 khs
k3 m3
c3
u (force on mass one)
Figure 12. MDOF Nonlinear System with Single Input Table 4. MDOF Nonlinear System Parameters Parameter
Value
m1
1
c1
5
k1
10
m2
1
c2
2
k2
1
khs (y2 – y3 > 0)
2000
m3
1
c3
4
k3 (cubic spring)
10
As with the SDOF example above with noise, noise was added to the outputs and the input, and were on the order of 80 dB SNR.
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The identification algorithm was first applied to y1 with the parameters listed in Table 5. The first four Chebyshev functions were used for both the basis set for y1 and y1 – y2. The coefficients for y1 – y3 were assumed to be constant, due to having insight into the system that these physical elements are indirectly connected to each other. Figure 13 contains the results from the identified model. It should be noted that when creating the results shown in Figure 13, only y1 was synthesized, the measured values for y2 and y3 were used.
Table 5. SI/MO NL-ARX-GLS Parameters used to Identify Output 1 Parameter
Value
P
2
Q
1
R Number of Chebyshev Polynomials in Basis for y1 Number of Chebyshev Polynomials in Basis for y1 – y2 Coefficients for y1 – y3 Constant Number of Points in Basis Vectors GLS Tolerance
8
1.0E-09
Maximum Iterations
100
4 4 N/A 1000
Figure 13. SI/MO Example with Noise: Synthesized Results Compared to Original for y1 using the Original y2 and y3 Results The identification algorithm was then applied to y2 with the parameters listed in Table 6. The best model was identified using 14 Block Pulse functions for y2, four Chebyshev Polynomials for y2 – y1 and 13 Block Pulse functions for the y2 – y3 basis sets. Figure 14 contains the results from the identified model and as with y1, the results shown are for only y2 being synthesized, the measured values for y1 and y3 were used.
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Table 6. SI/MO NL-ARX-GLS Parameters used to Identify Output 2 Parameter
Value
P
2
Q
1
R 8 Number of Block Pulse Functions in Basis 14 for y2 Number of 4 Chebyshev Polynomials in Basis for y2 – y1 Number of 13 Block Pulse Functions in Basis for y2 – y3 Number of Points in 1000 Basis Vectors GLS Tolerance 1.0E-09 Maximum Iterations
100
Figure 14. SI/MO Example with Noise: Synthesized Results Compared to Original for y2 using the Original y1 and y3 Results Finally, the identification algorithm was applied to y3 with the parameters listed in Table 7. The best model was identified using four Chebyshev Polynomials for y3, four Block Pulse Functions for y3 – y1 and 15 Block Pulse functions for the y3 – y2 basis sets. Figure 15 contains the results from the identified model and as with the other two outputs, the results shown are for only y3 being synthesized.
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Table 7. SI/MO NL-ARX-GLS Parameters used to Identify Output 3 Parameter
Value
P
2
Q
1
R 8 Number of Chebyshev Polynomials in Basis 4 for y3 Number of 4 Block Pulse Functions in Basis for y3 – y1 Number of 15 Block Pulse Functions in Basis for y3 – y2 Length of 1000 Basis Vectors GLS Tolerance 1.0E-09 Maximum Iterations
100
Figure 15. SI/MO Example with Noise: Synthesized Results Compared to Original for y3 using the Original y1 and y2 Results Applying the forcing function in Eq. (21) to the identified model, Eq. (20) with the coefficients found, outputs y1, y2 and y3 were synthesized simultaneously (i.e. original data not used). The synthesized results matched the original quite well, as can be seen in Figure 16.
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Figure 16. SI/MO Example with Noise: Full Synthesized Response Compared to Original using NL-ARX-GLS To demonstrate the validity of the identified model, both the original system and the model constructed from the data were driven by an input other than the one used for the identification process, see Figure 17. Comparison of these results, shown in Figure 18, shows the accuracy of the identified model.
Figure 17. SI/MO Example with Noise: Measured and Applied Force to Identified Model
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Figure 18. SI/MO Example with Noise: Response of Original and Identified System to Different Forcing Function Conclusion A method for performing system identification of nonlinear single-input/multiple-output systems has been developed (SI/MO NL-ARX-GLS). The method utilizes basis sets to form a pool of linear independent vectors. Coefficients are found, using least squares, to form linear combinations of the vector pool to generate ARX coefficients that are a function of the states (outputs) of the system. Using a filter created with the residuals, the effects of noise on the input and outputs are reduced. The process is then repeated by recalculating the coefficients using the filtered input and outputs until the residuals converge. This is performed on each output independently. The SI/MO NL-ARX-GLS identification method was demonstrated on a multiple degree-of-freedom nonlinear numerical example. The accuracy of the identified model was demonstrated by the close match of results from the truth model to the identified model using a different input than what was used for the identification. The benefits of the developed method are that it is capable of producing a state-dependent ARX model with discontinuous coefficients to reproduce the characteristics of a nonlinear system. This can be accomplished using corrupted input and output data.
References 1. 2. 3. 4. 5. 6. 7. 8.
Doebling, S.O, Farrar, C., Prime, M., and Shevitz, D., Damage Identification and Health Monitoring of Structural and Mechanical Systems From Changes in Their Vibration Characteristics: A Literature Review, Los Alamos National Laboratory Report LA-13070-MS (May 1996) Sohn, H., Farrar, C., Hemez, F., Czarnecki, J., Shunk, D., Stinemates, D. and Nadler, B., A Review of Structural Health Monitoring Literature: 1996-2001, Los Alamos National Laboratory Report, LA-13976-MS, 2004 Slotine, J.J.E. and Weiping, L. (1991), Applied Nonlinear Control, Prentice Hall, New Jersey Lu, S., Ju, H. and Chon, K., A New Algorithm for Linear and Nonlinear ARMA Model Parameter Estimation Using Affine Geometry, IEEE Transitions on Biomedical Engeineering, Vol. 48, No. 10, October 2001 Zou, R., Wang, H. and Chon, K., A Robust Time-Varying Identification Algorithm Using Basis Functions, Annals of Biomedical Engineering, Vol. 31, pp. 840-853, 2003 Lu, S. and Chon, K., Nonlinear Autoregressive and Nonlinear Autoregressive Moving Average Model Parameter Estimation by Minimizing Hypersurface Distance, IEEE Transactions on Signal Processing, Vol. 51, No. 21, December 2003 Zou, R. and Chon, K., Robust Algorithm for Estimation of Time-Varying Transfer Functions, IEEE Transactions on Biomedical Engineering, Vol. 51, No. 2, February 2004 Golub, G. and Van Loan, C., Matrix Computations, 3rd Edition, The Johns Hopkins University Press, Baltimore, MD, 1996
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9.
Bendat, J. and Piersol, A., Random Data Analysis and Measurement Procedures, 3rd Edition, Wiley Series in Probability and Statistics, John Wiley & Sons, Inc., 2000 10. Ljung, L., System Identification – Theory for the User, Prentice-Hall, New Jersey, 1999 11. Clarke, D., Generalized Least Squares Estimation of Parameters of a Dynamic Model, First IFAC Symposium on Identification in Automatic Control Systems, Prague, 1967
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Model Identification for a Modal State Estimator from Output-Only Data Stefan Engelke, Christoph Schaal and Lothar Gaul Institute of Applied and Experimental Mechanics, University of Stuttgart Pfaffenwaldring 9, 70569 Stuttgart, Germany
ABSTRACT Precise information of the current state of motion of a vibrating system is crucial for every effective active vibration control. Due to the limitation of the number of sensors in the final operational setup, a state estimator is needed to reconstruct the current state of vibration at important degrees of freedom from a small set of sensors. If the excitation of the structure (system input) cannot be measured directly, a sensor based estimator has to be used, which operates on response data only (system output). Core piece of this estimator is a reduced model of the vibrating system. If additionally some sensors are only capable to measure spatially distributed deflection instead of the deflection at single degrees of freedom (like piezo-electric foils), the reduced model must incorporate both those sensors as well as the important degrees of freedom as outputs. For structures which cannot be excited by a measurable excitation even in the development phase, the reduced model can still be identified by an output-only modal analysis. In this work, the identification technique is demonstrated for a curved plate submerged in water and exited by a surrounding sound field.
Nomenclature A, C, L Cd , Cs Q, R, S M, K, D λi ,ωi ,ζi ψi ,θi MA,i , MB,i , Qi ... ...∗ , ...T , ...∗T q, z, η xt yd , ys Λi , Σ, G E {...} δts
1
State transition matrix, output matrix, update matrix Partitions of output matrix for structural displacements and laminar sensor Covariance matrices of system and output noise Mass matrix, stiffness matrix, damping matrix Eigenvalue, natural frequency, damping ratio of mode i Full mode shape and mode shape mapped to output space of mode i Modal A, modal B, modal scaling factor of mode i Diagonal matrix with ... as diagonal elements Complex conjugate, non-conjugate transpose, complex conjugate transpose Vector of structural displacements, state vector of symmetric form, modal state vector State vector of discrete-time model Vector of measured structural displacements and laminar sensor Output covariance matrix, state covariance matrix, next-state output covariance matrix Expected value of random variable ... Kronecker delta.
Introduction
In this paper, the design of a state observer for estimating structural vibration is considered, for cases where the observed structure can only be analyzed by looking at system responses. As design technique for state estimation the well known Kalman filter is used. The technique of reconstructing a state sequence of a dynamical system by observing the output has been established as a powerful tool in control theory [1, 2, 3]. Due to the increased computing power of microcontrollers nowadays, these techniques become even more appealing in active vibration control. Reconstructing the state of vibration
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_115, © The Society for Experimental Mechanics, Inc. 2011
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may significantly reduce the number of needed sensors and even expand the field of applications, where sensors are difficult to place. Core piece of the estimator is a reduced model of the vibrating system. Modeling the structure may be done analytically for simple beam and plate structures, numerically by discretization techniques or identified from measurement data. To utilize the model for a state observer, the dynamics should be reduced to a minimal amount, to allow an execution in realtime. Specifying the dynamics of vibration by modal parameters is a commonly used and well-understood technique in structural dynamics. This representation is also suitable for reducing the whole dynamics to a required minimum, by selecting a range of frequencies and damping ratios to include. For identification of modal parameters from measurement data a large spectrum of tools for experimental modal analysis has been established [4]. Recently emphasis of research also includes techniques for extracting modal parameters from output-only data [5, 6, 7]. In output-only modal analysis only the structural response is measured, while the excitation is unknown. Under certain assumptions of the excitation, modal parameters can still be identified. Though the identified parameters do not include modal scaling factors, these methods are suitable for constructing a state-observer. In this work an overview of nomenclature for modal parameters is given by deriving the modal model in state-space form. The state-space form enables easy setup of an appropriate model for designing a Kalman filter. The proposed procedure is applied at a test structure for evaluation. For this purpose a ship-like structure is analyzed in the hydro-acoustic laboratory at the Institute of Applied and Experimental Mechanics, University of Stuttgart. The modal parameters of the structure are identified by an output-only modal analysis. For excitation an underwater loudspeaker is used. With the identified modal parameters a state observer is designed, which is capable of reconstructing structural displacements at a predefined measurement grid from only one sensor signal. Section 2 starts with a short overview of the Kalman filter, derives the modal state-space model of a vibrating structure and states the main idea behind balanced realization for output-only modal analysis. Section 3 applies the procedure at a test structure and discusses the results. Section 4 concludes the paper.
2 2.1
Theory State Estimation of Vibrating Structures
To estimate the current state of vibration with a state observer a dynamical model of the observed system is needed. This model is simulated in realtime and produces a virtual measurement signal. The difference between real and virtual measurements is feed back and updates the simulation. Figure 1 shows the signal structure of a state observer for a linear system without input. The corresponding equation writes xt + L(yt − C x ˆt ), x ˆt+1 = Aˆ
(1)
where A and C are the state transition and output matrix identified with the observed system and L is an update matrix. For the simple linear observer (1) several design methods have been established to calculate L in respect of an optimal update. The well known Kalman filter determines an optimal update matrix under the presence of a system and sensor noise wt , vt with zero mean and known covariances: T Q S xt+1 = Axt + wt E{wt } = 0 wt ws δts , (2) E = E{vt } = 0, yt = Cxt + vt , vt vs ST R From this system definition an optimal update matrix is calculated as L = (AP C T + S)(CP C T + R)−1 ,
(3)
where P is the solution of the algebraic Riccati equation P = AP AT − (AP C T + S)(CP C T + R)−1 (AP C T + S)T + Q.
(4)
Having a model of a vibrating structure in the form (2) enables the calculation of a state observer (1) with an optimal update matrix. Sections 2.2 discusses, how an output-only modal analysis can be utilized to setup an appropriate model.
2.2
Model Identification
In the last decade a variety of analysis techniques in frequency domain as well as in time domain has been developed to identify modal parameters of a structure from output-only data. Since the different algorithms provide their results in
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sensor model
simulation update
C
yˆt yt
+
-
L
+ xˆt+1 +
x ˆt
1 z
A system model Figure 1: Signal structure of sensor based, linear state observer. different notations, this section will briefly derives the dependencies. The dynamics of the vibrating structure is assumed to be modeled sufficiently at N discrete degrees of freedom M¨ q + D q˙ + Kq = u yd Cd = q ys Cs
q ∈ RN
(5a) (5b)
where the symmetric matrices M, K and D are the mass, stiffness and damping matrix, respectively. q is the vector of displacements and u the vector of excitation forces. The signals of interest are the structural displacements yd which may be defined at a different number of locations than degrees of freedoms, especially if the model is identified with an experimental modal analysis. The matrix Cd maps the nodes between those two meshes. Furthermore an additional sensor signal ys with its corresponding output matrix Cs is included, which models a general displacement sensor like a piezo-electric foil. This signal will be used to estimate the state of vibration in a later step. T Introducing the state vector z = q T q˙T system (5) can be written as a symmetric model of first order
D M K 0 I z˙ + z= u M 0 0 −M 0 yd Cd 0 = z. ys Cs 0
z ∈ R2N
(6a) (6b)
The corresponding generalized eigenvalue problem
D M M 0
φi λi +
K 0
0 −M
φi = 0
(7)
yield N pairs of complex conjugated eigenvalues {λi , λ∗i } and eigenvectors {φi , φ∗i }. Since each eigenvector is orthogonal with respect to the inner product induced by both system matrices (as discussed in [8]) the following factorizations are valid: D M K 0 T T Φ Φ = MA,i Φ Φ = MB,i , (8) M 0 0 −M where Φ = [ φ1 ... φN ] is the eigenvector matrix and MA,i , MB,i are diagonal matrices containing the so called modal A parameters MA,i and the modal B parameters MB,i . Modal A and modal B of the system depend on the norm of the eigenvectors, which may be scaled arbitrary. In classical theory of modal analysis there are different norms for scaling the eigenvectors. The most common one is a normalization of the modal A parameters. Dealing with output-only data no normalization can be done, which will become clear when looking at the modal representation of the system. Using the eigenvector matrix as transformation basis z = Φη, multiplying system equation (6a) with Φ∗T and using the fact, that
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the eigenvectors are composed as Φ =
Ψ Ψ λi
η˙
yield the modal representation of the system
MB,i − η MA,i
=
yd ys
λi
=
+
1 Ψ∗T u MA,i
(9a)
Qi
Cd Ψ η, Cs
(9b)
Θ
where Qi , the reciprocal of modal A is the modal scaling factor and Ψ = [ ψ1 ... ψN ] are the mode shapes. Θ = [ θ1 ... θN ] are the mode shapes mapped to the output space. From the set of equations (9) it becomes clear that the modal scaling factors can only be identified, if the input as well as the output is measured. This means that the mode shapes ψi as well as the mapped mode shapes θi are arbitrary scaled in the output-only case. However arbitrary scaled mode shapes together with the eigenvalue λi are sufficient to assemble a state-space system, needed for an observer design. The complex eigenvalues λi may be expressed by the natural frequencies ωi and the critical damping ratios ζi by λi , λ∗i = ωi −ζi ± i 1 − ζi2 . (10) With this parameters, identified by an output-only modal analysis, the system equations can be assembled. Since the observer design presented in section 2.1 is based on a discrete-time model, it is suitable to write system (9) without input as a discrete-time model: λi Δt xt+1 = e xt (11a) A yd = Θ xt , (11b) ys C
where Δt is the sampling time of the desired state observer. System (11) with sensor output ys is now used to design a state observer. The output yd is taken to calculate structural displacements from the estimated state xt .
2.3
Balanced Realization
As an example of an algorithm for output-only modal analysis the balanced realization is briefly introduced. This algorithm belongs to the group of covariance-driven identification methods, since it is based on the signal covariances instead of the raw time sequences. Thus the first step is the estimation of the output covariance matrices Λk from measured signals yk . An empirical estimate is given by Λk = E{yi+k yiT } =
N −k 1 yi+k yiT N −k
k = 0, 1, ....
(12)
i=1
On the other hand the covariance matrices can also be derived analytically from a model given in equation (2): ⎧ ⎨ for k < 0 : GT (Ak−1 )T C T T Λk = E{yi+k yi } = for k = 0 : C Σ C T + R ⎩ for k > 0 : CAk−1 G
(13)
T where the state covariance matrix Σ = E{xk xT k } and next state-output covariance matrix G = E{xk+1 yk } are given by
Σ = A Σ AT + Q T
G = AΣC + S
(14) (15)
As described in [2, 9, 10] this dependency can be utilized to identify the state-transition matrix A, output matrix C as well as the covariances of the system and sensor noises Q, R, S.
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500 mm
20 mm
460 mm
510 mm 355 mm
R=420 mm
0
10
−5
10
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Figure 3: Measured and synthesized power spectra densities for acceleration at measurement point 27 and PVDF signal.
3
Experiment
For experimental validation of an observer design with modal parameters of an output-only modal analysis, a reference structure is chosen as shown in Figure 2. It consists of a rectangular aluminum frame and a 1 mm curved plate of stainless steel. For the test setup the structure is partially immersed in water and excited by underwater sound. In [10] it has been shown, that this setup is suitable for an output-only modal analysis of the structure. Section 3.1 summarizes the results of the modal analysis, while in the following section 3.2 the identified modal parameters are used for the design of a state observer.
3.1
Modal Analysis
The output-only modal analysis is carried out by defining a measurement grid of 7 × 11 equally spaced measurement points on the inner side of the curved plate. Two acceleration sensors are mounted to fixed positions of the measurement grid, while one additional acceleration sensor with magnetic base is used as a roving sensor, measuring at a different position each setup. Doing so at all 75 remaining measurement points, the resolution of the identified mode shapes can be merged to span all 77 points of the measurement grid. Since the state observer should estimate the state of vibration from a simple sensor signal, a foil of polyvinylidene fluoride (PVDF) is pasted on the inner side of the plate (figure 4). The signal of this laminar sensor is treated as an additional measurement point in the grid, so the basis of the identified mode shapes also includes this virtual degree of freedom. To calculate the modal parameters from output-only data a balanced realization is utilized. As an advantage of this algorithm, it also estimates the covariance matrix of the excitation noise. This allows to synthesize the power spectrum density (PSD) of each output, which can used to visually validate the identification result. Figure 3 shows the measured and synthesized PSD for the acceleration signal at a selected measurement point 27 and for the PVDF sensor. The identified model contains 20 modes in the frequency range between 50 and 1000 Hz. Figure 5 shows the mode shape of one mode at 250 Hz as an example of the spatial quality.
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Figure 5: Mode shape of mode at 250Hz identified from output-only data. Figure 4: Laminar sensor pasted on the test structure. state observer test structure PVDF signal
+
sensor model
acceleration sensor excitation by underwater sound
structure model
output selection comparison for evaluation Figure 6: Test setup.
3.2
State Estimation
In a next step the identified modal parameters are used to assemble a state-space system given in equation (11). For designing a Kalman filter as described in section 2.1, specifications about the system noise wt and sensor noise vt are needed. In the special case of a balanced realization the corresponding covariance matrices are also identified and available for the observer design. If they are not available they can be seen as design parameters, though loosing guarantee of optimality. In this case the cross-covariance matrix S is set to zero and only the diagonal elements of the system noise covariance Q and sensor noise covariance R are estimated. Each diagonal element of Q gives a weighting factor of how fast a certain mode is updated, where a higher value updates the corresponding mode faster. Each diagonal element of R gives a weighting factor of how well each sensor can be trusted, where a higher value weights the corresponding senor less. Even though the balanced realization provides all covariances, in the following only the system noise covariance Q is used from the identification process. S is set to zero and R is used as design parameter, which is a single scalar parameter in the case of only one sensor. With this parameter the convergence of the observer can be adjusted to deal with non-modeled effects. For evaluation of the resulting state observer an independent measurement is taken. The setup for measuring evaluation data is shown in figure 6. Only the signal of the PVDF sensor and the acceleration at a selected point 27 on the measurement grid is captured. The PVDF signal is fed into the state observer, which estimates a sequence of modal states. With the corresponding output matrix the estimated acceleration at the selected point 27 is reconstructed. Figure 7 shows the measured signals overlaid by the estimated ones. Since the state observer starts with zero initial states the curves differ at the beginning. After 50ms a good convergence can be observed. This convergence time can be tuned by modifying the assumed covariance of the sensor noise R. Doing so one has to keep in mind, that the observer may become unstable for the chosen sampling time Δt. After the convergence phase the observer gives a good estimate of the structural acceleration at any point on the measurement grid.
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4
Conclusion
In this paper, state estimation has been discussed for vibrating structures with few, or in the optimal case, only one single sensor attached. It has been shown that a modal model identified by an experimental modal analysis is suitable for designing a Kalman filter. Even an output-only modal analysis, where only the structural response is analyzed, yields a model of sufficient quality. Model assembling from modal parameters has been shown by deriving the diagonal model from a finite-element formulation. To demonstrate the procedure a ship-like test structure is analyzed. Exciting the structure by underwater sound enables an output-only modal analysis from measurements of accelerations on a predefined grid. A laminar sensor of polyvinylidene fluoride is used as an additional sensor in the identification process. With a state observer, fed by the signal of this laminar sensor, structural accelerations at all grid points is estimated. For adjustment of convergence time the covariance of the sensor noise is used as a design parameter. Further work will deal with transformation and scaling of the observer equations for efficient execution on a microcontroller without floating-point unit.
References [1] Thomas Kailath. An innovation approach to least-squares estimation part i: Linear filtering in additive white noise. IEEE Transactions on Automatic Control, 13(6), 646-655, 1968. [2] Peter Van Overschee and Bart De Moor. Subspace identification for linear systems: theory - implementation - applications. Kluwer Academic Publishers, Boston/London/Dordrecht, 1996. [3] Leonard Meirovitch. Analytical methods in vibrations. Macmillan, New York, 1967. [4] D.J. Ewins. Modal Testing - theory, practice and application - second edition, volume 0. Research Studies Press Ltd., 2000. [5] G.H. James, T.G. Carne, J.P. Lauffer, and A.R. Nard. Modal testing using natural excitation. In Proceedings 10-th IMAC, San Diego, 1992.
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[6] L. Hermans and H. Van der Auweraer. Modal testing and analysis of structures under operational conditions: Industrial applications. Mechanical Systems and Signal Processing, 13:193–216, 1999. [7] B. Peeters and G. De Roeck. Stochastic system identication for operational modal analysis: A review. Journal of Dynamic Systems, Measurement and Control, 123:659–667, 2001. [8] Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. Van der Vorst. Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide. SIAM, Philadelphia, 2000. [9] Tohru Katayama. Subspace Methods for System Identification. Springer, London, 2005. [10] S. Engelke and L. Gaul. Output-only modal analysis of submerged structures excited by underwater sound. In 16th International Congress on Sound And Vibration, Krakow, 2009.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Data Acquisition for a Bridge Collapse Test Kurt Veggeberg, Business, Development Manager ([email protected]) National Instruments,11500 N. Mopac C, Austin, TX 78759 ABSTRACT - Researchers at the Ferguson Structural Engineering Laboratory successfully collapsed a 120-foot bridge with an intentionally fractured girder to study behavior and safety. It required three rounds of testing before the damaged bridge finally failed under an applied load of more than 360,000 pounds. The bridge, tested to determine its vulnerability to collapse following the fracture of a girder, withstood about 4.5 times the maximum legal truck load. Nearly 300 strain gauges and displacement transducers as well as recorded how the bridge reacted in this extremely damaged condition to incrementally increasing loads during the two rounds of testing. Wireless data acquisition modules were used to monitor the condition of cranes used to apply loads. This presentation is an overview of the methodology used to determine whether or not this design is fracture critical as is commonly assumed. Introduction The Texas Department of Transportation and the Federal Highway Administration funded a large-scale research project through the Ferguson Structural Engineering Laboratory at the University of Texas to develop methods for evaluating the redundancy of fracture critical steel bridges. (Figure 1) As part of the research project, a full-scale twin box-girder steel bridge representative of fracture critical bridges in Texas was decommissioned from the highway system in Houston, rebuilt at Ferguson Lab, and prepared for testing. (Figure 2)
Figure 1.Twin box-girder steel bridge (Ferguson Structural Engineering Laboratory) T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_116, © The Society for Experimental Mechanics, Inc. 2011
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1314 A series of three experiments were performed on the test-bridge to observe its response to a fracture of one of its bottom flanges. The AASHTO (American Association of State Highway Transportation Officials) Bridge Design specifications define a fracture-critical member as a component in tension whose failure is expected to result in the collapse of a bridge. To avoid the catastrophic collapse suggested by the specifications, bridges with fracture-critical members are subjected to frequent and stringent evaluation and inspection. Texas with over 166 twin box-girder steel bridge units has never found a fatigue related fracture. Instances of such two-girder bridges that have experienced fracture without collapse have prompted research to determine the level of redundancy that can be expected in twin box-girder bridges.The data gathered during the test were compared to the calculated response from the model to verify the predictive capabilities of the model. If able to predict response accurately, a computer model could be used during design to indicate the presence of redundancy and the decreased need for frequent inspection of this type of bridge resulting in significant savings to the bridge owner. The test set-up and corresponding models represented a worst-case scenario for loading on the structure at the time the simulated fracture was initiated.
Figure 2.Test Bridge at Ferguson Structural Engineering Laboratory
First Full-Scale Test The first test on October 21, 2006 used explosives to induce a complete fracture in one of the bridge’s bottom flanges. (Figure 3) Despite having a load equivalent to a 76,000 lb. truck positioned directly above the mid-span fracture location, the fracture did not propagate into the webs of the girder, minimal deflections were observed, and there was no significant degradation in the capacity of the structure despite the loss of a fracture critical element. (Figure 4)
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Figure 3.Explosive shaped charge applied to test bridge
Figure 4.Bridge fracture after explosion. As important as executing the full-scale test safely and to specification was ensuring that data were acquired during the experiment so that the behavior of the bridge could later be analyzed. The instrumentation plan was designed and implemented to measure deflections and material strains, which could then be used to help quantify material stresses along portions of the hypothesized redundant load paths. Strain gages attached directly to bridge components took measurements of material deformations. Wires connected to each piece of instrumentation were run to the southern end of the bridge where a small hut housed all of the data-acquisition equipment (Figure 5). For the first full-scale test, a high-speed data-acquisition system was configured for all 127 channels of instrumentation (Figure 6). Because the loading and subsequent bridge response was expected to be dynamic, it was important to sample the data rapidly, accurately, and in a synchronized manner. The system used for the test employed equipment from National Instruments and was set up to sample data simultaneously from all 127 channels, 1000 times each second. The wires from the 127 instrumentation channels were connected into sixteen National Instruments SCXI-1314 8-channel terminal blocks. Each terminal block hooked into its own SCXI-1520 8-channel universal strain module. Two SCXI-1001 12-slot chassis were used to
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1316 house the sixteen modules.. The two chassis were connected through a PC that was equipped with a National Instruments PCI-6250 data acquisition card. LabVIEW was used on the PC to view and organize the incoming data.
Figure 5.Wires connected to instrumentation
Figure 6.Data Acquisition System
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1317 The static deflections of the intact and fractured girders were relatively small. The mid-span of the intact girder deflected downward 0.5 in. under the initial application of the simulated truck live load, then deflected an imperceptible amount following the fracture of the opposite girder, and finally rebounded 0.25 in. when the live load was released. The deflection of the mid-span of the fractured girder follows a similar pattern. It first deflected downward 1.25 in. under the live load, then deflected a very small amount following the fracture of its own bottom flange, and finally rebounded 0.4 in. when the live load was released. A total deflection of 1 in. 2 in. across a 120 ft. span is considered small. The bridge performed exceedingly well relative to the AASHTO fracture critical designation. (Figure 7)
Figure 7.Girder deflections for test 1. The Second Full-Scale Test The second test shored the damaged girder while the fracture was manually extended to the full depth of the webs. Afterward, the same design load of approximately 76,000 lbs. used in the first test was placed above the location of the full-depth fracture. The shoring system was removed nearly instantaneously with the use of explosives, and the bridge was allowed to respond dynamically to its damaged condition. Substantial deflections and damage were observed, but the bridge resisted collapse and maintained complete serviceability. The data-acquisition system used for the 127 channels from the first test was expanded to accommodate the addition of 117 channels for the second test, bringingthe total number of channels to 244. National Instruments manufactured all of the hardware used. The two 12-slot SCXI-1001 chassis used in FullScale Test 1 were both filled to capacity with a total of 24 SCXI-1520 8-channel universal strain modules.
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1318 Two new SCXI-1000 4-slot chassis were used. Filling the eight new slots were five additional SCXI-1520 8-channel universal strain modules, and three SCXI-1121 4- channel isolation amplifiers. SCXI-1314 8channel terminal blocks were connected to each of the twenty-nine SCXI 1520 modules, and three SCXI1321 4-channel terminal blocks were connected to the SCXI-1121 isolation amplifiers. All four of the chassis were connected through the National Instruments PCI 6250 data acquisition card into the PC, configured with LabVIEW. The second full-scale test was designed to produce a dynamic response of the test bridge after it had been held in position while damage comparable to what it would have sustained in the event of an actual fracture was induced. Preparations for this test were extensive. A scissor-jack system was designed, constructed, and installed to raise the mid-span of the fractured girder 0.25 in. and support it while damage was induced on the bridge. (Figure 8) The support structure was also capable of immediate collapse when a critical link was severed with explosives. Coordinating for the appropriate and safe use of the explosives was an integral part of the test preparations. 244 channels of instrumentation equipment were prepared to gather data that would help characterize the response of the bridge to the simulated dynamic fracture event. (Table 1)
Table 1.Sensors used for the second full-scale test. The second full-scale test was executed and successfully loaded the test-bridge after it sustained a fulldepth fracture in one of its girders. The downward deflection of the fractured girder was as much as 7 in. in locations, and the bridge sustained significant damage to the shear stud connections between the top flanges of the fractured girder and the concrete deck. (Figures 9 and 10) Despite the displacements and damage sustained, the test-bridge resisted collapse, maintaining complete serviceability in its fractured state with the design truck load positioned directly above the fracture location.
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Figure 8. Scissor jack system used to shore up bridge for test 2
Figure 9.Fractures after release in test 2.
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Figure 10.Girder deflections in second full-scale test. Collapsing the Bridge in the Third Full-Scale Test Typical tests at a Structural Engineering lab push the unit under test to failure so it was decided to do this here. The third test incrementally over-loaded the bridge while the progressive failure mechanisms were closely observed. Loading continued until the ultimate load was reached and the bridge collapsed. Because the test-bridge was still capable of supporting additional loads following completion of the second full-scale test, a third full-scale test was planned as a follow-up to extend the results from the dynamic test. The goal was to observe the sequence of failure mechanisms and to determine the ultimate load required to induce a total collapse of the bridge. The experimental procedure for the third was designed as a loadcontrolled test, where additional load in excess of the design truck load was applied incrementally and without significant dynamic effects. The first two tests that deployed explosives were monitoring by agents from Homeland Security and the FBI. By moving to a load-controlled test, this approval wasn’t required. After placing the concrete girders on the bridge deck, additional load was applied by incrementally dumping material into and eventually around the bin. Road base was chosen as the loading material for its ease of acquisition, low cost, and relatively high density. (Figures 11 - 15) Obtaining a lifted weight measurement for each crane pass, from the placement of the concrete girders through the placement of each bucket of the road base, was critical. To measure the weights quickly and easily, a load-cell was attached to the crane load line above the lift bucket. A Wi-Fi transmitter was
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1321 connected to the load cell so that the load data could be easily read and recorded from across the worksite. After three days of loading 1500 pounds of road fill at a time, large portions of concrete at the mid-span expansion joint of the exterior railing began to spill when the total load applied to the bridge reached 360,200 lbs. After the onset of major material losses, three additional lift bucket loads were placed on the bridge before the bridge came to rest on the concrete bed below. As the load applied to the bridge increased over the course of the experiment, the bridge components experienced a series of failures. Following these intermediate failures, the bridge was able to redistribute the applied loads, suggesting the contribution of redundant load paths in maintaining equilibrium of the bridge in its progressively damaged state. (Figure 16)
Figure 11.Loading the bridge with road base.
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Figure 12.Bridge loaded to collapse.
Figure 13.Remote load cell monitoring.
Figure 14.Wi-Fi DAQ module with battery in housing box for measuring load cell
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Figure 15.Side view of intact girder.
Figure 16.Bridge Model Validation visualization with Avizo
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1324 Conclusion Supported by a number of its elements contributing to create a robust redundant load path, the testbridge performed extremely well and supported the application of over four times its design load after sustaining a full-depth fracture of one of its two girders. Results obtained from the finite element model indicated that adequate redundancy exists in the bridge design to maintain stability after the fracture of one girder. This was validated in the full-scale tests. The large concrete railing above the fractured girder transmitted force away from the fracture location when bridge deflections resulted in a closing of its expansion joints. The bridge deck also transferred significant loads in flexure, both transversely and longitudinally to the bridge span. . This research may lead to revisions to the current AASHTO specifications that a) can accurately predict the behavior of these bridges following the failure of a critical member, and b) subsequently prescribe appropriate inspection and maintenance requirements. Research at the Ferguson Structural Engineering Laboratory implies that the current requirement for bi-annual detailed inspections does not appear to be an effective use of labor or financial resources. Acknowledgements Thanks to the faculty and students of the Ferguson Structural Engineering Laboratory for sharing information about this application. The laboratory, named after Professor Phil M. Ferguson, is located on the Pickle Research Campus of the University of Texas at Austin and is an integral part of the Department of Civil, Architectural and Environmental Engineering. Students and faculty conduct largescale tests of a broad range of civil engineering structures in this facility. For more information on the “Methods of evaluating the redundancy of steel bridges,” visit their web site at : http://fsel.engr.utexas.edu/research/5498_webinar/index.cfm References American Association of State Highway Transportation Officials. (2004). AAHSTO LRFD Bridge Design Specifications. Washington, D.C. Hovell, Catherine. (2007). “Evaluation of Redundancy in Trapezoidal Box-Girder Bridges Using Finite Element Analysis.” Masters Thesis, University of Texas at Austin. Neuman, Bryce Jacob. (2009) “Evaluating the Redundancy of Steel Bridges: Full-Scale Destructive Testing of a Fracture Critical Twin Box-Girder Steel Bridge.” Masters Thesis, University of Texas at Austin.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Lake Mead Low Lift Pump Riser Failure Analysis Dirk H. Duffner, P.E., Duffner Engineering, Emerald Hills, California
ABSTRACT Premature failures – progressive damage from fatigue - of pump risers at a low lift pumping station located on Lake Mead in Southern Nevada were investigated. A typical pump riser was 30 inches in diameter, 100 to 130 feet long, manufactured from 0.375 inch thick steel and installed in a 54 inch diameter well casing. A pump was installed at the bottom of a typical riser connected by a drive shaft that ran the length of the riser to the electric motor mounted on the pumping station floor. Metallurgical analysis revealed that the mode of failure was fatigue. Stress concentration effects of local weld geometry and a taper in the riser structure were investigated. Strain gages, accelerometers, and ultrasonic position sensors were installed on recently repaired pumps, risers, and motor assemblies to verify analytical predictions and conduct modal analysis. A fatigue analysis found that the 100 foot risers accumulated significantly more damage than the 130 foot rises. The risers swayed at their natural frequency as a result of flow induced vibration of the pump suction. The bending stress was limited in the more flexible 130 foot riser resulting in a much longer fatigue life prediction. INTRODUCTION One year after the completion of a facility upgrade in 2000, cracks started to appear in the circumferential welds of steel riser pipes at the Low Lift Pumping Station located at the Alfred Merritt Smith Water Treatment Facility. The upgrade was undertaken in part to reduce energy consumption; among other changes, the riser lengths were shortened from 200 feet to 100 feet. The facility is located along Lake Mead in Southern Nevada and supplies water to the Las Vegas metropolitan area. The pumping station is part of the Southern Nevada water distribution system managed by the Southern Nevada Water Authority (SNWA). The Low Lift Pumping Station is composed of twenty pumps with a total capacity of up to 600 million gallons of water per day. The water is pumped from the lake through steel risers that are 30 inches in diameter and 0.375 inches thick. At the time of the failures the lake water level was 63 feet lower than the floor of the pumping station. The length of the risers was 100 feet, meaning that approximately 37 feet of riser including the pump was immersed in the water. The risers consist of 10 foot long sections of pipe with flanged ends bolted together, and along with top and bottom transition pieces, and short sections, bolted to the uppermost and lowermost riser sections respectively, make up the 100 foot length (Figure 1). The top transition piece has in inlet diameter of 30 inches to match the riser sections, but T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_117, © The Society for Experimental Mechanics, Inc. 2011
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tapers down to 24 inches to match the diameter of the discharge piping; it is 84 inches long, but tapered over only 46-½ inches of its length. The majority of the cracks were discovered in the upper circumferential weld of the top transition piece. These risers are installed in wells that have been drilled in the rock and are 54 inches in diameter. The pumps are bolted to the bottommost riser section located deep in the lake water. Drive shafts, the length of the riser, transmit power from the electric motors mounted on the pumping station floor to the submersed pumps below. The riser is not attached in any way to the well casing and is free to move inside the well. Bumpers have been installed around the circumference of the pump bowl to protect the pump against accidental impact with the casing. The riser acts like a cantilever beam rigidly attached at the top to the floor of the pumping station but completely free to move at the bottom. This 100 feet riser length was set during a facility upgrade performed between 1998 and 2000. Prior to this the risers had been nearly 200 feet long. The pumping station was originally put in service in the 1970’s (11 pumps) and in the 1980’s (9 additional pumps). The first riser failure occurred with pump No. 3 in January 2001. A crack at the edge of the weld near the top of the transition piece was observed. This transition piece is the last section of the riser inside the well and it connects to the support system on the pumping station floor. During the following months, Pumps No. 5, No. 2, and No. 8 failed. The failure on Pump No. 2 did not occur in the transition piece, like in the other cases, but in the section just below. This is a short straight section about 7 feet long with a circumferential weld about 2 feet from one of the flanges. In the case of pump No. 2, the crack occurred along the similarly located circumferential weld. In May of 2001, the concrete foundation for pump No. 5 was re-grouted and a new transition piece was installed. This transition piece failed again in November 2001, after six months of service. Due to the history with Pump No. 5, the riser of Pump No. 5 was instrumented for this investigation, as well as the riser of Pump No. 11. Pump No. 11 was chosen because it had been removed from its well in October 2001, due to observed excessive vibration of the motor and had not yet been re-installed by the time riser instrumentation started in December 2001. The investigation included metallurgical inspection of some of the sections from Pumps No. 5, No. 2, and No. 11. It involves the instrumentation of Pumps No. 11 and No. 5 set at riser length of 130 feet. Pump No. 5 was also instrumented at a riser length of 100 feet. This paper summarizes the data gathered in each one of these areas and draws some conclusions regarding the findings of the investigation. METALLURGICAL FINDINGS Three damaged riser sections were inspected; one from the top transition piece removed from pump No. 5, and two pieces are the short, top straight sections from pump No. 2 and No. 11. The large crack observed at the welded flange on the tapered end of the top transition piece from Pump No. 5 was caused by fatigue. This crack initiated at the
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outside surface of the riser, at the toe of the weld, and grew inward, penetrating the wall and extending a significant distance around the circumference. Beachmarks consistent with rapid growth were observed at the ends of the crack. Based on the size and spacing of the beachmarks, the crack was close to reaching a length at which catastrophic failure would have occurred had the crack not been noticed because of leaking water. The weld geometry at the flange on the tapered end of the pipe from Pump No. 5 appeared to be substandard with respect to AWS (American Welding Society) D.1.1. The gap between the pipe section and the flange, as seen in Figure 2, is a built-in flaw that can propagate a crack along the circumference of the riser. The crack observed in a circumferential weld in the 76-inch long riser (short) section removed from Pump No. 2 also appears to have been caused by fatigue. This crack initiated at the inner diameter of the pipe and grew toward the outer diameter. Metallographic samples through the cracked circumferential weld indicate welding problems, including a lack of fusion. Large weld defects at the inner diameter of the pipe were observed in sections away from the cracked area. These welding problems or defects probably contributed to the failure of the riser section. A well-defined, narrow band of rusted steel was observed along the inner diameter of the circumferential weld in the 76-inch long pipe removed from Pump No. 11. Upon sectioning of the circumferential weld, it was determined that this narrow band was not a crack, but an area of weld overlap that was not adequately covered by the internal paint, and therefore corroded. STRAIN MEASUREMENTS Seventeen strain gage bridges were installed on the risers in various configurations providing stress concentration, axial tension, bending in two directions, pressure, and torsion on the riser. Figure 3 shows an overall arrangement of the gages on the transition section, and Tables 1 through Table 3 show some typical results of the strain gage measurements. Each of the three tables is a summary of data taken over two minute windows for each of the following three pump conditions: Pump No. 5 at 100 feet, Pump No. 5 at 130 feet, and Pump No. 11 at 130 feet. These two-minute windows are representative of the several hours of data recorded for each pump configuration. Included in each table is data taken while the pump was running in a steady state as well as any pre or post-startup, or shutdown transient that may have been recorded. Prior to final assembly of the transition piece and diverter to the pump bowl and riser, all strain gage bridges were balanced or set to read zero. The initial bridge balance settings were not changed during the course of testing. The bridge balancing was carried out with the transition piece and diverter suspended from a crane. Time histories of the bending bridges (X and Y) are given in Figure 4 through 6. These time histories correspond to their respective amplitude and mean stress entries shown in Table 1 through 3. Results are shown for each of the three pump arrangements, Pump
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No. 5 installed at 100 feet, Pump No. 5 installed at 130 feet, and Pump No. 11 installed at 130 feet. The figures show a representative time slice taken from several hours of recorded testing. It is possible to make a correlation between the stresses at the top of the riser and the displacement of the riser at the bottom by estimating the moment of inertia of the riser and by assuming that the riser acts like a cantilever beam. Using this model and the stress amplitude in the case of Pump No. 5 installed at 100 feet, it is estimated that the bottom of the riser is oscillating with amplitude of about 3 inches (6 inches total travel). WELD STRESSES Circumferential fatigue cracks have been observed in the toe of the weld joining the tapered section to the 24-inch flange in the upper transition piece. Fatigue cracks in this location and in this orientation, grow under the influence of axial stresses. Review of the pump and riser design revealed two sources of axial stresses, bending and tension. Axial stresses resulting from bending are tensile on one side of the cross section and compressive on the other. They tend to decrease linearly away from sources of constraint such as the pump base. The bending stresses also decrease away from the pump base as the section modulus increases due to the effect of the transition piece taper. Axial stresses resulting from tensile loads are uniform throughout the cross section. The axial stresses from tensile loads, like the bending loads, decrease away from the pump base, in this case due to the increasing area of the tapered cross section away from the pump base. The bending and tensile stresses are amplified when approaching the transition piece upper flange. The magnitudes of these stress amplifications were determined analytically, and then verified experimentally. In addition to this amplification, the local weld geometry has a stress concentration factor associated with it. The stress amplification, as the weld toe is approached, is a combination of the global stress change due to the taper plus the local stress change due to the weld. The stress amplification at the weld toe includes the combined effect of the gradual taper along the transition piece, and the localized geometry of the weld. Table 4 shows the results of the stress amplification analysis. The stress amplification is different in bending than in tension. In the case of bending, the stress at the weld is proportional to the moment, the section modulus, and the localized stress concentration factor. Since the moment is known some distance away from the weld (bending bridge measurement location), it can be scaled up to include the effect of the taper and the weld. The taper amplifies the bending stress in three ways: the moment of inertia, and distance from the neutral axis to the extreme fiber are lower, and the moment arm increases toward the top of the taper. The equation used to predict the bending stress amplification is shown below, were L is the length of the riser, I the moment of inertia, c the distance from the neutral axis to the extreme fiber, and x is the length of the taper plus the distance of the bending measurement location from the end of the taper.
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Lc top K bend
I top K L x cbot weld I bot
The overall stress concentration includes, in addition to the effect of the taper, the local effect of the weld (Kweld). This was determined for the given weld geometry from the text Stress Concentration Factors by R.E. Peterson. The specific geometry used was “Stress Concentration Factor for a Stepped Round Tension/Bending Bar with a Shoulder Fillet.” The estimated radius at the toe of the weld used for the evaluation was 0.05 inches. In the case of tensile stresses, the amplification associated with the taper results only from the change in cross section area. This, combined with the localized stress concentration of the weld, gives the overall stress amplification in tension shown in the equation below, Abot K tens K weld Atop Abot is the area of the straight pipe cross-section, Atop is the area of the transition piece at the top of the flange, and Kweld is the local stress concentration factor of the weld. MEASURED STRESS AMPLIFICATION The analytical results shown in Table 4 were verified experimentally. The bending loads on the structure were measured with strain gages arranged and wired in such a way as to cancel out the influence of other loads such as tension or pressure. Likewise, the tensile loads on the structure were measured in a similar fashion. These measurements were taken in the uniform 30-inch nominal diameter section of Pump No. 5 and Pump No. 11 transition pieces below the taper, 49.75 inches below the bottom of the upper flange. Because of the cross section change associated with the taper, and the stress concentration factor at the upper flange weld, the stresses measured at the bending and tension bridge location 49.75 inches below the upper flange are significantly attenuated compared to those at the toe of the weld where the fatigue cracking had been observed. A line of strain gages was installed along one axis of the transition piece (+X). These strain gages were used to determine the level of stress amplification associated with the weld, taper, and bending measurement offset. Figure 7 through 9 show the strain gage measurements taken while running Pump No. 5 operating at 100 feet, Pump No. 5 operating at 130 feet, and Pump No. 11 operating at 130 feet. The gages are divided into the three regions shown as follows: Near Weld (strain gage channels 3 through 6), Taper (strain gage channels 7 through 10), and Straight (strain gage channels 13, 14, and 17 on Pump No. 5). The three strain gages in the Straight region include the two bending bridges (X and Y) at 49.75 inches, and the single axial gage located at 106.44 inches on Pump No. 5 only. The remaining eight strain gages are in the tapered section, and include four gages within 3
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inches of the weld, and four distributed over the remaining 43.5 inches of the taper. The stresses near the weld were extrapolated back to the toe of the weld. On Pump No. 11, the weld toe measured 0.833 inches from the bottom of the upper flange. The strain gages on Pump No. 5 were located in identical positions as those on Pump No. 11; however, no detailed measurement of the weld toe location relative to the strain gages was taken on Pump No. 5, so the location from Pump No. 11 was used. The strain gages used for the extrapolation were located at 1 inch, 1.14 inches, 2 inches, and 3 inches. The measured values of the bending stress amplification using this technique were 3.9, 4.4, and 5.2 for Pump No. 5 at 100 feet, Pump No. 5 at 130 feet, and Pump No. 11 at 130 feet, respectively. The amplitude of the tension stress measured by the tension bridge at 49.75 inches during the above test was 148 psi. Because this tension stress is so small, and the bending bridge output is very close to the output of axial strain gages nearby, the fluctuating stress during this test was caused predominantly by alternating bending loads, and not alternating tension loads. FATIGUE ANALYSIS Using the measured stresses and the metallurgical work performed during Phase 1 of the investigation that identified the steel as AISI-1020 (American Iron and Steel Institute), the fatigue life of the subject pumps was predicted. Table 6 gives the cyclic stress amplitude, the mean stress, and the effective stress amplitude used in the fatigue analysis for each of the pump tests performed, as well as the earlier August 21, 2001 test performed by SNWA. The cyclic stress amplitude for the SNWA test was read directly from their strain gage output plot titled “8/22/01 B-A Axial Microstrain.” The strain amplitude (one half of peak-to-peak) shown on this plot is 123.5 micro-inches per inch. As shown in Table 6, this measurement was taken 42 inches below the upper flange, on the 30-inch diameter, non-tapered section of the transition piece. Multiplying by Young’s modulus of 30,000,000 psi, and the bending stress amplification factor of 4.3 shown in Table 5, the corresponding stress amplitude at the weld toe is 15,932 psi. The cyclic stress amplitude for each of the three tests conducted by FaAA is taken from Table 5. The X-Bending stress in the pump running condition was used for each test. The mean stresses shown in Table were determined analytically. The results of the strain gage measurements suggest that the pumps and/or risers may be subjected to external loading from time to time somewhere along their lengths, thus lowering the measured mean stress near the top of the riser. For the fatigue analysis, the larger, predicted mean stress was used. The predicted mean stress is based upon the calculated weight of the riser and pump, plus the weight of the water above the well level (63 feet), and the tensile stress amplification factor of 3.3 shown in Table . The estimated weight of the 100-foot riser and pump is 21,150 pounds, and for the 130-foot riser and pump is 26,480 pounds. The weight of 63 feet of water, 29.25 inches in diameter is 18,344 pounds. The resulting mean stresses, 3,734 psi for the 100-foot risers, and 4,238 psi for the 130-foot risers are shown in Table 7.
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The last riser of Table shows the effective stress amplitude at the weld. The effective stress amplitude includes the effect of mean stress in the fatigue analysis. It is the stress amplitude multiplied by the ratio of the ultimate strength to the ultimate strength minus the mean stress. The ultimate strength for this material is 64,000 psi. A model of the riser was used to determine its natural frequencies. The model used assumes a cantilever beam and takes into account the mass of the water inside the riser. The results from the model are presented in Table 7 where they are compared to the measured X and Y bending response spectra shown in Figures 10 and 11. The similarity between the model and the experimental values indicate that the risers behave like cantilever beams subjected to broadband excitation. A vibration response frequency of 0.25 Hertz was used for purposes of the lifetime prediction. Figure 12 shows a stress-life, or S-N curve for this material, along with data points for each of the effective stress amplitudes shown in Table 6. The reference for the S-N curve, as well as the ultimate strength of this material is The Society of Automotive Engineers (SAE) “Technical Report on Low Cycle Fatigue Properties” – SAE J1099 JUN98. The predicted lives for each of the three test cases exceed 10 years. The test condition exhibiting the closest effective stress amplitude to the S-N curve was Pump No. 5 at 100 feet. This test condition displayed a similar time history on the bending bridge, as did testing conducted on the same pump by SNWA several months earlier on August 22, 2001. The earlier results exhibited higher stress amplitude, and thus would have predicted a relatively short life. Testing and examination of this pump in December 2001 revealed the presence of a crack near the upper transition piece weld of sufficient length and depth to allow water passage during pump operation. This transition piece had reportedly been newly installed just prior to testing on August 22, 2001. As shown in Figure 12, assuming continuous running of the pump, the life expectancy of Pump No. 5 at 100-feet using the August data is estimated to be about 4 months. This is probably close to its actual number of cycles between August and December 2001.
CONCLUSIONS 1. The cause of the upper section damage to the risers inspected in our laboratory was fatigue driven by cyclic bending stresses. The inspection of the cracks in the risers also revealed poor flange connection design and welding. 2. Bending stresses measured in the top transition piece of the instrumented 100 feet riser indicate fluctuating stresses that could potentially cause fatigue failure of the riser. 3. Measurements taken on the 100 feet riser and engineering analysis using beam theory indicate a bending oscillation of about 3 inches on either side of the centerline at the bottom of the riser. The maximum possible bending amplitude
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before the pump strikes the well casing is about 5 inches. Measurements performed by FaAA at 100 feet show no indication that there was any contact between the riser and the wall of the well due to these bending oscillations. 4. Bending stresses measured in the top transition piece of the two instrumented 130 feet risers indicate small fluctuating stresses that are well below the fatigue limit of the steel for these risers. 5. Lateral forces at the intake of the suction inlet are the most probable cause of the deflection at the intake, which causes bending stress in the transition in both the 100 and 130 feet risers. 6. The differences in the vibration modes and amplitude observed between the 100 feet and the 130 feet risers are due to the difference in stiffness of the risers. The 130 feet riser is sufficiently flexible so that the lateral force in the system forces the pump bowl against the wall of the well and limits the movement of the riser during operation. It takes less than half the force to horizontally move the pump bowl an equal distance for the riser of 130 feet as compared to 100 feet. REFERENCES 1. R.E. Peterson, Stress Concentration Factors , PUBLISHER, CITY, DATE, PAGES 2. Society of Automotive Engineers (SAE) “Technical Report on Low Cycle Fatigue Properties” – SAE J1099 JUN98
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Table 1.
Channel
Measured maximum amplitude and mean stress for each of the strain gage channels on Pump No. 5 operating at 100 feet*.
Description
Weld Toe
Distance From Flange (in)
Pump No. 5 100 Feet Running
Pump No. 5 100 Feet Pre-Shutdown
Pump No. 5 100 Feet Post-Shutdown
Amplitude (psi)
Mean (psi)
Amplitude (psi)
Mean (psi)
Amplitude (psi)
Mean (psi)
0.833
1
Rosette - Hoop
1
1588
-1995
1649
-2035
142
-2644
4
Transition 1
1
4466
1921
4624
1858
387
1067
2
Rosette - 45
1.07
3041
602
3179
559
273
-326
3
Rosette - Axial
1.14
4052
-101
4164
-151
331
-865
5
Transition 2
2
1978
771
2044
734
171
396
6
Transition 3
3
1214
847
1255
836
107
642
7
Transition 6
6
1818
926
1884
904
167
591
8
Transition 12
12
1913
1257
1985
1272
168
888
9
Transition 24
24
1623
601
1685
565
139
121
10
Transition 45
45
1391
1163
1434
1101
124
767
11
Hoop
49.75
430
456
496
367
63
-164
13
Bending - X
49.75
1511
155
1563
141
131
110
14
Bending - Y
49.75
1575
-111
1806
-26
264
132
15
Tension
49.75
148
520
138
528
26
213
16
Torsion
49.75
29
121
27
118
3
-36
17
Short Section Axial
106.4375
1198
284
1228
153
101
-136
12 Dummy 12 -20 12 -66 10 -67 * The running data was taken over a 2 minute window on December 14, 2001 at 9:08.16 a.m. The shutdown data was taken over a 2 minute window on December 14 at 7:53.37 p.m. No startup was available for this pump and configuration.
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Table 2.
Channel
Measured maximum amplitude and mean stress for each of the strain gage channels on Pump No. 5 operating at 130 feet*
Description
Weld Toe
Distance From Flange (in)
Pump No. 5 130 Feet Pre-Startup
Pump No. 5 130 Feet Post-Startup
Pump No. 5 130 Feet Running
Amplitude (psi)
Mean (psi)
Amplitude (psi)
Mean (psi)
Amplitude (psi)
Mean (psi)
0.833
1
Rosette - Hoop
1
15
450
129
1005
142
730
4
Transition 1
1
18
1735
298
2209
341
2045
2
Rosette - 45
1.07
19
978
230
1594
260
1447
3
Rosette - Axial
1.14
15
1300
286
1511
308
1332
5
Transition 2
2
10
597
137
596
148
724
6
Transition 3
3
11
399
79
378
94
599
7
Transition 6
6
16
410
129
552
146
561
8
Transition 12
12
11
922
130
1099
149
1163
9
Transition 24
24
8
801
105
1004
126
1070
10
Transition 45
45
6
852
92
1173
108
1218
11
Hoop
49.75
5
-464
45
106
48
220
13
Bending - X
49.75
3
58
86
41
109
74
14
Bending - Y
49.75
3
411
68
374
77
360
15
Tension
49.75
4
307
31
631
31
643
16
Torsion
49.75
1
11
26
172
29
187
17
Short Section Axial
106.4375
4
-271
77
3
87
-14
12 Dummy 10 11 20 10 13 -30 * The startup data was taken over a 2 minute window on December 20, 2001 at 6:48.23 a.m. The running data was taken over a 2 minute window on December 19 at 2:53.12 p.m. No shutdown data was available for this pump and configuration.
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Table 3.
Measured maximum amplitude and mean stress for each of the strain gage channels on Pump No. 11 operating at 130 feet.*
Description Channel
Weld Toe
Distance From Flange (in)
Pump No. 11 130 Feet Pre-Startup
Pump No. 11 130 Feet Post-Startup
Pump No. 11 130 Feet Running
Amplitude (psi)
Mean (psi)
Amplitude (psi)
Mean (psi)
Amplitude (psi)
Mean (psi)
0.833
1
Rosette - Hoop
1
15
-43
342
-39
448
151
4
Transition 1
1
20
214
967
-75
1260
390
2
Rosette - 45
1.07
24
-14
634
96
817
427
3
Rosette - Axial
1.14
16
110
834
-268
1111
132
5
Transition 2
2
10
-174
376
-302
473
-73
6
Transition 3
3
8
-44
214
-98
253
11
7
Transition 6
6
14
333
347
203
435
345
8
Transition 12
12
7
878
347
795
455
869
9
Transition 24
24
8
1332
300
1224
405
1394
10
Transition 45
45
8
2486
274
2486
375
2585
11
Hoop
49.75
5
-137
100
160
103
183
13
Bending - X
49.75
4
517
249
176
330
285
14
Bending - Y
49.75
3
-642
261
-429
284
-349
15
Tension
49.75
5
-748
58
-417
29
-383
16
Torsion
49.75
2
-66
51
92
27
126
17
Short Section - Axial
106.4375
12 Dummy 10 -363 16 -368 13 -453 * The startup data was taken over a 2 minute window on December 11, 2001 at 2:44.57 p.m. The running data was taken over a 2 minute window on December 11 at 11:32.46 p.m. The shutdown data was taken over a 2 minute window on December 11, 2001, at 11:55.04 p.m.
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Table 4.
Results of stress amplification analysis.* Pump No. 5 8/22/01
Pump No. 5 - 100 ft
Pumps No. 5 / 11 - 130 ft
100
100
130
0.375
0.375
0.375
23.25 / 29.25
23.25 / 29.25
23.25 / 29.25
27.833 / 34.901
27.833 / 34.901
27.833 / 34.901
I Top / Bottom (in )
1942 / 3829
1942 / 3829
1942 / 3829
C Top / Bottom (in)
12 / 15
12 / 15
12 / 15
Taper Length (in)
38.75
46.5
46.5
Gage Dist From Taper (in)
3.25
3.25
3.25
2.6 / 2.6
2.6 / 2.6
2.6 / 2.6
Kbend
4.3
4.3
4.2
Ktens
3.3
3.3
3.3
Length (ft) Nominal Thickness (in) ID Top / Bottom (in) 2
Area Top / Bottom (in ) 4
Kweld Bend / Tension
* August 22, 2001 Strain gage testing conducted by SNWA
Table 5. Extrapolated stresses at the toe of the weld for each of the three pump configurations. X Bending Pump/Elevation (ft), Operation
Y Bending
Tension
Amplitude (psi)
Mean (psi)
Amplitude (psi)
Mean (psi)
Amplitude (psi)
Mean (psi)
5/100, Running
6498
667
6774
-477
488
1716
5/100, Pre-Shutdown
6723
608
7768
-112
456
1741
5/100, Post Shutdown
563
473
1137
569
86
703
5/130, Pre-Startup
14
244
13
1725
13
1013
5/130, Post-Startup
359
174
287
1569
103
2083
5/130, Running
457
312
322
1513
101
2121
11/130, Pre-Startup
16
2173
14
-2698
15
-2468
11/130, Post-Startup
1046
741
1097
-1802
191
-1377
11/130, Running
1384
1197
1194
-1465
95
-1264
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Table 6.
Weld Stresses Measured by SNWA
Pump ID
Pump Depth (feet)
Test Date
Cyclic Stress Amplitude at Weld (psi)
Mean Stress at Weld (psi)
Effective Stress Amplitude at Weld (psi)
5
100
8/1/2001 by SNWA
15,932
3,734
16,920
5
100
December, 2001
6,498
3,734
6,901
5
130
December, 2001
457
4,238
489
11
130
December, 2001
1,384
4,238
1,482
Material - Steel, 1020 Hot Rolled Sheet Stress Concentration at Weld - 2.6
Table 7.
Measured and modeled natural frequency modes
Mode
100 Feet Model (Hz)
100 Feet Measured (Hz)
130 Feet Model (Hz)
130 Feet Measured (Hz)
1
0.28
0.23
0.17
0.375
2
1.7
1.8
1.0
1.1
3
4.8
5
2.9
3.1
4
9.5
10.2
5.7
5.8
5
15.7
15.9
9.4
9.7
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Pump Discharge Motor
+X
+Y Top Transition Piece Wall of Well Top Riser Section
Z
100 ft 130 ft Bottom Transition
Pump
Suction Inlet 3 ft
Figure 1. Schematic of the instrumented pump with table showing instrumentation location on Pump No. 11.
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Weld
Gap
Crack Flange
Pipe Weld
Figure 2
Cross section of the riser at the flange-pipe connection.
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Figure 3.
Strain gage instrumented transition piece for Pump No. 11 shown here with diverter installed (looking at +X side of transition piece).
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Figure 4.
X and Y bending stress on Pump No. 5 operating at 100 feet. The data were taken on December 14, 2001 at 9:08.16 am during normal pump running. The bending stress measurement location is 49 ¾ inches below the upper flange on the straight section of the transition piece.
Figure 5.
X and Y bending stress on Pump No. 5 operating at 130 feet.
This data was taken on December 19, 2001 at 2:53.12 pm during normal pump running.
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Figure 6.
X and Y bending stress on Pump 11 operating at 130 feet. This data was taken on December 11, 2001 at 11:32.46 pm during normal pump running.
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Pump 5 at 100 Feet Measured Bending Stress Amplification = 3.9 Extrapolated Stress Amplitude At Weld Toe = 5728 psi
1000
1575 psi
1330 psi
Stress Amplitude (psi)
Near Weld y = 4592.3x
-1.2097
1511 psi
Taper
Straight y = -5.5277x + 1786.1
y = -12.708x + 1962.4
Average Stress Amplitude At Bending Bridge Location = 1472 psi
100
Weld Toe (0.833 in) 10 0.1
1
10
100
Distance From Upper Flange (in)
Figure 7.
Measured stress amplitude versus distance from upper flange on transition piece. This plot is for Pump No. 5 during normal running operating at 100 feet.
1000
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Pump 5 at 130 Feet Measured Bending Stress Concentration Factor = 4.4 Extrapolated Stress Amplitude At Weld Toe = 434 psi
Stress Amplitude (psi)
1000
108 psi
100
109 psi
Taper Near Weld y = 348.47x
y = -1.0752x + 155.46
0.1
77 psi
-1.2053
Average Stress Amplitude At Bending Bridge Location = 98 psi
Weld Toe (0.833 i )
10
Straight y = -0.1032x + 97.834
1
10
100
1000
Distance From Upper Flange (in)
Figure 8.
Measured stress amplitude versus distance from upper flange on transition piece. This plot is for Pump No. 5 during normal running operating at 130 feet.
10000
Pump 11 at 130 Feet Measured Bending Stress Amplification = 5.2 Extrapolated Stress Amplitude At Weld Toe = 1710 psi
Stress Amplitude (psi)
1000 375 psi
Taper y = -1.8868x + 458.56
330 psi 284 psi
Straight
Near Weld
100
y = 1304.4x-1.4818 Average Stress Amplitude At Bending Bridge Location = 330 psi
Weld Toe (0.833 in) 10 0.1
1
10
100
1000
Distance From Upper Flange (in)
Figure 9.
Measured stress amplitude versus distance from upper flange measured on Pump No. 11 operating at 130 feet during normal running.
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Figure 10.
X and Y bending spectrum for Pump No. 5 at 100 feet
Figure 11.
X and Y bending spectrum for Pump No. 5 at 130 feet.
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100,000
Pump 5 100 Feet SNWA 8/22/1 Testing
10,000 Pump 5 100 Feet
Pump 11 130 Feet
100 1,000,000
13 Years at 1/4 Hz
1.3 Years at 1/4 Hz
1,000 4 Months at 1/4 Hz
Effective Stress Amplitude, Vaeff (psi)
SN Curve for 1020 Hot Rolled Sheet (Ref: SAE J1099)
10,000,000
100,000,000 Life, N (cycles)
Figure 12.
Fatigue Analysis Results
Pump 5 130 Feet
1,000,000,000
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Beamforming for quality control in industrial environment
P.Castellini, A.Sassaroli, N.Paone Università Politecnica delle Marche, via Brecce Bianche, Ancona, Italy [email protected] ABSTRACT This work presents the application of beamforming for the quality control of washing machine in industrial application. The objective is to substitute sensors like accelerometers or Laser Doppler Vibrometers in order to acquire vibration emission of a machine and use that information for the diagnosis of the quality status. The main problem is related to the industrial environment in which the noise level, due to production process or the test of other machine, reduce the Signal-to-Noise of acoustic based techniques. In this paper a small microphone array is used to acquire acoustic emission of the machine. Beamforming algorithm is used to focus the microphone sensitivity and to avoid disturbance of surrounding environment and other machines. The performance of the proposed technique is also evaluated by comparison with LDV measurement in controlled conditions in an anechoic chamber and in a reverberant environment, with different level and kind of disturbing noise in order to qualify the proposed technique. 1. Introduction I the industrial production the quality control is more and more important in order to guarantee a product that can satisfy customer needs. This control is achieved mainly by two approaches: the quality control at the end of the production line; the monitoring of the machine for all the life of it (for expensive or safety critical devices) or in accelerated tests on samples of product. In both cases the efficiency of the control process is of fundamental importance in order to avoid time consuming or instrumentation intensive approaches. Contactless sensors allow to simplify the sensors placement and installation and are becoming more and more interesting respect to traditional devices like accelerometers. This is the reason of the increasing success of laser Doppler vibrometers (LDV) now frequently installed also in production lines. The main limit of LDVs is the cost. Another interesting solution is the use of microphones. Acoustic sensors allow not only to take information about the health of the machine contactless, but are also less expensive and gives information directly related to the noise produced by the machine. Nowadays microphones can be applied only in laboratory, or under well controlled acoustic conditions (like insulated boxes or separated rooms). The objective of this work is to study the possibility to apply Beamforming Algorithms [1] [2] [3] to make the acquisition of an array of microphones directive and selective in such a way to capture the noise emitted by a single machine even if this is close to disturbing sources of noise, like other similar machines present on a production line or in testing room where several machines operates at the same time without insulation. 2. Experimental set-up The specific application of the beamforming for the quality control was highly conditioning the set-up.
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_118, © The Society for Experimental Mechanics, Inc. 2011
1347
BookID 214574_ChapID 118_Proof# 1 - 23/04/2011
1348 In fact the idea to apply acoustic systems instead of more traditional solutions (like accelerometers or Laser Doppler Vibrometers) want to obtain the non-contact detection of vibration spectra or patterns with a cost as small as possible if compared with state-of-art solution. This specification affects the selection of the number of microphones and acquisition channels. Considering that usually acquisition devices have block of 8 channels, we define available for the array 14 channels, leaving 2 additional channels for monitoring signals. The system is based on 14 microphones Bruel & Kjaer 4951 and acquisition system PXI with NI 4472 boards (8 syncrhonous channels, 24 bit). In addition even system size must be as small as possible in order to make the application in an industrial environment easier. The array must stay in a volume that was defined about 300x300x200 mm. The frequency range is in the range 1÷4kHz, while the distance between the array and the machine surface is in the range 0.1÷0.6 meters. Quite original must be also the software approach. Instead of usual beamforming, in which maps at a specific frequency range (usually in third of octave spectrum), in this application is much more interesting to obtain the complete spectrum at a specific location. Maps, which are too heavy and time-consuming results for diagnosis, will be used only in the development of the procedure in order to understand the performances of each solution. The performances of the implemented solution were done in simulated conditions. The machine under test is installed in a semi-anechoic room and reverberation and disturbs are simulated by a series of loudspeakers located at the same distance and at different angular locations. In this way is possible to simulated different disturbs and different performances of the array. In particular, loudspeaker in: position 1 stress the capability of direction selection; position 2 stress the sensitivity to reflections; position 3 stress the capability of resolution; position 4 stress the capability of focusing; Loudspeakers are driven by white noise while the tested machine is running in operating conditions. The level of disturb are set at different levels. Z 4
1.00
3
0.32
3
2
1
2
1
Figure 1
Lay-out of testing facility in anechoic conditions
3. Results A test campaign has been done in order to compare the developed array with state-of-art solution for the mapping of acoustic emission of machines. These techniques cannot be applied on industrial cases due to cost and/or time needed for measurement. In particular, different approaches have been tested to be compared with beamforming: 1. Acoustic Intensimetry 2. Acoustic holography STSF; 3. Acoustic holography HELS;
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1349 Tests were performed in a semi-anechoic chamber as described before with the machine running at 1200 RPM in steady conditions. Hereafter is reported the comparison of acoustic power maps for each technique at 4 more interesting frequencies. Due to the differences of output of different techniques and of differences in data plotting, only qualitative comparison have been made, just to determine the capability of each techniques to individuate the lobe of emission and to focus on it. In this way, it also possible to compare the main features of each techniques and then evaluate possible alternatives.
Figure 2 Comparison of results with different techniques at 650 Hz
The comparison among different techniques is also more difficult because of the different behaviour of each technique at different frequency. As example, for HELS and the intensimetry the problem is related to the spatial resolution of the sampling (it is related to the time to take the measurement) and then it is not possible to obtain data at the higher frequencies. The limits in array size and number of microphones limit the dynamic range of the beamforming and then the minimum frequency that can be tested. As expected, the behaviour of the beamforming improves a lot when the frequency increases, with a much better spatial resolution and noise rejection. The gain of lateral noise is from 5 to 7 dB lower then that in front of the array where it is focused.
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1350 Much more interesting is the comparison between the acoustic maps obtained with and without the noise, with disturbing loudspeaker in position 2 and with an acoustic level similar to that of the machine. This position was chosen because it is the most representative case in operating conditions. These results are shown in Figure 3. The maps with and without disturbing noise are almost the same, and that highlight that the beamforming algorithm is able to select the direction of arrival of the acoustic signal. Without disturb
With disturb
1000 Hz
Z
1250 Hz 1.00
0.3 m
0.32
3
2
1600 Hz
11 m 2
Position 2 2000 Hz
Figure 3 Comparison of beamforming maps with and without disturbing noise
In order to have a direct comparison between the spectrum measured by beamforming and the “actual” spectrum on the machine surface, an additional test was performed. With the beamforming algorithm, the spectrum in a control point on the machine surface was calculated starting from far field acquisition, as in the previous experiments. This spectrum was compared an acquisition done synchronously with a single microphone physically installed very close with the surface in the same location. The test was performed with the disturbing source in position 2 and 3, and with focusing point in the front of the machine, as shown in Figure 4.
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Position 3 1m
1.00 0.32 0.3 m
3
2
11m 2
Position 2
Figure 4 Comparison between beamforming and near-field microphone: disturb and focusing point position
Figure 5 Comparison between spectra obtained with beamforming and near-field microphone: no disturb
The comparison of spectra presented in Figure 5 shows that the reconstruction of spectrum with beamforming is accurate without the disturbing noise, as expected. The difference between the levels is due to the scaling of beamforming output. The amplitude in beamforming algorithm is still an open question [4] and it depends to the scaling procedure used. In any case, in for the application in diagnostic, this feature is not useful. In Figure 6, Figure 7 and Figure 8 is shown the comparison among spectra without the disturb and with the disturb in position 2 and 3. The reconstructed spectrum is not very affected by the disturb and then beamforming can be used as far-field system for machine diagnostics.
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Figure 6 Comparison between spectra without the disturb and with the disturb in position 2 and 3 in the range 1÷2 kHz
Figure 7 Comparison between spectra without the disturb and with the disturb in position 2 and 3 in the range 2÷3 kHz
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Figure 8 Comparison between spectra without the disturb and with the disturb in position 2 and 3 in the range 3÷4 kHz
4. Conclusions In this paper the possibility to apply beamforming technology to obtain an acoustic sensor for the quality control in an industrial environment. The array lay-out has been optimised, while the performances of beamforming algorithm was verified under controlled conditions. Limits and advantages were quantified in terms of SNR. 5. Acknowledgements The Authors would like to acknowledge Mrs Cristina Cristalli for their precious support. The presented research activity was partially supported by Region Marche within the project “Robot Diagnostici – POR MARCHE FESR 2007-2013”. 6. 1 2 3 4
References S. Oerlermans, B. Mendez Lopez; Acoustic Array Measurements on a Full Scale Wind Turbine; NLR-TP2005-336. R.P. Dougherty, Beamforming in Acoustic Testing in Aeroacoustic Measurement, Springer P.Castellini, M.Martarelli, Acoustic beamforming: analysis of uncertainty and metrological performances, Mechanical Systems and Signal Processing, 22, pp.672–692, 2008, ISSN 0888-3270 S. Oerlemans, P. Sijtsma; Determination of absolute levels from phased array measurements using spatial source coherence; Proceedings of the 8th AIAA/CEAS Aeroacoustics Conference & Exhibit, 17-19 June 2002, Breckenridge, Colorado AIAA 2002-2464.
BookID 214574_ChapID FM_Proof# 1 - 23/04/2011
BookID 214574_ChapID 119_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Noise source localization on washing machines by conformal array technique and near field acoustic holography
a
Paolo Chiariottia, Milena Martarellia, Enrico Primo Tomasinia and Ravi Beniwalb Department of Mechanical Engineering, Università Politecnica delle Marche, Via Brecce Bianche, 60131 Ancona, Italy b SenSound, 440 Burroughs St., Suite 170, Detroit, MI 48202, USA
ABSTRACT The acoustic emission of a washing machine has been deeply studied by comparing three different techniques, which are: - conventional acoustic intensity, - planar near-field acoustic holography and – conformal array technique based on the Helmotz Equations Least Squares method. These techniques have been used to measure the front of a washing machine, i.e. the more critical side from the acoustic comfort point of view in the working environment. The acoustic intensity measurement has been taken as reference for the comparison of the two other techniques. The sound intensity probe has been scanned over a grid of several discrete positions and the acoustic intensity and pressure on the measurement plane have been determined. For both the conformal and planar near-field acoustic holography techniques an antenna of 30 microphones has been employed scanning over several positions in order to cover the entire washing machine front with a spatial resolution of 2.5 cm (maximum frequency 13720 Hz). Advantages and limitations of the noise source location techniques have been examined thoroughly.
1. Introduction The home appliance market is very large world-wide; practically every family in any country is a customer of appliances. The washing machine represents a type of appliance which is rapidly embodying high level technology and is sold in million items per year in the world. Nowadays acoustic comfort is becoming a key objective of washing machine manufacturers, thus it is necessary to accurately locate main noise sources. A washing machine is an electromechanical system, basically composed of a metal cabinet which hosts internally a tub which in turn contains the rotating drum, driven by an electric motor, in most cases through a pulley and with a control board which manages the washing cycles. Because of such a structure both noise sources coming directly from the drive system and those produced by the external cabinet mode shapes have to be expected. Water in the tub also produces the classical splash noise which can be annoying especially for long washing programs. Understanding the spatial distribution of noise sources on washing machine thus becomes an attractive challenge for acoustic measurement techniques. Standard measurements on such appliances are usually performed through the use of sound intensity probes: in this paper advanced array techniques like Near field Acoustic Holography (NAH) and Helmotz Equation Least Square (HELS) have been used to precisely identify noise sources on the front side of a standard washing machine, while intensity probe measurements have been performed in order to compare results with a classical technique. For the first time NAH and HELS have been applied to a real test case such as a running washing machine. Up to now the performances of the two techniques have been compared only on laboratory test cases, [1]. 2. Basic theory review 2.1. Sound intensity measurements The usefulness of measuring sound intensity is directly related to its vectorial nature. This allows testing to overcome problems coming from standard pressure measurements, which are often misleading if not performed
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_119, © The Society for Experimental Mechanics, Inc. 2011
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1356 in controlled environments (e.g. anechoic rooms). The knowledge of both the amplitude and the direction of an acoustic field allows testing to locate the sources of sound. The most used method of measuring sound intensity in air is the “two microphone” (or p-p probe) method. This approach uses two closely spaced and phase-matched microphones and relies on the finite difference approximation to the sound pressure gradient. The distance between microphones define the measurable frequency range. Several configurations are possible: (i) measurement at discrete points or, (ii) measurement by continuously scanning the measurement plane with the sound intensity probe. If it can be defined an hypothetical surface that completely encloses the noise source (in conjunction with an acoustically rigid and continuous surface, i.e. if the source lies on the floor), the sound intensity measurement on this surface can be used to determine the sound power level of the source itself (according to the standard 9614-Part 1 for measurement at discrete points and 9614-Part 3 for scanning measurement [2,3]). From sound intensity measurements the acoustic quantities such as pressure, active and reactive intensity and power can be calculated. 2.2. Near Field Acoustic Holography Near Field Acoustic Holography (NAH) is an efficient method for the determination of the acoustic field generated by a noise source on the 3D space surrounding the source itself. The technique is based on near field measurement via microphones array. From the acoustic pressure data measured on the array surface (the hologram) 3D acoustic field can be reconstructed, with high spatial resolution, depending on the spatial resolution of the array. The NAH basic assumption [4] is that the sound field can be described as a combination of planar and evanescent waves (with amplitude decreasing with the distance from the source). These latter have amplitude and direction represented by their wavenumber k (kx,ky,k z) defined along a generic direction as the ratio between the frequency and the propagation speed. The plane waves represent the sound field portion that propagates to the far field, while the evanescent waves describe the complex sound field which exist close to the emitting surface and is damped out in the far field. A particular application of NAH is the planar NAH where the measurement surface, i.e. the microphones array, is planar. From the sound pressure data p(rM,t), measured in the near field by the planar array, i.e. at the distance rM from the emitting surface, the sound field can be reconstructed at any plane parallel to the measurement one. The total 3D acoustic field p(r,t) can be determined at arbitrary distance r from the surface radiating the noise. One implementation of the planar NAH is the The Spatial Transformation of Sound Fields (STSF) technique developed at Brüel& Kjær [5,6]. The STSF is based on the measurement of the pressure cross-spectra over a planar surface close to the source. The acoustic field can be reconstructed at any plane parallel to the measurement one by performing a convolution of the pressure cross-spectra with a known Greem function (
e
j(r rM ) k 2 k x2 k y2
4π r rM
). The power of this technique on reconstructing the acoustic field and the particle velocity
directly on the radiating surface has been demonstrated in several applications, like [7]. In particular, measurement of high spatial dense surface vibration velocity performed via scanning laser Doppler vibrometry has been used to validate the particle velocity calculated via STSF on the emitting surface. 2.3. Helmotz Equation Least Square In this paper the results of the Helmholtz Equation Least Squares (HELS) method implemented by SenSound Acoustic Imaging are reported, [8,9]. HELS is based on the measurement of the sound using a conformal array (i.e. reproducing the shape of the radiating surface) placed near the surface itself, Figure 1 (a). The measured sound field is then curve-fitted using spherical wave functions, Figure 1 (b), this allows the visualization of pressure, intensity and velocity on the emitting surface, Figure 1 (c). Mathematically, HELS can be written as J
p(r , f ) C j ( f ) j (r, f )
(1)
j 1
where the expansion functions j are the particular solutions to the Helmholtz equation, at any distance r from the surface. The expansion coefficients Cj can be determined by requiring the assumed-form solution to satisfy the boundary condition at the measurement points, distant rM from the emitting surface:
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1357 J
p( rM , f ) C j ( f ) j ( rM , f )
(2)
j 1
Figure 1 HELS process Therefore the expansion coefficients are determined by matching the assumed-form solution j to the measured data. Finally the errors in this approximation are minimized and optimized using the least squares method. Once the expansion coefficients are determined, the acoustic pressures anywhere in the space and on the source surface can be reconstructed. One unique feature of the HELS method is that reconstruction of the radiated acoustic pressure is not based on spatial sampling, but on synthesis of spheroidal functions. For example, if the source radiates a pure dipole sound, then theoretically reconstruction can be done exactly with no more than four expansion terms or equivalently, four measurements regardless the frequency. For an arbitrarily vibrating structure, the radiated acoustic pressure may be quite complex. Nevertheless, this acoustic pressure field can be expressed as a multipole expansion. Moreover, at low-to-mid frequencies the major contributions are from the first few expansion functions. The higher-order terms represent the small-scale effects and can be neglected. 3. Measurement set up The measurement object is a washing machine of principal dimensions sketched in Figure 2, together with the reference axis system. The front panel has been measured via sound intensity probe, STSF and HELS. The washing machine was driven at a constant rate of 1200 rpm (electric motor drive revolutionary rate, 20 Hz) in order to obtain a steady state excitation condition.
Figure 2 Washing machine sketch
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1358 3.1. Sound intensity measurement set-up Sound intensity measurements have been performed through B&K 3595 p-p probe. A scanning grid of 0.70m×0.98m (x and y direction respectively) with 10 rows and 14 columns was chosen to locate noise sources 2 on washing machine, see Figure 3. Segments of 0.0049m have thus been used to calculate sound power over the measurement area. The scanning grid was placed 0.250 m far away from the front panel of the washing machine. A time acquisition of 30s was chosen to evaluate sound intensity on each measurement segment.
Scan direction
Scanning Grid: Number of Rows=10; Number of Columns=14; Segment Area=0.07m×0.07m=0.0049 m 2; Grid Area=0.70m×0.98m; Measuring distance=0.250m; Acquisition time (T)=30s.
y x Figure 3 Sound intensity measurement and acquisition parameters 3.2. STSF measurement set-up STSF measurements have been performed through B&K 7688 system. All measurements have been performed using an array of 30 microphones (B&K 4935) placed at 0.025 m from the front panel and automatically moved through a Cartesian X-Y moving robot, see Figure 4, and a set of 6 reference microphones fixed at several positions around the washing machine.
(c) (a) (b) Figure 4 STSF microphones array fixed on the Cartesian X-Y moving robot (a) placed in front of the washing machine (b) and scanning positions (c) Several interlaced positions (4x4) have been acquired, see Figure 4(c), in order to have the possibility of covering an area bigger than the frontal surface of the washing machine and of calculating the sound field at sufficient low frequency range. The interlacing allowed to increase the spatial resolution of the microphones from 0.010 m (real distance between microphones) to 0.005 m in order to increase also the high frequency range for the sound field calculation.
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1359 Six reference microphones have been used to perform a Principal Component Analysis (PCA) once all scan had been acquired. This procedure enables the end user to separate correlated and uncorrelated sources during postprocessing phase.
bearing
pump motor
Rear
right side Figure 5 Reference microphones positions
front
3.3. HELS measurement set-up The HELS measurement has been performed with an array of 6x5 microphones, see Figure 6, that was moved over the washing machine front panel in order to reproduce a sufficient conformal measurement surface. The measured patches (6x5) are shown in Figure 7. The blue meshes correspond to the microphones array’s positions and the light blue mesh represents the calculation positions, i.e. located on the washing machine front panel surface. In order to process the patches all together, since they have been measured at different time instants, a reference microphone has been used to realign them in time. The microphones arrays distance from the washing machine surface was of about 0.005 m.
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Reference microphone Figure 6 Microphones array and reference microphone
Figure 7 Microphones arrays positions (blue patches). 4. Discussion of results The sound intensity maps measured by the sound intensity probe at 0.250 m from surface have been compared with the maps calculated by the STSF in the far field at 0.225 m from the measurement plane, it being located at 0.025 m from the emitting surface. In Figure 8 two maps at 400 Hz and 1600 Hz are given, the latter being the
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1361 frequency at higher emission around the tub area. The sound intensity level and distribution calculated by the STSF system is equal to the intensity measured by the intensity probe, with an improvement on spatial accuracy due to the large number of measurement microphones. dB (dBref 10pW/m 2) 400 1/3 octave band
400 20 Hz banwidth
1600 1/3 octave band
1600 20 Hz banwidth
Figure 8 Intensity maps at 0.250 m from the washing machine front panel. The results of the two holographic systems have been compared in terms of Sound Pressure Level (SPL) and air particle velocity calculated on the emitting surface, the washing machine front panel. Figure 9 shows the comparison between the SPL distribution for three frequencies: - at 800 Hz the maximum pressure is radiated by the soap drawer, - at 1200 Hz the noise amplification due to the reflection from the floor is evident, - at 1600 Hz the emission from the porthole is clear. Figure 10 gives the air particle velocity distribution and the active intensity over the washing machine front panel at the most interesting frequency, i.e. at 1600 Hz where the maximum emission is located on the basket aperture. 5. Conclusions Two kind of near field acoustic holography, the planar one based on Spatial Transformation of Sound Fields (STSF) and the spherical one based on conformal array measurements and Helmotz equation least-square method (HELS) have been applied for the first time to a real test case and their results compared. The location of the noise source is very accurate with both the techniques either in terms of SPL, sound intensity and air particle velocity. For an almost planar test case, as it is the washing machine front panel, the test implementation (array microphone preparation, noise source geometry definition) is very simple in the case of STSF, and more complex for the HELS system, however, that complexity is indispensable for 3D objects where conformal arrays are necessary and planar holography is not appropriate. The results of the STSF have been validated also by means of a traditional scanning technique based on sound intensity measurements with a p-p probe following the international standards for noise source location.
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1362 dB (dBref 20Pa) 876 Hz
880 20 Hz banwidth
1196 Hz
1200 20 Hz banwidth
1600 Hz
1620 20 Hz banwidth
Figure 9 SPL maps on the washing machine front panel.
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1363 1600 Hz (m/s)
1620 20 Hz banwidth (dB)
Air particle velocity 2
1600 Hz (W/m )
1620 20 Hz banwidth (dB)
Active intensity Figure 10 Air particle velocity maps on the washing machine front panel. References [1] J. Gomes, F. Jacobsenn, M. Bach-Andersen , “Statistically optimised near field acoustic holography and the Helmholtz equation least squares method: a comparison” [2] ISO 9614-1:1993 Acoustics-Determination of sound power levels of noise sources using sound intensity-Part1: Measurement at discrete points. [3] ISO 9614-3:1993 Acoustics-Determination of sound power levels of noise sources using sound intensity-Part2: Measurement by scanning. [4] J. D. Maynard, E. G. Williams, and Y. Lee, “Near-field acoustic holography. I: Theory of generalized holography and the development of NAH”, J. Acoustical Society of America 78 (4), 1395-1413, 1985; [5] J. Hald and K. B. Ginn, “Spatial Transformation of Sound Fields: principle, instrumentation and applications”, Proceedings of the Acoustic Intensity Symposium, Tokyo, 1987; [6] Spatial Transformation of Sound Fields Software Type 7688 – User Manual, Brüel & Kjær; [7] M. Martarelli, G. M. Revel, E. P. Tomasini, “Laser Doppler Vibrometry and Near-Field Acoustic Holography: different approaches for surface velocity distribution measurements”, Fifth International Conference on Vibration Measurements by Laser Techniques: Advances and Applications, SPIE, 4827, Ancona, 2002; [8] Z. Wang, S. F. Wu, “Helmotz equation-least squares method for reconstructing the acoustic pressure field”, J. Acoustical Society of America 102 (4), 2020-2032, 1997. [9] S. F. Wu, “Methods for reconstructing acoustic quantities based on acoustic pressure measurements”, J. Acoustical Society of America 124 (5), 2680-2697, 2008.
BookID 214574_ChapID FM_Proof# 1 - 23/04/2011
BookID 214574_ChapID 120_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
DESIGN OF AN ACTIVE SEAT SUSPENSION FOR AGRICULTURAL VEHICLES Francesco Braghin , Federico Cheli , Alan Facchinetti , Edoardo Sabbioni
Department of Mechanical Engineering Politecnico di Milano, Campus Bovisa, Via La Masa 1, 20156 Milano, Italy e-mail: [email protected], [email protected], [email protected], [email protected]
ABSTRACT Agricultural vehicle operators are exposed to intense vibrations mainly induced by soil unevenness. Since most of tractors are not equipped with any chassis suspension, the seat is the only system able to reduce the vibrations experienced by the operator. Traditional passive seats amplify vibrations at frequencies close to their natural frequencies. The first natural frequency of typical passive seats with an air spring and a hydraulic shock-absorber is between 1.5 and 4Hz. Thus their efficiency is poor for low frequencies and high amplitudes. In order to improve comfort of operators, active or semi-active suspensions for the seat can be introduced. An active suspension system for the seat of an agricultural vehicle relying on an active air spring is presented in this paper. The capability of the system of improving comfort of operators has been evaluated through simulations carried out with a validated model of the entire vehicle. Results of the proposed active suspension system are compared with ones provided by the passive suspension and the ones provided by an active system where the traditional passive shock-absorber is substituted by a controllable hydraulic actuator.
INTRODUCTION As known, operators of earth moving machines/agricultural vehicles are exposed to prolonged and intense vibrations in the frequency range 0-20Hz mainly induced by road/soil irregularity. Particularly critical from the operator comfort/health point of view are the low frequency vibrations, which are usually characterized also by large amplitudes. The consequences of excessive exposure to vibrations may be loss of concentration, tiredness, decrease of effectiveness of the work being performed and eventually injuries (e.g. back pain). One possibility for reducing vibrations is through the vehicle suspension system. It is however to point out that even nowadays a lot of tractors still are not equipped with any suspension system except the seat suspension. Unfortunately, traditional passive seat suspensions amplify low frequencies vibrations, being their natural frequency usually between 1.5 and 4Hz. In order to improve the seat suspension efficiency (i.e. the capability of the seat of isolating the operator from vibrations) at low frequencies semi-active and active systems have been introduced ([1],[2],[3],[12]). Semi-active seat suspensions mainly rely on MagnetoRheological (MR) fluid dampers (substituting the traditional ones, [2]), while active suspensions may be implemented either by replacing the traditional shock-absorber with an hydraulic actuator ([12]) or by controlling the suspension air spring ([3]). An active seat suspension is presented in this paper adjusting the force provided by the suspension air spring through a servo-valve regulating the inlet/outlet flow rate in the air spring itself. The control strategy mainly relies on the sky-hook theory ([16]). Acting the air spring as an integrator (as it will be explained later on) a control force opposing the seat vertical speed (and thus suppressing seat vibrations) can simply be obtained by feeding back the servo-valve regulating the flow rate in the air spring with a control signal proportional to the vertical acceleration of the seat. The control system has been developed based on a nonlinear model of the passive seat. Ad hoc experimental tests have been carried out on a instrumented seat in order to evaluate the parameters of the model. The capability of the model of suppressing road/soil induced vibrations has been assessed through numerical simulations carried out with the implemented nonlinear seat model. Simulations have been performed both considering the seat model alone and integrating the seat model into a full vehicle model (experimentally validated [13],[14]) able to reproduce the tractor vertical dynamics. This has allowed to evaluate the effectiveness of the implemented active seat suspension in presence of simulated working conditions. For all the simulations, the results provided by the proposed active seat suspension have been
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_120, © The Society for Experimental Mechanics, Inc. 2011
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1366 compared with the ones of the traditional passive seat and with the ones of an active seat suspension in which the traditional shock-absorber has been replaced with a controllable hydraulic actuator.
TESTED SEAT The passive seat shown in Figure 2 has been tested in order to identify the seat model parameters. The tested seat is equipped with a typical scissor suspension system allowing only the seat vertical motion. The suspension system consists of an air spring, a hydraulic shock-absorber and end-stop buffers (Figure 1). Tests have been carried out at CRA-ISMA research centre using a servo-hydraulic actuator to impose the desired displacement to the seat base (Figure 2).
Accelerometers
Air spring Shockabsorber
Figure 1: Suspension system of the seat.
Figure 2: Tested seat on the hydraulic shaker.
[deg]
2
2
[(m/s )/(m/s )]
Two different test campaigns have been carried out. Aim of the first series of tests was to evaluate the capability of the seat to isolate the operator from vibrations. Tests have been performed according to the ISO 5007 consisting in a broadband random excitation signal with an acceleration power spectrum approximately in the range 2-4Hz ([4]). Sweep sine tests at different imposed amplitudes 1.5 (2, 4mm) have also been carried out. Sweep sine Experimental Numerical tests consist in a chirp signal with a constant 1 imposed amplitude and frequency increased from 0.5Hz to 15Hz. During all the tests, an additional 0.5 load has been fixed on the seat in order to simulate 0 the operator weight. According to [4], two different 2 4 6 8 10 12 14 loads have been considered: 40 and 80kg. During 200 the tests, the seat has been instrumented with two accelerometers placed on the seat plane and on 100 the load fixed on the seat in order to measure their 0 vertical acceleration. The vertical acceleration at -100 the seat base and the displacement imposed by the actuator have also been measured (Figure 2). -200 2 4 6 8 10 12 14 As an example of the obtained results, Figure 3 Frequency [Hz] shows the experimental Frequency Response Function (FRF) between the seat plane and the Figure 3: Numerical-experimental comparison: seat base acceleration when a load of 40kg is fixed FRF between the seat plane and the seat base on the seat (blue curve). A resonance peak at acceleration. 2.78Hz is clearly visible showing a dynamic amplification almost equal to 1.3. The second series of test was aimed at assessing the seat suspension force characteristic. At this purpose, the following tests have been performed: sine wave test at a frequency of 0.5Hz and imposed amplitude of ±75mm in order to identify the end-stop buffers characteristics; sine wave test with an imposed amplitude of ±2mm and different frequencies (1Hz, 2Hz, 5Hz) in order to evaluate the suspension hysteresis;
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1367 The scheme of the measurement set-up adopted during these tests is shown in Figure 4. A force transducer has been placed between the upper part of the seat and a fixed frame, while the deflection of the suspension has been measured through an LVDT. As an example of the obtained results, Figure 5 shows the suspension force vs. deflection characteristic during a sine wave test at 1Hz. Frame
100
Force transducer
80 60 40
z
20
t
[N]
Seat suspension system
0 -20
z
-40 -60
Hydraulic actuator
Experimental Numerical
-80
Displacement transducer
-100 -3
Ground
Figure 4: Experimental set up for evaluating the suspension force.
-2
-1
0 [m]
1
2
3 -3
x 10
Figure 5: Numerical-experimental comparison: force vs. displacement.
THE PASSIVE SEAT MODEL On the basis of the previously described tests, a nonlinear model of the seat has been implemented. The model takes into account the air spring force (Fa), the damping force provided by the shock-absorber (Fd), the end-stop buffers force limiting the maximum displacement (Fb) and the suspension hysteresis (Fh). the equation of motion of the system (shown in Figure 6) can thus be written as:
my Fa Fd Fb Fh
0
(1)
where y is the displacement of the seat, while m represents the seat mass including the load eventually placed on it. The air spring force is given by:
Fa
1
Ga
Aef pa p0
(2)
being Ga the spring ratio, Aef the air spring effective area and (pa-p0) the relative pressure. Assuming an adiabatic transform, the actual value of the pressure in the passive air spring is defined by the following first order differential equation ([7]):
p aVa kpaVa
0
(3)
where Aa is the air spring wall surface, Va the air spring actual volume varying according to eq. (4) and k is the air specific heat ratio (assumed equal to 1.4):
Va
§ y ys · ¸¸ © Ga ¹
Aef ¨¨
Aef 'y
(4)
The nonlinear damping force characteristic of the shock absorber as a function of the relative speed has been provided by the manufacturer and it is shown in Figure 7. According to the Bouc-Wen model, the friction force describing the hysteretic properties of the suspension system can be introduced as ([8]):
Fh
kh y y s J y y s Fh E y y s Fh
(5)
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1368 where kh influences the magnitude of the hysteresis cycle, while J and E identify the shape of the hysteresis cycle and represent the visco-elastic properties of the seat suspension when movement direction changes. 500 400
y
m
300 200 100
Fb
Fd
[N]
Fh
0 -100
ys
Fa
-200 -300 -400 -500
Figure 6: Seat physical model.
-0.1
-0.05
0 [m/s]
0.05
0.1
0.15
Figure 7: Force vs. relative seed shock-absorber characteristic.
Finally, the end-stop buffers are modeled as a quadratic nonlinear stiffness acting only when the deflection ys is exceeded:
Fb
kb O y ys
2
O ° ®O ° ¯O
1
y t ys
0
y ys
1
y d ys
(6)
The unknown model parameters have been identified by minimizing the differences between the experimental and the numerical data. An iterative optimization routine (subspace trust region method) based on the interior-reflective Newton method has been used at the purpose ([9],[10]). As an example of the results, the force vs. deflection characteristic of the numerical model is compared with the experimental one in Figure 5 during a sine wave test at 1Hz and imposed amplitude of ±2mm (red curve). The FRF (amplitude and phase) of the numerical model is instead shown in Figure 3 for a sweep sine test with imposed amplitude of 4mm and an additional load of 40kg (red curve). A very good agreement can be seen between the numerical model and the experimental data in both cases.
THE ACTIVE SUSPENSION SYSTEM A control strategy for an active air spring suspension aimed at reducing the seat vertical acceleration and speed has been designed. The equation of motion of the active seat suspension has a similar form as eq. (1), but in this case the force Fa provided by the air spring can be actively controlled by means of a servo-valve regulating the air inlet/outlet flow rate into the air spring. In the following, the model of the pneumatic system for actively controlling the air spring and the design of the regulator are presented. Air spring pneumatic system model In presence of a servo-valve regulating the inlet/outlet air flow into the air spring (Figure 8), eq. (3), becomes:
p aVa kpaVa
kGv RTa
(7)
Eq. (7) has been obtained by applying the equation of perfect gas and the continuity equation for the air spring volume:
paVa
M a RTa
(8)
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d U V GV dt a a
(9)
where Ua and Ta are respectively the density and temperature of air inside the air spring, Ma is the mass of air inside the air spring and Gv is the mass flow rate through the servo-valve. The further assumption of an adiabatic transform has been made:
pa
U
k a
p0
(10)
U0k
where p0 and V0 are the initial pressure and the initial volume of the air spring. This assumption is justified by the fast dynamics of considered process, allowing to neglect thermal exchanges between the system and the environment.
'yA ǻz
FF Aa
ppAa, ,V VAa
G GvV
SVSv
Servo-valve
Air spring
Figure 8: Scheme of the active air spring. The mass flow rate through the servo-valve, Gv, is evaluated according to the characteristic adopted by standard ISO 6358 ([11]), discarding the temperature correction term: ° Gv ° ° ® ° °G ° v ¯
CV p1
if
§ p1 bV ¨ p CV p1 1 ¨¨ 2 © 1 bV
· ¸ ¸ ¸ ¹
p1 d bV p2
2
(11)
if
p1 ! bV p2
where p1 = pAlim, p2 = pa and Gv = |Gv| when the servo-valve is inflating the air spring, and p1 = pa, p2 = p0 and Gv = -|Gv| when the servo-valve is deflating the air spring (where pAlim stands for the air intake pressure and p0 for the atmospheric pressure). The valve sonic conductance CV is considered as a linear function of the servo-valve command Sv, while the critical pressure ratio of the valve bV is assumed constant for the different valve operating conditions:
CV
CV SV
bV
const
(12)
Once pressure inside the air spring is known, the air spring force Fa can be determined by applying eq. (2). It is to point out that now the force provided by the air spring is a function of both the spring deflection 'y and the servo-valve command Sv:
Fa
Fa 'y, Sv
(13)
Design of the regulator In order to design the regulator for the active air spring control, the previously described model of the seat and of the controlled air spring have been linearized:
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my ceq y y s
Fa
(14)
being ceq the damping coefficient associated with the linearization of the force produced by the shockabsorber. The linearization of the air spring force requires instead the linearization of the differential equation describing the pneumatic system (eq. (7)):
p aV0 kp0 Aef 'y
kRTa Gv
(15)
The valve mass flow rate can be considered as a function of the valve command signal SV only, neglecting at this stage the dependence from air spring pressure: GV SV
GV
(16)
Moreover the relationship between the valve flow rate and the command signal can be approximated to a linear one, accounting for the valve bandwidth by considering a first order response. Thus, moving to Laplace domain, the force applied by the air spring can be expressed as:
Fa s
kRTa Aef Gv 0 kp0 A2 kRTa Aef Gv 0 SV s 2 ef Y s Ys s S s K eq Y s Ys s G aV0 s 1 W v s G a V0 G aV0 s 1 W v s V
(17)
where Wv represent the valve time constant and Gv0 is the valve mass flow rate for unit command. As it can be seen, discarding non-linear terms, the air spring force results from two separate contributions, the first one related to the servo-valve command and the second one due to the air spring deflection. In particular, the system composed by the air spring and the servo-valve reacts: to the command signal as an integrator, with a high frequency pole related to the valve bandwidth; to the air spring deflection as a pure gain, corresponding to air spring stiffness (keq). By substituting eq. (17) in eq. (14) and passing to the Laplace domain, the linearized equation of motion for the seat can be obtained:
ms
2
ceq s keq Y s
kRT A G
ceq s keq Y s G V sa 1ef W v 0s SV s a
Ys s
ceq s keq
(18)
s
0
v
+
1 ms 2 ceq s keq
-
kRTa Aef Gv 0 V0G a s
Sv
1 Wvs 1
Sv
Y s
k p 1 Td s
s
Figure 9:Block diagram of the feedback control system. Based on the lienarized model of eq. (18), a Proportional-Derivative (PD) regulator acting on the servo-valve regulating the air flow rate to the air spring has been designed. Aim of the regulator is the reduction of the seat speed:
SV s k p 1 Td s sYref s sY s
k p 1 Td s sY s ;
kd
k pTd
(19)
where kp and kd are the regulator gains, while the reference speed ( y ref ) has been set to zero. It is to point out that, acting the air spring as an integrator, the derivative contribution has the same effect of a sky-hook
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1371 control ([16]). In view of a practical implementation of the proposed control system, it must be pointed out that, although in theory the proportional contribution of the regulator provides a control on the position of the seat, this might result not sufficient for avoiding the seat drift. Thus an additional contribution proportional to the relative displacement between the seat plane and the seat base must be added in eq. (19) in order to maintain the seat oscillations bounded about the static equilibrium position (the sensor needed by the control system are therefore an accelerometer to measure the seat plane acceleration and a position transducer to measure the seat suspension deflection). However, since the seat drift has a slow dynamics, the associated contribution in the control action is expected to be small (and thus not affecting the control system stability) and consequently negligible in the regulator design. Under this assumption, the block diagram of the implemented feedback control system is shown in Figure 9. 1
80
0.9 60 0.998
0.8 0.7
20
0.6 700
0
600
500
400
300
200
100
[m]
Imaginary axis
40
0.4
-20
0.3
-40 -60
0.5
0.2
0.998
0.1
-80 -800
-700
-600
-500
-400 -300 Real axis
-200
-100
0
Figure 10: Root locus of the feedback control system.
0 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
t [s]
Figure 11: Feedback control system to a unitary step input.
The root locus of the closed loop transfer function for variable proportional gain kp and fixed derivative time constant Td is shown in Figure 10. The poles corresponding to the selected regulator gains are represented by dark green triangles. The choice of the regulator gains derives from a compromise between performances and air consumption. Figure 11 is relevant to the step response of the feedback control system. It can be seen that the proposed regulator is able to almost halve the seat vibration and that the overshoot and consequent oscillations are very small.
SIMULATION RESULTS In order to assess the performance of the implemented active suspension system, it has been applied to the seat nonlinear model previously described. Both simulations with the seat alone and integrated into a full nonlinear vehicle model have been carried out. Simulations with the seat nonlinear model alone have allowed to assess the active seat transmissibility, while through simulations with the full vehicle model the performance of the proposed active system in real working conditions could be evaluated. The results of the designed active air spring suspension are compared with the ones of the passive suspension system and the ones of an active suspension system where the traditional passive shock-absorber is substituted by a controllable hydraulic actuator. The control strategy for the hydraulic actuator active control has been implemented on the basis of the literature ([12]). In particular, a PID regulator has been implemented acting on the servo valve of the hydraulic actuator and trying to minimize the seat vertical speed (an approach similar to the one previously described has been followed in the regulator design). Input to the control system are the measured seat plane vertical acceleration and the relative displacement between the seat plane and the seat base. The regulator gain and the hydraulic actuator bandwidth have been selected in order to achieve the same results claimed in [12], i.e. a reduction in the seat vertical RMS acceleration of about 66% with respect to typical air seat suspensions. Vibration tests on the seat nonlinear model The capability of the implemented active seat suspension of isolating the operator from vibrations has been evaluated by imposing at the seat model base (displacement ys, see Figure 6) a random excitation bandlimited in the frequency range 0.5-15Hz with maximum amplitude of ±20mm. A mass of 80kg has been added to the seat. The FRF of the passive seat suspension (blue curve), of the active seat suspension with controlled hydraulic actuator (AHAS, green curve) and of the active seat suspension with controlled air
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1372 spring (AASS, red curve) are reported in Figure 12. As it can be seen, the two active suspension systems show similar performances and they are both able to significantly reduce vibrations. In particular, the seat efficiency at low frequencies is significantly increased and the seat resonance peak is completely damped out. Comparing the different active systems, the AHAS shows an higher isolation capability at low frequencies, while the AASS becomes more effective as the frequency increases (higher then 6Hz). 0.2
AHAS AASS Passive suspension
1
Passive AHAS AASS Excitation
0.18 0.16
2
2
[(m/s )/(m/s )]
2 1.5
0.14
0.5
0.12 4
6
8
10
2
2
[m/s ]
0
0.1 0.08
[deg]
100
0.06
0
0.04
-100
0.02 2
4
6 Frequency [Hz]
8
0
10
Figure 12: Numerical FRFs: comparison between passive and active seats.
2
4
6
8 [Hz]
10
12
14
Figure 13: Seat acceleration spectra during a simulation wit the tractor moving perpendicularly to the ploughing direction at 7km/h. Comparison between the seat suspension systems.
Vibration tests with the seat+full vehicle model As anticipated, in order to assess the performance of the implemented seat active suspension in real working conditions, the seat nonlinear model has been integrated into a validated 7dofs vehicle model able to reproduce the vertical dynamics of a tractor ([13],[14]). The vehicle model is made of three rigid bodies, i.e. the tractor frame, cabin and front axle. These bodies are connected between them and with the ground by means spring damper elements representing the vehicle suspensions and the tires radial stiffness and damping respectively. Thus, the model accounts for the roll motion of the front axle, the roll, pitch and heave motions of the tractor frame and cabin. In order to reproduce real working conditions, the accelerations measured during in-field tests have been imposed to the vehicle model hubs. Experimental tests (carried out within the Italian research project VIBRAMAG, [15]) have been performed on different types of ground (deformable soil and asphalt) and at different speeds. On deformable soil, tests have been carried out both along and perpendicularly to the ploughing direction. When testing on deformable soil, vehicle speed has been varied between 7 and 12km/h while on asphalt vehicle speed has been changed from 30 to 40km/h. Table 1: Percentage reduction of the vertical seat acceleration RMS during simulations of real working conditions. Comparison between the different seat suspension systems. Terrain type
Deformable soil
Asphalt
Direction of motion
Speed [km/h]
Parallel to ploughing direction Perpendicular to ploughing direction
10 12 7 10 30 40
Percentage reduction of the seat vertical acceleration RMS [%] AHAS AASS 64% 63% 66% 65% 46% 41% 54% 52% 37% 35% 41% 37%
During the experimental tests, the vertical accelerations of the tractor hubs have been measured. As an example of the obtained results, Figure 12 shows the spectra referred to the different seat suspension systems during a maneuver with the vehicle running perpendicularly to the ploughing direction at 7km/h. The spectrum of the acceleration at the seat base (i.e. the excitation) is also reported (black pointed-dashed curve). As it can be seen, the results previously obtained on the seat model alone are confirmed. Both AHAS
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1373 and AASS are able to increase the seat efficiency at low frequencies with respect to a traditional passive seat and their performances are comparable. The results of the complete simulation set are reported in Table 1 in terms of percentage reduction of the seat vertical acceleration RMS with respect to the passive seat suspension. A reduction of the RMS higher than 40% is obtained for all the on-field maneuvers and of about the 35% for the on-road maneuvers by both the active suspension systems and their performances are always comparable. It must be pointed out that the performance of the proposed active air spring suspension system are not the maximum achievable (see Figure 10). In fact, as anticipated, the regulator gains have been selected also taking into account the air consumption. In particular, during the regulator design, a maximum air consumption of about 400A.N.R./h has been assumed to be acceptable during the most onerous working conditions on deformable soil (which are more critical than asphalt working conditions). The control system performances can thus be increased/decreased at the cost of higher/lower air consumption.
CONCLUDING REMARKS In order to improve the comfort of agricultural vehicle operators, an active suspension system for the seat has been proposed in this paper, based on a controllable air spring. In particular, a PD regulator aimed at minimizing the seat vertical speed has been implemented. The controller acts on the servo-valve regulating the inlet/outlet flow rate in the suspension air spring. The performances of the implemented active suspension system have been evaluated through simulations carried out with a nonlinear model of the seat alone or integrated into a full vehicle model in order to reproduce real working conditions. Simulations results have shown that the capability of the seat of isolating the operator from low frequency vibrations can be significantly improved by the implemented control system. This lead to reductions of about 50% in the seat vertical acceleration RMS during any kind of working condition (deformable soil or asphalt) with respect to a traditional passive suspension seat. The performances of the proposed active suspension system have also been compared with the ones of an active seat suspension where the traditional shock absorber has been replaced by a controllable hydraulic actuator. Comparable results have been provided by the two different active suspension systems for all the performed tests.
ACKNOWLEDGEMENTS The authors wish to kindly acknowledge the Ministero dell’Agricultura for having founded this research project and all the partners of the project VIBRAMAG, particularly the CRA-ISMA for having supported the experimental tests.
REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
D. Drakopoulos: A review of the current seat technologies in agricultural tractors, National AgrAbility Project-Assistive Technology, 8(3), Spring 2008. S.J. McManus, K.A.St. Clair: Evaluation of vibration and shock attenuation performance of a suspension seat with a semi-active magnetorheological fluid damper, Journal of Sound and Vibration, 253(1), pp. 313-327, 2002. I. Hostens, K. Deprez, H. Ramon: An improved design of air suspension for seats of mobile agricultural machines, Journal of Sound and Vibration, 276, pp. 141–156, 2004. International Standard, ISO 5007: Agricultural wheeled tractors-Laboratory measurement of transmitted vibration, 1990. J. S. Bendat, Nonlinear systems techniques and applications. John Wiley & Sons, 1998. J. S. Bendat, A. G. Piersol, Engineering applications of correlation and spectral analysis. John Wiley & Sons, 1993. P. Beater: Pneumatic drives, system design, theory and calculation, Springer, Berlin, Heidelberg, 2007. T.P. Gunston, J. Rebelle, M.J. Griffin: A comparison of two methods of simulating seat suspension dynamic performance, Journal of Sound and Vibration, 279, pp. 117-134, 2004. T. F. Coleman,. Y. Li. An interior, trust region approach for nonlinear minimization subject to bounds. SIAM Journal on Optimization, 6, 418-445, 1996. T.F. Coleman, Y. Li. On the convergence of reflective Newton methods for large-scale nonlinear minimization subject to bounds. Mathematical Programming, 67(2), 189-224, 1994. International Standard, ISO 6358: Pneumatic fluid power-Components using compressible fluidsDetermination of flow rate characteristics, 1989. TM D.L. Dufner: John Deere active seat : a new level of seat performance, Proc. of AgEng European Conference, Budapest, Hungary, 2002. F. Braghin, F. Cheli, A. Genoese, E. Sabbioni, C. Bisaglia, M. Cutini, Experimental modal analysis and numerical modelling of agricultural vehicles, Proc of IMAC XXVII A Conference and Exposition on Structural Dynamics, Orlando, Florida, USA, 9-12 February, 2009.
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1374 [14] F. Braghin, F. Cheli, M. Colombo, E. Sabbioni, C. Bisaglia, M. Cutini: Characterization of the vertical dynamic behaviour of an agricultural vehicle, Proc. of Multibody Dynamics 2007, ECCOMAS Thematic Conference, Milan, Italy, June 25-28, 2007. [15] F. Braghin, F. Cheli, E. Sabbioni, J. Ventura, M. Cutini, C. Bisaglia: Sensitivity analysis of MB agricultural vehicle model parameters for driver comfort optimisation, Proc. of XXXII CIOSTA & CIGR Conference, Nitra, Slovakia, September 17-19, 2007. [16] D. Karnopp: Active Damping in Road Vehicle Suspension Systems, Vehicle System Dynamics, 12, pp. 291-316, 1983.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Calibration and Processing of Geophone Signals for Structural Vibration Measurements
Rune Brincker, Professor Bob Bolton, Associate professor Anders Brandt, Associate Professor Department of Industrial and Civil Engineering, University of Southern Denmark, Niels Bohrs Allé 1, DK-5230 Odense M, Denmark
NOMENCLATURE x(t) base displacement y(t) coil displacement M moving mass k suspension stiffness c suspension damping ratio f frequency Z cyclic frequency damping ratio 9 H V G
transfer function voltage transduction constant
ABSTRACT Geophones are highly sensitive motion transducers that have been used by seismologists and geophysicists for decades. The conventional geophone's ratio of cost to performance, including noise, linearity and dynamic range is unmatched by advanced modern accelerometers. However, the problem of this sensor is that it measures velocity, and that the linear frequency range is limited to frequencies above the natural frequency, typically at 4-12 Hz. In this paper an instrument is presented based on geophone technology. The sensor is aimed at low vibration level measurements on large civil structures, thus the problem of correcting the bad frequency response becomes essential. The instrument is based on a digitally wired system principle where time synchronization is obtained by GPS, and a good frequency response is secured by calibration and subsequent correction using inverse filtering techniques.
1. INTRODUCTION This sensor type has several advantages. Because of the simple construction, the sensor element is robust and cheap. The sensor is simple to apply in long cable or wireless systems because the passive sensor element does not require power supply. Further, if the sensor is well engineered, it has an excellent linearity
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_121, © The Society for Experimental Mechanics, Inc. 2011
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1376 and a large frequency range. Since normally not only a single coil is used, but two coils in differential coupling, and since the sensor does not include any active elements to introduce potential additional noise, the sensor has an extremely low noise floor. Mainly because of the bad frequency response, the sensor has not been used much in modal applications like Operational Modal Analysis (OMA). However, using a measurement system that samples the signal at the source, transmits the signal trough a digital transmission line, and correcting the signal digitally after sampling, the geophone sensor element is a good alternative to more complicated sensors like sensors based on a force balance principle. In this paper a digital system is introduced that allows the user to perform OMA of very large structures, since two independent systems can be used, each synchronized through GPS, allowing the user to perform OMA using one system for reference measurements and moving the other system around to acquire data for different data sets.
2. SENSOR ELEMENT The problem of the geophone sensor is that the linear frequency range is limited to frequencies above the natural frequency, typically at 4-12 Hz. Also, some would add that it might also be a problem that it measures velocity. However, that can also be seen as an advantage, since if one might need to obtain displacement by integration, the velocity signal needs to be integrated only once. Further, for OMA it does make any difference if the measured signal is velocity or acceleration, thus, in this paper, the main problem is considered to be the non-linear frequency response. The geophone sensor consists of a coil suspended around a permanent magnet, see figure 1. Describing the suspended coil as a one DOF system and using the Faraday law, the frequency response function between velocity of the base and the relative velocity between coil and magnet can be found to, Brincker et al [1]
(1)
H
Z2 Z 02 2 jZ 0Z9 Z 2 0
See figure 1. The natural frequency
Z0
(2)
k M
9
Z0 and the damping ratio 9 c 2 kM
It is useful to note that the following results for the phase
Z0
of the suspended system is given by
M
can be obtained at the natural frequency
2Sf 0 , Brincker et al [1]
M
(3)
S /2
dM df
1 9f 0
These relations can be used for identification of the natural frequency and damping of the sensor element. The actual geophone sensor element has the following mean properties, G
9
0.56
28.8 Vs / m , f 0
4.5 Hz ,
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y(t), coil displacement
x(t), base
Figure 1. Left: Definition of base and coil displacements, Right: Theoretical transfer function for the geophone sensor element.
The main advantage of the geophone sensor element is its simple and robust sensor configuration and the long term experience with the sensor from many geophysical applications. However, also the noise properties are outstanding. For the actual sensor the inherent noise floor is specified by the vendor to
U noise
0.1 nm / s Hz corresponding to an electrical signal of V noise
3 nV / Hz . With proper sealing
and electromagnetic shielding, this noise floor can be assumed to be the noise floor of the applied sensor.
3.
MEASUREMENT SYSTEM
The measurement system is purely digital and consists of a client/computer, a sensor base, and a measurement chain of sensor nodes, see figure 2. The sensor base serves as a hub for all communication to and from the sensor nodes along with GPS handling and time stamping of data and communication of measurement and status information to the client. The base station uses primarily GPS time for time synchronization; however it is also equipped with an internal clock, which ensures that all sensors connected to the base remains synchronized even in the absence of a GPS signal. A typical measurement setup is shown in figure 2. The time synchronization error is smaller than 0.5 ms. The digital technology of the A/D converter is similar to the one described in Brincker et al [2]. The sensors have been produced by CAP2 ApS, Denmark, and further information can be found in the data sheet, [4].
4. CALIBRATION Sensor elements can be easily calibrated if a 3D shaking table is available, in this case, the movement of the shaking table can be measured by a laser, and the frequency response function can be estimated as described in Brincker et al [1]. The main problem in this procedure is to estimate the time delay that might be present in the applied laser, to deal with the inherent non-linearities in many industrial lasers, and finally, to ensure enough movement of shaking table in the low frequency region in order to obtain a good estimate of the FRF in this region.
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Figure 2. One measurement system with base and GPS, and one measurement chain that can contain up to 32 nodes, each housing 3 geophone sensor elements in a 3D configuration and local A/D converters.
When an FRF is estimated, the sensor element properties ,
G , f 0 , 9 can easily be found either by simple
means like using Eq. (3) or by fitting a parametric model to the FRF for instance by using the MATLAB Signal Processing toolbox, [5]. An alternative to calibrating each sensor element separately using a laser to measure the exact movements of the sensor base like described in Brincker et al [1], is to calibrate “on the site” putting all sensors close to each other and using a reference accelerometer.
5. SIGNAL CORRECTION The measured data u y x are divided into data segments and taken from the discrete time domain to the discrete frequency domain by the Fast Fourier Transform (FFT). In the frequency domain the measured signals are corrected using the inverse transfer function given by Eq. (1)
(4)
X
U / H 0
and the corresponding time signals are then obtained by an inverse FFT transform and added by a similar procedure as described in Brincker et al [3]. After such correction and assuming that the noise floor is constant and limited to the inherent noise, the dynamic range of the sensor element can be found to be as shown in figure 3, Brincker et al [1].
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Figure 3. Dynamic range estimated for the considered geophone sensor element after digital correction.
6. CONCLUSIONS A system has been developed that is capable of being used for OMA on large structures. The system supports high accuracy synchronization using several measurement stations far apart from each other, it supports high sensor counts, and takes full advantage of the high sensitivity and low noise floor of the geophone sensor element.
7. REFERENCES [1] Brincker, R., Lagö, T., Andersen, P., Ventura, C.: Improving the Classical Geophone Sensor Element by Digital Correction. In Proc. of the International Modal Analysis Conference. 2005, Orlando, FL, USA, Jan 31 - Feb 3, 2005. [2] Brincker, R, Larsen, J.A, Ventura, C.: A General Purpose Digital System for Field Vibration Testing. In Proc. of the International Modal Analysis Conference. Orlando, FL, USA, Feb. 19-22, 2007. [3] Brincker, R. Brandt, A., Bolton, R.: FFT Integration of Time Series using an Overlap-Add Technique. In Proc. of the International Modal Analysis Conference. 2010, Jacksonville, FL, USA, Feb 1- 4, 2010. [4] Data sheets: CAP2 Geophone Sensor Node, and CAP2 Sensor Base, CAP2 ApS, [email protected] [5] MATLAB Signal Processing Toolbox, Mathworks Inc., www.mathworks.com
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BookID 214574_ChapID 122_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
WAKE PENETRATION EFFECTS ON DYNAMIC LOADS AND STRUCTURAL DESIGN OF MILITARY AND CIVIL AIRCRAFT Wolfgang Luber EADS Defence and Security, - Military Air Systems Rechliner Str., 85077 Manching, Germany E-Mail: [email protected] Keywords:
Wing Tip Vortices, Wake Penetration, Dynamic Loads, Structural Design
Abstract: The effects on the structural design of military and civil aircraft caused by dynamic loads resulting from the flight through high wake velocities which are generated by different types of aircraft have not been sufficiently investigated in the past. Military aircraft might experience this impact during formation or squadron flight or during combat manoeuvres. Civil aircraft could be affected during start and cruise through wakes from other aircraft. Passing through the wake the safety of the aircraft might be critical by wrong guidance, uncontrolled movements or by induced dynamic loads which might cause failure of structure or structural fatigue. The design of aircraft structure accounting for wakes is not state of the art. The standard design includes dynamic loads from PSD gust analysis and buffet or tuned gust analysis, where the intensities of the gust velocities are defined by military or civil specifications and buffet intensities are defined from wind tunnel test results (Ref. 1). Predictions by analysis of wake velocity fields of different aircraft indicate however that the known maximum gust velocities are exceeded and the time history of the velocities experienced by the affected aircraft during penetration is different for example to the 1-cos gust in the discrete analysis. Moreover flight test results of the dynamic aircraft response during wake penetration produced evidence for its criticality in several flight regimes. This contribution concentrates on military aircraft and demonstrates several examples of predicted critical wake fields and results from calculated dynamic response during different kind of penetration. Also examples from flight dynamic responses are discussed. Finally some recommendations for future research and activities are given which should lead to a wake and wake penetration specification for military and civil aircraft required for future structural design and clearance.
1.
INTRODUCTION
Wake penetration can endanger the aircraft safety by resulting uncontrolled aircraft movements/motions, loss of control or by induced static and dynamic loads which might cause failure of structure or structural fatigue. A number of accidents are known from military T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_122, © The Society for Experimental Mechanics, Inc. 2011
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and civil aircraft. One severe accident of a civil aircraft shall demonstrate as an example the present situation, Ref. 4. American Airlines and aircraft manufacturer Airbus have reportedly blamed each other for a 2001 crash that killed 265 people in New York. American Airlines flight AA-0587 crashed on November 12th 2001 a few minutes after takeoff from New York's John F. Kennedy Airport. The Airbus A300-600 aircraft was carrying 260 people when it passed through two wakes generated by a Japan Airlines aircraft that had departed from the same runway two minutes earlier. About 85 seconds into the flight the aircraft reportedly carried out two quick rudder swings to the right and then one all the way left. About five seconds later the rudder movements, the aircraft fin broke off, causing the flight to crash in New York Queens. The tail fin and rudder were discovered about half a mile from the wreckage in Jamaica Bay. Five people were killed on the ground. The airline has blamed the crash on the aircraft's flight control system, while Airbus said the pilot was improperly trained. In general a number of regulations or instructions exist to avoid penetration into safety critical wake environment during take off and cruise. The regulations on ground describe especially the time between the take off of the proceeding aircraft, where the time is depending of the type of the proceeding aircraft, in flight the regulation describe the horizontal and vertical distance to the proceeding aircraft as function of aircraft type, for example Ref. 3. In addition flight safety technologies are under development especially for civil aircraft (Ref. 4, 5), for example an Aircraft Wake Safety Management (AWSM) system, Ref. 4 has been tested using SOCRATES and LIDAR sensors. The Aircraft Wake Safety Management (AWSM system is being developed to provide a total airport system solution to the need for increased airport capacity with enhanced safety. SOCRATES is an airport based laser acoustic wake vortex sensor for the detection and tracking of wake vortex turbulence. UNICORN is an airborne radar for collision avoidance using state of the art components to achieve low cost, small size. For military fighter aircraft it would be of benefit for aircraft safety to design the aircraft structure such that in flight wake encounters of maximum wake velocities from all possible wake generating aircraft are covered for possible short distances. This contribution should highlight only the problems for structural design and not the guidance problems.
2.
WAKE DESCRIPTION
Wake turbulence, see definition from Ref. 2, is turbulence that forms behind an aircraft or helicopter as it passes through the air. This turbulence includes various components, the most important of which are wingtip vortices and jet-wash. Jet-wash refers simply to the rapidly moving gasses expelled from a jet engine; it is extremely turbulent, but of short duration. Wingtip vortices, on the other hand, are much more stable and can remain in the air for up to three minutes after the passage of an aircraft. Wingtip vortices make up the primary and most dangerous component of wake turbulence. Wake turbulence is especially hazardous during the landing and takeoff phases of flight, for three reasons. The first is that during take-off and landing, aircraft operate at low speeds and high angle of attack. This flight altitude maximizes the formation of dangerous wingtip vortices. Secondly, takeoff and landing are the times when a plane is operating closest to its stall speed and to the ground - meaning there is little margin for recovery in the event of encountering another aircraft's wake turbulence. Thirdly, these phases of flight put aircraft
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closest together and along the same flight path, maximizing the chance of encountering the phenomenon. Wing tip trailing vortices behind an aircraft see Figure 1 have many parallels in nature, in very small as well as in very large scales. For examples, the well known galaxies in astronomy and the hurricanes and tornados in the meteorology. One major characteristic for the description of the vortex flow is its tangential velocity which decreases with the distance from the core or centre. At or near the centre, the tangential velocity changes the sign. Depending on the total diameter of the vortex, there is a region inside the core (singularity) with undefined (zero) velocity. In the case of a hurricane, the typical diameter of the eye is in the order of 10 to 100 km, while it is in the order of one meter only for tornadoes. For wing tip vortices, it depends on the distance behind the aircraft, the speed, the size (mass, span) and type (wing sweep angle, taper ratio) of the aircraft, starting in the order of mm's only to 100 m's, see Figure 2.
Figure 1:
Formation flight and maneuvering of military aircraft and vortex generation by different transport aircraft – Boeing 757, 777, 747, Ref. 10
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Figure 2:
Wake vortex development, from Ref. 10
2.1 Analytical Prediction Dynamic load predictions due to uncertainty in aerodynamics are described in detail in Ref. 1. The analytical prediction of wake velocity profiles can be performed using classical approaches or modern numerical aerodynamic CFD tools. For example a classical approach is described in Ref. 2 and 3. In the references the distribution of vortex induced velocities is described by analytical vortex models for known circulation and core radius. From flight measurements it was concluded that for the description of the tangential velocity Vt of a single vortex the formulation of BURNHAM-HALLOCK leads to reasonable results as illustrated in the figure below. For example Ref. 8 documents wake vortex advanced prediction. A number of predictions treat the far field vortex location and decay. Simple models for the prediction of trajectories and decay of the wake vortices have been investigated. These have been implemented into wake warning systems. Large eddy simulation of wake vortices has been developed. Advanced numerical computational models and algorithms exist to study the effect of various atmospheric conditions on wake vortex motion and decay. 2.2
Verification of Predicted Wake Velocities by Different Analytical Approaches and Flight Test Results The predicted results of wake velocity profiles as function of aircraft configuration, flight condition (Mach number, altitude, angle of attack, etc.) may be verified by comparison to predictions of different aerodynamic approaches and by comparison to flight measured flow sensor signals.
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An example of the comparison of predicted and flight measured results of vortex induced velocities from Ref. 2 is shown below in the Figure 3. More evidence of the validity of other analytical predictions is however required for different configurations and flight conditions.
Figure 3: 3.
Vortex induced velocity distribution, from Ref. 2
WAKE PENETRATION DESCRIPTION
The wake profile behind a generating aircraft is characterized by up-wash and downwash areas in span wise direction. These up- and downwash areas can be penetrated in arbitrary manner. The wake velocity field can be crossed in perpendicular way to the path of the wake generating aircraft or in direction of the proceeding wake of the generating aircraft by different heading angles, Figure 4 Thus the structural load factor, the altitude and climb/sink rate and the roll angle and roll rate of the following aircraft may change and vibrations of the elastic aircraft can occur which produce dynamic loads. The wake induced effects can result in degradation or loss of aircraft guidance and the high structural vibrations will lead to corresponding high dynamic loads or even to structural failure.
Figure 4:
Wake profile behind generating aircraft (Ref. 9)
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4.
ANALYTICAL PREDICTION OF AIRCRAFT RESPONSE
The analytical prediction of aircraft response and dynamic loads is possible through the prediction of wake velocities by wake generating aircraft followed by an analytical dynamic response calculation and dynamic load calculation of the wake receiving following elastic aircraft using a flexible aircraft model. The wake generating aircraft produces a wake profile, which depends on the flight condition of the generating aircraft and the distance to the wake receiving following aircraft and its heading position, see Figure 5 below.
Figure 5: 4.1
Wake profile from wake generating aircraft, from Ref. 11
Analytical Prediction of Wake Velocities for further Aircraft Response Analysis
Wake velocities have been analytically predicted for further flexible aircraft response calculations. The wake velocities generated by an Eurofighter aircraft and a civil aircraft VFW 614 have been derived at a distance of 800 ft between the two Aircraft. The flight conditions of the wake generating aircraft are described in the table 1 below.
V h AoA Nz
[ KCAS ] [ ft ] [°] [-]
Profile #1 Fighter Type 350 10 000 20 -
Profile #2 Tanker Type 116 10 000 1
Table 1: Flight Condition of Wake generating Aircraft, from Ref. 11
Figure 6:
VFW 614 wake prediction; VFW614 speed: (Mach 07; 10 kft), 116KCAS
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The generated wake velocity by a VFW614, by Eurofighter and by A310 at a distance of 800 ft is shown in Figures 6, 7 and 8. Maximum speed for VFW614 wake velocity is 14 m/sec for Eurofighter 55 m/s and Airbus A310 42 m/s. A comparison of wake by VFW614 and Eurofighter to the maximum gust velocity applied by the discrete gust analysis has been, demonstrates that the wake velocities are higher than the gust velocities. In the structural dynamics analysis the term "discrete gust analysis" is used. The discrete gust analysis is defined in the time domain by the (1-cos) gust. If the gust length varies the analysis is called "tuned gust analysis". In the design process the maximum gust velocity is defined by 66 ft/s, whereas the maximum wake velocity is x x x
187 ft/s with Eurofighter as generating A/C 138 ft/s with Airbus A310 as generating A/C 46 ft/s with the commuter A/C VFW614 as generating A/C
Vertical Induced Velocity [m/s] 30 20 10 0 -50
-40
-30
-20
-10
-10
0
10
20
30
40
50
-20 -30 -40 -50 -60
Figure 7:
Calculated wake generation by Eurofighter 350 KCAS, 20 deg. AoA; 10 000 ft a distance of 800 ft for further dynamic response calculation of flexible A/C
Figure 8:
Airbus A310 wake prediction max vertical velocity 42 m/s; generating aircraft A310 Speed: 325 KCAS
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4.2
Analytical Predictions of Dynamic Response and Loads
The dynamic response of the Eurofighter in the wake of the VFW614/ATTAS and in the wake of a Eurofighter prototype has been calculated (Ref. 11). This activity was initiated in order to derive advice for the flight test with respect to the distance to the wake generating aircraft and the speed of the wake receiving aircraft. The conditions for the prediction of the wake receiving aircraft crossing speed is 350 KCAS, altitude = 10000 ft, Mach 0.64, Nz = 6 at a distance to the wake generating aircraft of 800 ft. The accelerations of the forward, centre and rear tip pod station, the forward and rear station of the foreplane and two fuselage stations, the location of which are demonstrated in Figure 10, had been predicted. The acceleration reaches an almost critical high value of x x x
69.6g at the rear tip pod station 87.6g at the rear foreplane station and 20.6g at the front fuselage
The loads obtained at the different monitoring stations of the aircraft are inside the allowable loads envelopes for the wake profile of VFW614. For the fighter wake profile the loads exceed the allowable loads envelopes at some of the wing and foreplane monitoring stations. From the calculated dynamic responses and loads it was concluded, that for flight test the distance of 800 ft had to be increased to a distance > 1500 ft at 350 KEAS and 10000 ft. The influence of the heading angle of the wake crossing aircraft was found to be not very significant due to the short wake wave length. It can be concluded that the requirements for the structural design using the discrete gust specifications do not cover the wake environment produced by military fighter aircraft at high speed and high g levels for distances below 1500 ft. Since during dog fight the speeds of wake producing and wake receiving aircraft might be higher than 350 KEAS and distances to the wake receiving aircraft can be lower than 1500 ft. Therefore during future design of military aircraft structure it would be beneficial to include requirements for dynamic loads from wake velocities. 5.
EXAMPLES OF FLIGHT TEST RESULTS
Flight test results are available from a wake receiving military aircraft (Eurofighter -Typhoon) due to the wake generation by a Eurofighter and a VFW 614. Analytical predictions of wake velocities are available from wake generating aircraft see Figures 5 to 7 for a distance of 800 ft, i.e. an Eurofighter prototype, a VFW614 and a tanker Airbus A310, configurations see Figure 9 below. The flight test results of the wake receiving aircraft are local acceleration signals at different locations, wing tip stations, outer foreplane stations and fuselage stations.
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Figure 9:
Wake producing aircraft Eurofighter, VFW 614 and Airbus 310
The wake penetration flight tests have been performed recently. The wake penetration was performed by the application of different maneuvers which are described in Figures 11 and 12. Maneuver 1 consists in a 1g straight and level flight, maneuver 2 is a horizontal crossing in a g-turn and maneuver 3 is a vertical crossing. The response of a Eurofighter aircraft was tested due to the wake generated also by an Eurofighter prototype aircraft and due to the wake of a transport type VFW614 aircraft. The flight test results have been analyzed w. r. t. local accelerations on foreplane, outer wing, (the location of which is described in Figure 10), and on fin. It was intended to investigate the flight test data to have a preliminary rough estimate of the maximum local accelerations by extrapolation of flight tested data and by application of wake predictions for VFW614 and AIRBUS A310. A310 wake predictions besides the Eurofighter predictions have been applied to derive maximum wake conditions. The present evaluation of flight test results performed for 250 KCAS and a distance of 2500 ft shows already high levels of accelerations in terms of foreplane acceleration (90g) and wing tip acceleration (30g) for the flight tested conditions. Since the flight tests have performed at moderate wake velocity conditions at distances > 1500 ft, it is expected that the mentioned values of acceleration will increase significantly with reduced distances to the wake generating aircraft and higher speeds of the receiving aircraft. Therefore the existing flight test data have been extrapolated to the maximum possible wake velocities. Aerodynamic models are available which are able to predict the maximum velocities in terms of v and w. For the further evaluation of the flight test results w. r. t. maximum condition x x
5.1
the velocities v and w due to wake used in flight simulation for the flight conditions of existing flight tests (for example Eurofighter against Eurofighter) and the maximum velocities due to wake at the aircraft points used in flight simulation have to be calculated.
Manoeuvres for Wake producing and Wake receiving Aircraft
Different flight maneuvers had been defined for the wake penetration flight tests. In Figure 11 the flight maneuver 1 is illustrated, both the lead and the following test aircraft, the wake receiving aircraft, are in 1g straight level flight condition. Results of test 16 of table 2 are discussed in the following chapter 5.2.
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Figure 12 demonstrates the maneuver 2 which is characterized by horizontal crossing in a gturn. Maneuver 3 is not shown here, but it is the vertical crossing of the wake receiving aircraft.
Figure 10:
Accelerometer installation for flight test
Number
Start [sec]
Stop [sec]
Distance [ft]
Speed [KCAS]
1
53650
53660
1500
180
1g
2 3
53690 54010
53720 54020
500 3000
180 180
1g 20
4 5 6
54090 54170 54300
54100 54180 54310
1500 1000 3000
180 180 300
20 20 3g
7 8 9
54390 54540 54655
54403 54550 54665
2000 1200 5000
300 300 250
3g 3g 1g
10 11 12 13 14 15 16
54804 54934 55106 55215 55300 55456 55544
54814 54944 55116 55224 55310 55466 55555
4500 3000 3500 3000 2500 2000 2500
250 250 250 250 250 250 250
1g 1g 1g 1g 1g 1g 1g
Table 2:
nz/AoA Flight condition of [g/deg] Wake generating aircraft 1g, 180KCAS of Euro fighter 20AoA, 180KCAS of Euro fighter
3g, 300KCAS of Euro fighter
max g-turn, 400KCAS of Euro fighter
Flight conditions of wake receiving Eurofighter aircraft
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5.2 Time histories of flight measured dynamic responses on Eurofighter due to wake generation by Eurofighter and by VFW614 (tanker- transport type aircraft) Time histories of the responses at Typhoon (Eurofighter) due to Eurofighter prototype wake generation are described in the Figures 13 to 18. The flight condition for wake generating Eurofighter prototype at maximum g-turn was 400 KCAS, the flight condition of the wake receiving Eurofighter was 250 KCAS at 1g at a distance of 2500 ft. In figure 19 the Eurofighter response due the wake velocities generated by the VFW614 is demonstrated.
Straight and Level 1g Figure 11: Maneuver for wake penetration flight tests from ref. 7
AoA/nz (vertical)
AoA/nz (horizontal) Horizontal crossing in g-turn Figure 12: Maneuver for wake penetration flight tests from ref. 7
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time histories left O/B-flap 20
34 29
15
24 10
[deg]
0 55549,200
14
55549,300
55549,400
55549,500
55549,600
55549,700
55549,800
55549,900
-5
9 55550,000 4
load [g]
[deg/s]
19 5
-1 -10 -6 -15
-11
-20
-16
time [s]
delta ob [deg] g accel. wing tip aft LH
delta ob flap rate [deg/s] g accel. wing tip fwd LH
q pitch rate [deg/s]
Figure 13: Left O/B flap angles and rate; q and acceleration of fwd wing tip (The right y-axis belongs to the g-acceleration loads and the left one to the other states) - flight test results of the receiving aircraft at 250 KCAS, distance 2500 ft, 1g, Eurofighter as wake generating aircraft (400KCAS at max turn rate)
time histories left I/B-flap 35 45 30 25
15 5 55549,200
[deg]
20
10 55549,300
55549,400
55549,500
55549,600
55549,700
55549,800
55549,900
-15
55550,000 5
load [g]
[deg/s]
25
0 -5
-35
-10 -15
-55
-20
time [s]
delta ib [deg]
Figure 14:
delta ib flap rate [deg/s]
q pitch rate [deg/s]
g accel. wing tip aft LH
Left I/B flap angles and rate; q and acceleration of aft wing tip
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20
100
15
80
10
60
5
40
0 55549,200
55549,400
55549,600
55549,800
55550,000
55550,200
[deg]
-5
20 55550,400
load [g]
[deg/s]
time histories
0
-10
-20
-15
-40
-20
-60
-25
-80
-30
-100
time [s] eta [deg]
Figure 15:
eta rate [deg/s]
q pitch rate [deg/s]
g accel. FPL tip aft LH
Left foreplane angle and rate; acceleration of foreplane tip;
time histories
3
100
1
80
55548,000
55549,000
55550,000
55551,000
55552,000
55553,000
55554,000 60
40
-3
20 -5 0 -7 -20 -9
-40
-11
-60
-13
-80
-15
-100
q pitch rate [deg/s]
Figure 16:
nz normal accel. [m/s2]
time [s] g accel. FPL tip aft LH
Time history of q and nZ and acceleration of foreplane tip
load [g]
[deg/s]
[m/s2]
-1
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time histories rudder 35
20
25
15
15
10
55549,200 -5
5 55549,300
55549,400
55549,500
55549,600
55549,700
55549,800
55549,900
55550,000
0
load [g]
[deg/s]
5
[deg]
-15 -5 -25 -10
-35
-15
-45
-20
-55 -65
-25
time [s]
Figure 17:
g accel. of fin tip aft
r yaw rate [deg/s]
delta rud rate [deg/s]
delta rud [deg]
Rudder angle and rate; acceleration of fin tip
time histories
2
30
1
20
55549,000
55550,000
55551,000
55552,000
55553,000
55554,000
10
-1 0 -2
load [g]
[deg/s] [m/s2]
0 55548,000
-10 -3 -20
-4
-5
-30
time [s] r yaw rate [deg/s]
Figure 18:
ny lateral accel. [m/s2]
g accel. fin tip aft
Time history of yaw rate, lateral nY and acceleration of fin tip due Eurofighter wake
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DA1 - VFW614 wake flight, 56 [g] case at fpl UCAS 400[kts], 1g
15
60
40 10
20
[deg]
0
[g]
[deg/s]
5
0 30165,000
30165,200
30165,400
30165,600
30165,800
30166,000
30166,200
30166,400
30166,600
30166,800
30167,000
-20
-5 -40
-10
-60
time [s] eta fpl angle [deg]
Figure 19: 5.3
eta rate [deg/s]
q pitch rate [deg/s]
accel. at fpl tip RH [g]
accel. at fpl tip LH [g]
Foreplane angle and rate; acceleration of foreplane tip LH/RH
Comparison to Predicted Responses
5.3.1 Evaluation of the flight test results Eurofighter against Eurofighter The flight test results of Eurofighter flight with another Eurofighter as wake generating aircraft have been evaluated for the flight condition 250 KCAS, distance 2500 ft, 1g (see table 2). The wake generating Eurofighter was operating at maximum speed. Time histories of Eurofighter responses due to Eurofighter as wake generator have been analyzed. The following evaluation was carried out for flight case 16 (highlighted in Table 2), since for that time period the maximum loads were observed compared to the other flight cases. In Figure 17 the time history of the vertical acceleration of the LH rear and forward wing, the corresponding outboard flap deflection in degree and the outboard flap rate are shown together with the pitch rate. The LH outer wing acceleration reaches a maximum value of 30 g at rear wing and 21.5g at the forward position, resulting from the response of the first wing bending mode (6.7 Hz) and the wing torsion mode around 30 Hz. The damped pitch rate response q shows a 6.7 Hz vibration due to wing bending mode response at the IMU station. Also the outboard flap and flap rate response shows the motion due to first wing bending mode due to feedback of notch filtered pitch rate. In Figure 14 the vertical acceleration of the LH rear wing, the corresponding LH inboard flap deflection in degree and the LH inboard flap rate are shown together with the pitch rate. The LH outer wing acceleration reaches a maximum value of 30g as shown in Figure 13 and 14, resulting from the response of the first wing bending mode (6.7 Hz) and the wing torsion mode around 30 Hz. The damped pitch rate q response shows a 6.7 Hz vibration due to wing bending mode response at the IMU station. Also the inboard flap and flap rate response shows the motion due to first wing bending mode due to feedback of notch filtered pitch rate.
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In Figure 15 and 16 the vertical acceleration of the foreplane, the corresponding foreplane deflection in deg. and the foreplane rate are shown together with the aircraft pitch rate. The foreplane acceleration reaches a maximum value of 90 g, resulting from the response of the foreplane bending mode response. The damped pitch rate q response shows a 6.7 Hz vibration due to wing bending mode response at the IMU station. The foreplane deflection and foreplane rate response shows the motion due to first foreplane bending mode at around 30 Hz due to feedback of notch filtered pitch rate. Figure 17 shows the time history of Nz. In Figure 22 the lateral fin tip acceleration, the corresponding rudder deflection in deg. and the rudder rate is depicted. The fin tip acceleration reaches a max value of 18 g, resulting from the response of the fin bending mode response at around 13 Hz. Since the wake receiving aircraft was flying at 1g and did not perform a roll maneuver, the lateral acceleration is believed to be not fully representative for maximum combat conditions w. r. t. wake induced accelerations. From the comparison of vertical and lateral predicted velocities w and v, it could be detected, that the deviations in w and v are not very high, therefore the actual fin acceleration would reach similar values for instance at a bank angle of 90 degrees. Figure 18 shows the yaw rate and lateral acceleration. Conclusion of the test results of wake generating Eurofighter against wake receiving EF2000. The outer wing acceleration and therefore corresponding dynamic loads and the foreplane response is quite high at the evaluated moderate wake flight test results and will reach significant higher values for maximum wake velocities and higher dynamic pressure and shorter distances < 2500 ft to the generating aircraft. The Eurofighter wake chosen for the evaluation was at maximum turn rate at 400 KCAS. Therefore the maximum lift coefficient CA was reached at a near maximum speed of a combat maneuver. Consequently the extrapolated response according to the extrapolation to maximum possible speed described below is only dependent on the ratio of dynamic pressure and on angle of attack due to the wake velocity w. At constant Mach number the response is proportional to the rate of the dynamic pressure
U
V 2 for maximum velocity and dynamic pressure of flight test and proportional to ratio of 2 angle of attack Į at max. velocity and of angle of attack Į for flight test velocity of the receiving aircraft due to wake w wake of the generating aircraft, which produces an angle of attack at the receiving aircraft Į receiving aircraft due to the w wake of the generating aircraft (i.e. the enemy), therefore wwake gener AC wwake D receive AC and D wake ; wwake is proportional to V c A (D , Mach) Vreceive AC V of the wake generating aircraft. Therefore the factor for extrapolation of flight tested responses (for example accelerations) is: Vmax receive AC wwake gener AC V flight test recieve AC w flight test gener AC For example for the receiving Eurofighter aircraft at 250 KCAS, 1g and the generating Eurofighter aircraft in flight conditions for max turn rate and 400 KCAS, assuming that maximum conditions are already reached with max turn rate and 400 KCAS (i.e. flight test w generated is equal to max w), then the extrapolation is performed with the ratio of the maximum and flight tested velocities of the receiving aircraft.
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Assuming a maximum velocity of the receiving aircraft of 500 KCAS (flight test 250 KCAS), then the max outer wing acceleration would be 60 g and the max foreplane acceleration would be 180 g. These results already indicate the criticality of the wake penetration effects on flexible aircraft response. Even if the speed of the receiving aircraft would be 400 KCAS the outer wing dynamic loads might become critical, dynamic load assessment have therefore to be performed. This first preliminary guess of aircraft wake penetration dynamic response of flexible aircraft has to be confirmed / updated with further more detailed investigations which should be based upon detailed wake velocity information and refined assumptions of maximum wake conditions and flexible aircraft dynamic load calculations. 5.3.2 Evaluation of flight test results of Eurofighter due to generating aircraft VFW614.
Data for the foreplane are presented in Figure 19 showing foreplane deflection and rate and LH/RH foreplane tip acceleration. The maximum value of acceleration was 56 g at RH foreplane tip. The receiving Eurofighter speed was 400 KCAS Extrapolation of these data can be performed using predicted velocities. Preliminary extrapolation of existing Eurofighter accelerations using wake information of ATAS/VFW614 and of maximum wake definition for A310. VFW614 speed was at Mach 07, 10 kft KCAS=116; for A310 the maximum predicted vertical wake velocity of 42 m/s exceeds the maximum discrete gust speed of 66 ft/s by about a factor of 2. However the development of the wake impulse with time is different to the 1cos gust shape and the duration smaller than a 3c discrete gust. Therefore only high frequency elastic modes will be excited which would lead to different dynamic loads compared to discrete gust design conditions. The ratio of max vertical LH wing wake velocity (from A310, ca. Mach 0.85, 30kft) to ratio of max vertical LH wing wake velocity about 42/12 The extrapolation factor for wing acceleration by
Vmax receive AC wwake gener AC V flight test recieve AC w flight test gener AC
can only be generated if the wing tip data are available. 6.
SAFE FLIGHT IN WAKE ENVIROMENT
Safe flight in wake environment may be achieved through the determination of safe distances and additional consideration of wake velocity profiles during the design and clearance of new air vehicles. 6.1
Definition of Distance from generating to receiving Aircraft
An example is presented in Figure 20 below, demonstrating the variation of wake velocity with longitudinal x and lateral distance y as function of incidence for the Eurofighter.
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The decrease of wake velocity in x-direction can be used as indicator for the safe distance definition, if the aircraft is designed by the application of dynamic local loads from tuned gust analysis using a 66 ft/sec gust velocity, the safe distance for wake penetration shall be derived from the extrapolated wake velocity diagram w versus x- distance.
Figure 20: 6.2
Wake velocity behind Eurofighter
Calculation for static and dynamic loads for wake penetration
The static loads due to wake penetration shall be based upon the dynamic response of the rigid aircraft with flight control system using the flight dynamic model of the aircraft generated by the wake excitation. The dynamic loads on arbitrary local aircraft monitoring stations shall be calculated using the analytical model of the flexible aircraft. The derivation of local inertia and unsteady aerodynamic forces of the vibration modes together with the wake induced unsteady aerodynamic forces form the dynamic wake induced loads. 6.3
Definition of allowable loads envelopes for Design and clearance
For the design and clearance allowable load envelopes at local structure monitor stations have to be defined which include: x
Manoeuvre loads
x
Dynamic gust loads from tuned gust analysis
x
Dynamic buffet loads
x
Dynamic impact loads
x
Landing loads
x
Wake penetration loads
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The local load envelopes in terms of shear force Fz versus torque My and bending moment Mx versus torque My have to be defined. To prove that the structure withstand the gust and buffet loads as well as the wake penetration loads, the actual calculated or measured loads must be inside the Ale's (Allowable Loads Envelope). 7.
CONCLUSION
The results of the first preliminary assessment of flexible aircraft dynamic response from wake penetration flight tests already indicate the criticality of the wake penetration effects on flexible aircraft wing, fuselage and fin dynamic loads. The dynamic loads of the wake receiving aircraft should always include buffet loads from high angle of attack manoeuvres. The first preliminary guess of aircraft wake penetration dynamic response of flexible aircraft for assumed max conditions has to be confirmed / updated with further more detailed investigations which should be based upon detailed wake velocity information and refined assumptions of maximum wake conditions. The following future actions are recommended: x x x x x x x x x
8.
Comparison of predicted accelerations and dynamic loads from wake penetration and from discrete tuned gust for a representative set of military aircraft Refined investigation of max. dynamic response using detailed wake velocity information for a number military aircraft as wake generators Definition of wake velocities from existing wake penetration flight testing (for example DLR VFW614 and Eurofighter aircraft) and other military aircraft Clear definition of predicted max. wake velocities of generating aircraft Definition of max flight conditions of attacking aircraft in wake condition during combat maneuvers in terms of speed, Mach number , g- conditions Preparation of an analytical flexible aircraft model calculations to treat wake penetration using wake information from Performance of analytical dynamic response calculations for the generation of sectional dynamic loads on aircraft monitoring stations for comparison to Ale's. Definition of times spent in high, medium and low wake velocity conditions of a different military aircraft in 600 flight hours (wake – combat maneuver correlation for fatigue life prediction). Performance of fatigue life assessments for outer wing, fin and foreplane based on the definition above RECOMMENDATION
Finally some recommendations for future research and activities are given which should lead to a wake and wake penetration specification for military and civil aircraft required for the structural design and clearance.
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8.1
Future Aerodynamic Research
Although a big variety of analytical investigations are available from past aerodynamic research, as for instance documented in Ref. 1-3 and 8 it is recommended in order to establish safe aerodynamic predictions to perform intensive aerodynamic wake vortex research campaigns also including results of comparisons of different CFD tools and validations using wind tunnel and in flight measured wake velocities for the validation of the tools. 8.2
Future Flight Test Programs of Wake Penetration of Different Aircraft
The knowledge of flight tested wake characteristics of different civil and military aircraft with respect to velocity intensities as function of speed, Mach number, altitude and AoA and nz of the wake generating aircraft and the distance to the aircraft is very limited. Flight test programs should therefore be initiated in order to establish a sufficient broad data basis for future validation of analytical wake velocity predictions and for definitions of aircraft design requirements followed by the formulation of corresponding design and clearance specifications. The data acquisition shall be performed by a wake receiving aircraft equipped with flow sensors. In addition to the flight measurement of wake vortex velocity characteristic also the dynamic response at different locations of the tested aircraft shall be measured for future validation of analytical dynamic response calculations which should include the flight measured wake velocities. 8.3
Verification of wake predictions
The verification of the analytical wake velocity predictions shall be performed through comparison with flight measured characteristics. The verification of the analytical dynamic response predictions shall be performed through the comparison of flight measured local aircraft accelerations. 8.4
Definition of Guidelines for Safe Flights in Wake Environment
From the results of the proposed investigations on wake penetration guidelines with respect to aircraft structural aspects shall be summarized for the safe flight of military and also civil aircraft in wake environment. The guidelines shall be part of military and civil specifications for static and dynamic loads and vibrations for aircraft design and flight clearance of aircraft structure and equipment.
9. ACKNOWLEADGEMENT The author would like to thank Dr. Ing. Jürgen Becker for countless useful discussions extending over many years, where we worked together and for his encouragement and support for preparing this paper
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10.
REFERENCE
1
W. Luber, J. Becker The Impact of Dynamic Loads on the Design of Military Aircraft IMAC XVI, 16th International Modal Analysis Conference; Santa Barbara, CA USA, January 1998
2
C. Schwarz, K.-U. Hahn Gefährdung beim Einfliegen in Wirbelschleppen, DLR Braunschweig, Institut für Flugsystemtechnik, DGLR_2003-242.
3
K. - U. Hahn Coping with Wake Vortex, 23rd International Congress of Aeronautical Sciences, Toronto, Canada ICAS Proceedings, 2002 K. - U. Hahn Safe limits for wake vortex penetration, AIAA-2007-6871
4
Flight Safety Digest, Flight safety Foundation, March-April 2002, Data show that U.S. Wake-turbulence Accidents are most frequent at low altitude and during approach and landing
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Flight Safety Technologies, Inc. Completes Initial Aircraft Wake Safety Management Milestone Business Wire, March 5, 2007 - AIRLINE INDUSTRY INFORMATION(C)1997-2004 M2 COMMUNICATIONS LTD
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US Patent 7333030 Method and system for preventing an aircraft from penetrating into a dangerous trailing vortex area of a vortex generator
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M. Hinterwaldner, J. Schwab Wake penetration- a tumultuous farewell of 1st Typhoon Prototype aircraft, EADS Military Air Systems, Internal Report
8
W. Jackson, Wake Vortex Prediction, Transportation Development Centre, Transport
Canada, TP 13629E, 2001
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S. Andrew. S. Carten, Jr. Aircraft Wake Turbulence An Interesting Phenomenon Turned Killer, Equipment Engineering and Evaluation Branch, Aerospace Instrumentation Laboratory, Air Force Cambridge Research Laboratories, Tufts University, Document created: 04 May 2004
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www.AviationExplorer.com What causes aircraft turbulence and vortex effects
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Carlos Maderuelo-Munoz EF2000 – Loads Evaluation for Wake-Penetration EADS-CASA, Getafe Spain, Internal Report
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Numerical and experimental modeling for bird and hail impacts on aircraft structure
M.-A. Lavoie*, A. Gakwaya*, Marc J. Richard*, D. Nandlall**, M. Nejad Ensan*** and D.G. Zimcik*** *Department of Mechanical Engineering Laval University 1065 avenue de la Médecine Québec, QC, G1V 0A6 **Defense Research Development Canada Valcartier, QC ***Institute for Aerospace Research Building U66A, Uplands Ottawa, ON, K1A 9R6 Abstract Aircraft bird-strike events are very common and dangerous. Hailstone impacts represent another threat for aircraft structures. As part of the certification process, an aircraft must demonstrate the ability to land safely after impact with a foreign object at normal flight operating speeds. Since experimental studies can be cost prohibitive, validated numerical impact simulation seems to be a viable alternative. Modelling of these soft body impacts still represents a challenge, involving modelling of both the target and the projectile. Here the smooth particle hydrodynamics method (SPH), which has been used successfully in ballistic applications involving bird strike scenarios, is extended to hail impact. The paper thus presents the meshless SPH numerical method as a novel modeling approach. The method is applied to model bird and hail impacts which are problems that traditional FEM based modeling methods typically struggle to solve because of involved mesh distortion problems. The numerical results are then evaluated by comparing with the data collected during recent experimental tests. The data acquisition methods are also described and evaluated for applications where the short duration of the impact presents a challenge. The accuracy of the numerical results allows us to conclude that the models developed can be used in the certification and/or design process of moving (aircraft) and stationary (wind turbines) composite structures subject to bird and hail impact. Keywords: bird & hail impact, SPH method & experimental validation 1. Introduction Nowadays, predictive numerical methods are an intrinsic part of aircraft and other high performance design. This has been strongly motivated by the relatively low cost of simulating events before conducting destructive tests and by the always increasing accuracy and efficiency of numerical methods. When modeling an impact event, the components are typically classified into two categories: the projectile and the target. A great deal of finite element work has been performed to develop element and material models that can accurately predict the behaviour of metallic and composite materials under large deformations induced by high velocity loadings. Although less frequent, impacts with soft bodies also pose a threat to structures. However, the amount of information and number of tools available to produce an accurate numerical model of such projectiles are usually limited. Because of its low computational cost, its reasonable precision and stability compared with finite elements method (FEM) and, more importantly, because of its ability to handle large distortions by avoiding the need for intensive FEM remeshing, SPH is a competitive approach compared to FEM and is increasingly being used in
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_123, © The Society for Experimental Mechanics, Inc. 2011
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1404 some fast-transient dynamics problems [1]. Several authors have proposed to couple FE and SPH which seems a reasonable approach in order to benefit from the advantages of both formulations [2,3]. A simple way to describe the SPH method is to imagine a structure finely meshed with solid elements where the elements themselves are discarded and only the nodes are kept. The connectivity between the nodes no longer depends upon the mesh but rather on the proximity of the neighbouring nodes, now called particles. All information such as stress, displacement, mass and density are now computed at each node. It also includes the contribution of each neighbouring particle which is proportional to the proximity of each given neighbour. Suitable information on the mathematics of meshless methods can be found in Nguyen et al. [4]. In a first attempt to model birds and hail as projectiles, it was observed that the experimental information was relatively limited. Hence, bird strike and hail impact modeling was sponsored by the Consortium for Research and Innovation in Aerospace Quebec (CRIAQ) and bird and hail tests were respectively conducted at the Defence Research and Development Canada (DRDC) and the National Research Council Canada (NRC) air gun facilities. This paper deals with the experimental studies of bird and hail projectiles that have been performed in order to increase the experimental data available as well as provide validation with an efficient numerical method, the smoothed particles hydrodynamics (SPH). 2. Bird impact tests In recent years, efforts were increased to model the bird impact event and predict the viability of aircraft structures prior to the mandatory expensive destructive certification procedures. Different modeling techniques were studied [5] as well as ways to evaluate the reliability of the obtained numerical results. Moreover, since the available experimental data were collected over thirty years ago with the available instrumentation [6], new tests were conducted and results were published by Lavoie et al. [7]. This paper describes briefly the developed numerical SPH bird model and compares the obtained results with those from new experimental data. During the experimental set-up, a 1 kg gelatine bird substitute impacted a rigid 12.7 mm thick steel plate at a velocity of 95 m/s (185 knots). The plate was of 0.3048 m by 0.3048 m side dimension with an elevated edge of 12.7 mm wide by 6.35 mm thick. A high-speed video camera was used to capture the behaviour of the projectile during the impact and frames were taken at a frequency of 3000 frame per second (fps). The numerical model was created in LS-DYNA and included a steel target modeled with solid elements and the bird, modeled with approximately 4500 SPH particles. An automatic nodes-to-surface contact controls the interaction between the projectile and the target. The developed numerical model is shown in Figure 1. Since theory tells us that the expected behaviour of the bird during impact is similar to a fluid, an elastic material model is used for the plate and an elastic-plastic-hydrodynamic material model with a polynomial equation of state is used to model the bird. The physical properties of the gelatine are a density of 950 kg/m3, a shear modulus of 2 GPa, a yield stress of 0.02 MPa and a plastic hardening modulus of 0.001 MPa.
Figure 1
Numerical model for the SPH impact on a rigid flat plate
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1405 Snap-shots are compared for the obtained experimental and numerical results in terms of deformations from the beginning of the impact in Figure 2 at time intervals of 0.0066 s. A very good correlation can be observed between the two sets of data.
Figure 2
Impact at 0q at time intervals of 0.66 ms, video camera and SPH method
Moreover, the deceleration of the end of the projectile and the increase of the diameter of the projectile were measured to provide a more quantitative approach. Those are plotted in Figure 3 where an experimental curve is given for three typical tests conducted. Although the time intervals are relatively large for the experimental data, the trend of both numerical and experimental results is very similar.
Figure 3
Variation of velocity (left) and diameter of the projectile (right) for perpendicular test
The accuracy of the SPH bird model can be assessed in other ways. For instance, it is possible to incline the target and measure the diameter and velocity of the projectile. It is also quite current to compare the pressure reading at the center of impact and the radial pressure distribution with the analytical and experimental values. This was done using carbon gages. However, the results were very inconclusive and are not presented here. For
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1406 more information on the analytical approach and the experimental approach the reader is referred to Lavoie et al. [5] and Lavoie et al. [7], respectively. The general conclusion drawn from the results presented here is that the SPH method yields results that are sufficiently reliable for it to be used in a more complex numerical simulation where the unknown are related to the structure and not the projectile. Moreover, there is no mass loss or significant decrease in the timestep as the deformation of the projectile increases. These properties make the SPH method very well suited for secondary impacts where damage caused by rebound on nearby structures is of concern. 3. Hail impact Another important threat to aircraft is when they are stationary and hail storms occur. In such instances, the velocity of hail impacting is approximately of 25 m/s, which is the velocity of free fall for a 0.04 m hail ball. So the minimal velocity at which the tests/simulations should be performed is 25 m/s [8]. Although hail is an easier projectile to manufacture when compared to a gelatine bird, very little theory is found in the literature as to the expected behaviour during impact. One interesting point is that in general, the density of hail is lower than that of ice. The nominal density of fresh water ice is 917 kg/m3 whereas the average density of hail is 846 kg/m3 [9-12]. As opposed to bird impact, no theory was found regarding the impact behaviour of hail. However, it is reasonable to assume that the behaviour is brittle upon impact. According to the literature [10,11], the elasticplastic with failure model is suitable for such an application. The purpose of the hail tests was to conduct preliminary tests. Moreover, given the difficulties encountered using the carbon gages to measure the pressure during the bird impact, it was suggested to use pressure sensitive film instead. The hail tests made it possible to evaluate the performance of pressure sensitive film for pressure acquisition. The hail itself is made by compressing snow until it turns into ice and reaches the desired density. This allows a better control of the density and it is easier to obtain the proper shape. The concept of the 0.04 m die is shown in Figure 4.
Figure 4
0.04 m die to create hail
The pressure sensitive film used to measure the impact was purchased from Fuji Film. Microballoons containing dye explode at a given pressure and the concentration of the coloration indicates the pressure reached. Only one pressure is obtained for the whole event and corresponds to the peak pressure of the impact. The pressure was determined by comparing the color concentration over an area of 0.5 mm by 0.5mm to a provided scale.
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1407 In addition, a high speed video camera was used during these tests. Issues encountered with the digital camera made it impossible to present results of the hail impacting. This was mainly due to the lack of resolution of the camera used. Since the impact only lasts 1.2 ms, a minimum acquisition speed of 10,000 fps would be required to have about 10 frames during the impact. However, the camera available was limited to 5,000 fps and at such a speed, the actual window captured was very small. It was nevertheless possible to observe that the hail is brittle upon impact and does not spread the way gelatine does. It is interesting to note that the zone of impact for which a pressure was captured by the pressure film is less than the diameter of the projectile. For the various tests, the diameter of the area of impact varied between 1.8 mm to 2.3 mm which can be compared later with the numerical results. Pressure results are shown for a 35 gr hail projectile impacting the rigid target at a velocity of 45 m/s. A high pressure sensitive film was used since the medium film seemed to be saturated and did not indicate the proper maximum pressure. Figure 5 shows, on the left, the actual pressure sensitive film after impact and on the right, the dye concentration measured by the computer. The pressure along the horizontal and vertical centerlines is plotted in Figure 6.
Figure 5
Pressure sensitive film and measurement
Figure 6
Horizontal & Vertical line profile
The SPH model created to simulate the hail uses the elastic plastic with failure material model. The density was set to 846 kg/m3, the elastic shear modulus to 3.46 GPa, the yield strength to 10.30 MPa, the hardening modulus to 6.89 GPa, the bulk modulus to 8.99 GPa, the plastic failure strain to 0.35 and the tensile failure pressure to 4.00 MPa [10,11]. A hail stone of 40 mm. was modeled using 587 particles. It was impacted on a target 0.2 by 0.2 m and 6.35 mm thick. The target had three elements in thickness and the mesh was refined to a size of 2.5 by 2.5 mm in the zone of impact. The properties of aluminum were used to model the plate with an elastic
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1408 material model. The resulting mesh includes 600 SPH elements for the projectile and 5000 solid elements for the target. The projectile and the point of impact on the plate are illustrated in Figure 7.
Figure 7
Refined mesh for target
To compare the numerical results with the experimental results, the maximum interface pressure was plotted along the centerlines. In order to obtain a valid comparison with the experimental pressure, for which the area used was of 0.5 mm by 0.5 mm, the mesh in the impact area was made as small as possible for the simulation to work properly and lowered to 2.5 mm by 2.5 mm. This results in the area over which the pressure is calculated being 25 times larger for the numerical simulation, and this will have to be taken into consideration when analyzing the results. The pressure along the centerline is plotted in Figure 8. The right order of magnitude is reached for the value of the pressure. The fact that it is lower is explained by the fact that the pressure is calculated over a larger area which has an averaging effect when compared with the experimental data. Moreover, the diameter of the area where the pressure larger than the cut-off pressure of 50 MPa is 15 mm, which compares very well with the experimental results.
Figure 8
Maximum pressure reading across centerline
Snapshots of the impact at time intervals of 0.25 ms are given in Figure 9. Although it is impossible to compare with the snapshots from the video camera, it is in agreement with the fact that the hail is brittle and it correlates very well with observations from Kim et al. [13].
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Figure 9
Hail impact at time intervals of 0.25 ms
Overall, the performance of the pressure sensitive film proved satisfactory and reliable. The provided data is limited to the maximum pressure and does not provide any information with respect to time, but it is relatively simple and inexpensive to use. Depending on the results expected from various tests, it can prove very adequate. Future tests would obviously require a better high speed camera and could include impacts on metallic and composite deformable structures. The obtained numerical results are in good agreement with the experimental data in terms of the pressure reached and the size of the zone of impact. Although it was impossible to correlate the deformations of the hailstone with experimental data, the numerical simulation results tend to agree with the current understanding and observation of the phenomenon by other authors [13]. The SPH method has the additional advantage that the modeling of multiple projectiles requires less computational and time efforts. However the mesh refinement for the hailstone and zone of impact is required for a more accurate reading of the pressure, which could become computationally expensive. 4. Conclusions We have shown in this paper how the SPH method can be used to obtain accurate results for bird and hail impact simulations. Many of its features make the method suitable to simulate complex impact events. Given the success of SPH method in the current project, it should be used in subsequent work involving more complex fluid-solid interaction as in aircraft ditching simulation. Future work would be to perform complex simulations and hopefully conduct experiments in parallel to evaluate the performance of the SPH projectile and the performance of the metallic/composite material model used for the structure. In such cases, additional valuable data would be found in the final deformations of the structure. 5. Acknowledgements We would like to thank the CRIAQ for their financial support for this project as well as Laval University, DRDC Valcartier and NRC for their close collaboration. Thanks also go to our industrial partners and Mr. Jacques Blais and Ron Gould for their expertise during the test programs. 6. References 1. Marie-Anne Lavoie, Soft body impact modeling and development of a suitable meshless approach, Université Laval, 2008, 131 pages 2. Alastair F. Johnson, Martin Holzapfel, Modeling Soft Body Impact on Composite Structures, Composite Structures, Vol. 61, pp. 103-113, 2003 3. M. A. McCarthy, R. J. Xiao, C. T. McCarthy, A. Kamoulakos, J. Ramos, J. P. Gallard, V. Melito, Modeling Bird Impacts on an Aircraft Wng- Part 2 Modeling the impact with and SPH bird model, International Journal of Crashworthiness, Vol. 10, n.1, pp. 51-59, 2005 4. Vinh Phu Nguyen, Timon Rabzuk, Stéphane Bordas, Marc Duflot, Meshless methods: A review and computer implementation aspects, Mathematics and computers in simulations, 79, 208, 763-813
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1410 5. M-A Lavoie, A. Gakwaya, M. Nejad Ensan, D.G. Zimcik, Validation of Available Approaches for Numerical Bird Strike Modeling Tools, International Review of Mechanical Engineering, 1, 2007, 380-389 6. James S. Wilbeck, Impact Behavior of Low Strength Projectiles, Air Force Materials Laboratory, 1977 7. M-A Lavoie, A. Gakwaya, M. Nejad Ensan, D.G. Zimcik, D. Nandall, Bird’s substitute tests results and evaluation of the available numerical methods, International Journal of Impact Engineering, 39, 2009, 12761287 8. D.A. Paterson, R. Sankaran, Hail impact on building envelopes, Journal of Wind Engineering and Industrial Aerodynamics, 53, 1994 9. Kelly S. Carney, David J. Benson, Paul Du Bois, Ryan Lee, A High Strain Rate Model with Failure for Ice in LS-DYNA, 9th International LS-DYNA Users Conference, Detroit, Michigan, June 4-6 2006 10. Hyonny Kim, Keith T. Kedward, Modeling Hail Ice Impacts and Predicting Impact Damage Initiation in Composite Structures, AIAA Journal, 38, 2000 11. Marco Anghileri, Luigi-M. L. Castelletti, Fabio Invernizzi, Marco Mascheroni, A survey of numerical models for hail impact analysis using explicit finite element codes, International Journal of Impact Engineering, 1, 2005 12. Q. Monsen, C. M. Ehresman, S. N. B. Murthy, Hail Ingestion Simulation Tunnel (HIST) for Inlet and Rotor Studies, 31st AIA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, San Diego, CA, July 10-12 1995 13. Hyonny Kim, Douglas A. Welch, Keith T. Kedward, Experimental investigation of high velocity ice impacts on woven carbon/epoxy composite panels, Composites Part A: applied science and manufacturing, 34, 2003, 25-41
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
A High Frequency Stabilization System for UAS Imaging Payloads Katie J. Stuckel, William H. Semke, Nicolai Baer, and Richard R. Schultz Unmanned Aircraft Systems Engineering Laboratory School of Engineering and Mines University of North Dakota Grand Forks, ND 58202 ABSTRACT A high frequency stabilization mount to compensate for small attitude fluctuations is developed for enhanced imaging and pointing systems on Unmanned Aircraft Systems (UAS). This system consists of a custom camera mount, piezoelectric actuators, and a digital controller to actively control flight vibrations. Payload designs that acquire views of the Earth surface and stationary or moving targets require stable cameras for precision viewing. Placing cameras onboard these small aircraft are vital to the Intelligence, Surveillance, and Reconnaissance (ISR) mission of many payload designs. It is necessary to have real time precision viewing imaging systems while in flight, but it is increasingly difficult as the small planes reach higher altitudes. A slight change in the camera angle at high altitudes results in a large shift from the designated target. This project focuses on high frequency analysis for small oscillations rather than large attitude changes that are accomplished with a gimbal. The two systems work together to handle both the high frequency oscillations due to engine vibration and turbulence as well as large low frequency attitude changes. Results will include laboratory testing and simulation data to further prove the effectiveness of this specialized stabilization system.
NOMENCLATURE αT α β θ φ E ∆ δ F L x y z Xa Ya Za Xc Yc Zc ωn f k keq m Vapp Vmax
= = = = = = = = = = = = = = = = = = = = = = = = = =
Coefficient of Thermal Expansion in per Kelvin ( /K) rotation about x-axis of stabilization system rotation about y-axis of stabilization system pitch roll Modulus of Elasticity in Pascal (Pa) displacement in the z-direction, usually pertaining to actuators displacement due to expansion/contraction of actuator in meters (m) or micrometers (μm) Force in Newton (N) length in meters (m) x axis (or coordinate) of the stabilization mount y axis (or coordinate) of the stabilization mount z axis (or coordinate) of the stabilization mount x axis of the plane coordinate system y axis of the plane coordinate system z axis of the plane coordinate system x axis of the camera coordinate system y axis of the camera coordinate system z axis of the camera coordinate system 2 2 natural frequency in radians/second (rad/sec ) frequency in Hertz (Hz) stiffness of spring in N/m equivalent k component for natural frequency calculation mass in kilograms (kg) applied voltage maximum voltage
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_124, © The Society for Experimental Mechanics, Inc. 2011
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1412 I. Introduction A high frequency stabilization mount is presented for use with unmanned aerial systems (UAS) to enhance imaging and pointing systems during flight. The active vibration control mount uses piezoelectric stack actuators to move the camera mount as needed. As the plane vibrates, the sensors onboard detect the slight change in tilt or displacement, send a signal to the actuators, causing them to move in the opposite direction for stability. It is the goal of the vibration compensator to steady any imaging or pointing system used during flight. This allows for precision viewing and higher quality systems onboard. Figure 1 shows the platform and the prototype of the stabilization system which will be flown. The Bruce Tharpe Engineering (BTE) Super Hauler is a custom built “small” unmanned aerial vehicle (UAV) designed for multiple types of payloads and flown by the Unmanned Aircraft Systems Engineering (UASE) laboratory. One of the primary payload development areas of the UASE is target imaging and tracking for use in Intelligence, Surveillance, and Reconnaissance (ISR) missions [1,2]. The active vibration control mount enhances these categories and fits into the body of the plane with the camera nadir pointing towards Earth.
Figure 1. Unmanned Aerial Vehicle High and Frequency Vibration Mount Precision pointing is vital to certain missions flown on UAVs for real time viewing and post analysis. The high frequency vibrations from the engine, as well as many other sources including wind buffeting, prove to be a challenge during flight. Image streams become grainy and unfocused the more the UAV shakes. The camera also moves and tilts, sometimes losing track of the target. This becomes more common as the UAV gains altitude and speed. Because of the slight tilt in the camera, the projected view has a large shift in the pointing of the camera. When zoom is also included in the system, the shift may be enough lose the target in the field of view completely, causing a failure in the mission. It is the goal of the high frequency stabilization mount to rectify this problem and help improve imaging and pointing capabilities. To begin development of the stabilization mount, it was necessary to analyze accelerometer data from a UAV during flight. Figure 2 shows example data from the Piccolo II autopilot, by Cloud Cap Technologies in which a range of the acceleration in the vertical direction is observed. The data is centered about Earth’s gravitational 2 constant at a value of 9.806 m/s and at any given time during flight data analyzed, the typical magnitude of 2 acceleration is less than one g (9.806 m/s )[3]. Based on a FFT analysis of the data, one can see the need for a high frequency stabilization system as the plane shows an unstable platform in the high frequency range. This vibration may interfere with other systems onboard or cause complications in imagery. 0
July 31, 2008
-2
August 1, 2008
Acceleration (m/s2)
-4 -6 -8 -10 -12 -14 -16 -18 -20 0
100
200
300
400
500
600
700
800
Time (sec)
Figure 2: Sample acceleration data in the vertical direction comparison between Jul 31 and Aug 01 flights
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1413 The designed system with its coordinate system, angles and orientations is shown in Figure 3. The active vibration control mount is defined by aligning the coordinate system of the mount with that of the airplane. This simplifies the algorithms used by keeping the coordinate systems aligned and translating the axis of the vibration mount along the Xa-axis of the plane coordinate system. The Xa-axis is the roll axis of the plane while the Ya-axis is the pitch axis of the plane. The Za-axis then becomes the direction of interest in which the actuators move. The vibration mount works in tandem with a gimbal system for the large angle rotations which the small, while high frequency angles are controlled with the active vibration control mount.
x
y
z
Figure 3. Stabilization mount and aircraft coordinate system The intent for this small, lightweight system includes high voltage DC-DC converters, software written in C code, a digital controller and displacement sensors to complete the active vibration control stabilization mount. As the displacement sensors detect a change in position, signals will be set through the controller to the actuators instantaneously to counteract this movement. In depth laboratory testing and simulation is used to support the analysis of the high frequency stabilization mount. Accuracy of the system is based on the measure of current deflections and rotations, with a future measure of the stability of the support ring. Both laboratory and simulation testing results are compared to hand calculations to demonstrate the effectiveness of the system. II. Description and Design of Model The high frequency stabilization mount is made from 6061-T6 aluminum for stiffness and weight attributes. Since the platform for use of the mount is a UAV, it is necessary to make the stabilization mount as light as possible. Table 1 also shows the dimensions of the piezoelectric compensators to show the use of active vibration control on a small scale design. A primary use will be for mounting cameras up to 2 kg. Mathematically, a plane can be defined by three points, which allows the system to properly orientate to any rotation angle. From this plane, a normal vector can be found which we will define as the line of sight vector. This is placed on the z-axis of the viewing angle of the camera. The goal of the system is to move the actuators in the z direction on the stabilization mount to keep the camera zc-axis pointing steadily at the target [4]. The actuators used in this system are purchased from APC International LTD with specified maximum stroke of 20 μm with 0 to +150 V, as shown in Table 1. These actuators are able to supply and receive a load up to 800 Newton and are vital in the high frequency stabilization mount design. The three actuators located at equal distances around the ring provide the necessary movement for prescribed rotations to help actively control the vibration. A static deflection test was performed on four actuators to verify the specifications. From this test, it was determined that the actual stroke of the actuators is greater than the specified deflection. Figure 4 shows the deflection of the actuators versus the amount of voltage applied. The predicted value is also plotted on this graph. The first three actuators tested were very similar in response, while the fourth actuator showed a greater distance traveled. Because of this, the three similar actuators are used in the stabilization mount to increase the reliability of performance. The updated actuator curve is used in the analytical calculations and ANSYS models to better match the experimental results. This is further explained in the Analytical Calculations and Mathematical Modeling section.
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1414 Table 1. Mechanical and Electrical Specifications for APC International Piezoelectric Stack Actuators Mechanical and Electrical Specifications Length 3.5 mm Width 3.5 mm Height 18 mm Voltage Range 0 to +150 V Max Stroke 20 μm Capacitance 200 nF Resonance Frequency 50 kHz Stiffness 25 N/μm Blocking Force 800 N Max Load Force 800 N
25.0
∆μm 1 Predicted Value ∆μm 2 ∆μm 3 ∆μm 4
20.0
Displacement (μm)
15.0
10.0
5.0
0.0 0
20
40
60
80
100
Voltage Input (V)
Figure 4. Comparison of actuators by user voltage input A ring was chosen for the mount to help simplify the design and reduce material. It is necessary to have a hollow shape to mount the camera without blocking the field of view, but also necessary to have sufficient material for mounting and allow for ease of manufacturing. It is critical to have spring/actuator pairings such that the ring remains firmly in place against the actuator so no impulsive loads are transmitted and the ring and actuator motion remain in phase. The rods next to each spring and actuator set allows for the system to move up and down freely while constraining the motion to the vertical direction. The springs have a stiffness of 63,550 N/m. The stiffness for the springs was chosen based on the systems natural frequency. Data is received from the sensors and fed into the system at 100 Hz, so it is important to have the natural frequency of the system, as well as the natural frequency of each set of spring and actuator greater than this to ensure the appropriate response. The natural frequency can be found using Eq. 1 and the conversion 2 from ωn (rad/s ) into Hz. (1) The keq required needed to be stiffer than 27,240 N/m. With the chosen springs based upon availability and acceptable dimensions, the natural frequency of the system is calculated at 177.66 Hz.
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1415 III. Analytical Calculations and Mathematical Modeling To better understand what is happening with the high frequency stabilization mount, it is necessary to perform analytical calculations to compare with simulated and experimental results. To calculate the deflection of a single actuator, the ratio of the voltages simply needs to be multiplied by the maximum deflection, Eq. 2. (2) For calculations of the deflection at the actuator of the system including the spring, it is necessary to taken into consideration the stiffness of the actuator as well. As shown in Table 1, the stiffness of the actuators is 25 N/μm. This in combination with the deflection of the actuator at the designated voltage and the deflection equation for a spring (Eq. 3) will give the final deflection as the actuator pushes against the spring. (3) Displacement of actuator including stiffness is shown in Eq. 4. (4) Combining these two equations and solving for the force will give the final results for the deflection. By setting ∆spring=∆act, we are able to achieve this. Since the spring and actuator are working against each other, equilibrium will be reached and the force will be the same in each. Putting the found force back into Equation 3 or 4, the final deflection can be found using Eq. 5. (5) At 100 V, the calculated force is 0.938 N, which gives a final deflection of 14.76 μm for each actuator. For simulation and experimental results, this will be the comparison standard.
IV. Finite Element Analysis A finite element analysis and simulation was done to verify the equivalent rotation and displacements of the spring, as well as complete a modal analysis to confirm the natural frequency of the system. The ring, actuators and springs were all modeled in ANSYS, with the appropriate geometry, material definitions, properties and element settings were all set according to each item. The model was meshed and the necessary constraints applied for each analysis. The analyses performed included both static and a modal analysis. To begin, the ring mount was modeled using SOLID 45 elements, which is an 8-node 3-dimensional structure. The spring was modeled using the COMBIN 14 element with a spring constant of 63,5000 N/m and a damping constant of 0. The piezoelectric actuators were modeled using LINK 1. To allow for the actuators to act as a piezoelectric material with a voltage applied, the analysis in ANSYS was performed using a thermal analysis, comparing the rise in temperature to the rise in voltage. Equations 6 and 7 were to find the necessary constants, E and α T, to simulate the piezoelectric expansion. (6) (7) Table 2 has complete details on material element types, real constants and material properties. The ring was modeled using keypoints 120° at equal radius apart from the origin. Lines were created from the keypoints, areas from lines and volumes from areas. Meshing was completed of the ring with 60 elements on each arc, 20 elements in the width and 5 elements in the height. The meshing used was a tetrahedral, 4 node mesh. Displacement constraints were applied in all directions and all rotation to the tops of the springs and bottom of the actuators, allowing the ring to move freely up and down, simulating what is happening in the actual system. Figure 5 shows the meshed structural and boundary conditions.
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Element Type
Table 2: ANSYS properties and element types Real Constants Material Properties
SOLID 45
N/A
COMBIN 14
Spring Constant: 63500 N/m Cross Sectional Area: -5 2 1.225 x 10 m
LINK 1
2
Density: 2700 kg/m Modulus of Elasticity: 69 GPa Poisson’s Ratio: .36 N/A Modulus of Elasticity: 36.73 GPA Poisson’s Ratio: .33 Thermal Expansion Coefficient: 8.222 x 10-6 /K
Figure 5. ANSYS constraints and meshing for high frequency stabilization mount Since this simulation is used to compare to the experimental and hand calculations, only one actuator was activated in ANSYS, and was done so at 100 V. Tests were performed both with the springs attached and without. The results from the final ANSYS simulation with the springs is in Figure 6 and a summary of results are listed in Table 3. The first tests without the springs showed very promising results with percent errors of 0% and 0.07%. The predicted results with the springs calculate 14.76 μm to be the deflection at the actuator with the springs and ANSYS gives 14.75 μm. Because of the nature of the system, it was not possible to measure the displacement at the exact location of the actuator, so measurements had to be taken at the edge of the ring, also shown in Figure 6. The parallel triangle theorem was used while the ring was assumed to rotate about the fixed axis of the other two actuators. This gives a displacement of 14.72 μm at the actuator. These similar deflections, 0.0% and 0.27% errors also show promising results for the tilt compensator.
Figure 6. ANSYS results with one actuator activated and experimental set-up (laser measurement point highlighted red)
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1417 Table 3. Summary of Simulated, Experimental and Hand Calculation Deflection of Stabilization Mount (μm) Displacement ANSYS Experimental Calculations % Difference % Difference w/o Spring 14.79 14.8 14.8 0.07% 0.07% w/Spring 14.76 14.72 14.76 0.27% 0.0% V. Modal Analysis The next analysis performed was the ANSYS modal analysis to compare the numerical natural frequencies to those predicted. In this analysis, only the ring and springs are included in the model. Using Equation 1, the analytical calculations showed the natural frequency to be 177.66 Hz. As shown in Figure 7, ANSYS calculates the axial natural frequency to be 175.299 Hz. This gives an error of 1.3%, which is determined to be acceptable. Table 4 shows a list of the first eight natural frequencies of the system calculated, listed from 0 to 5000 Hz. Modes 1 and 2, 4 and 5, and 6 and 7 are paired symmetric natural frequencies. The second mode in each pair is an orthogonal rotation of the first mode in each pair. The lower two natural frequencies show a tilt of the support ring and the axial natural frequency is the third mode shape. Higher mode shapes show a ring deflection for the shape. Figure 7 illustrates the first four mode shapes showing only one of each set of modes. Table 4: Natural frequencies for the stabilization mount Mode Frequency (Hz) 1 169.52 2 169.52 3 175.29 4 1410.3 5 1412.3 6 3514.6 7 3604.2 8 4449.5
a.
b.
c. d. Figure 7: a. Mode 1, f=169.52 Hz b. Mode 3, f=175.29 Hz c. Mode 4, f=1410.3 Hz d. Mode 6, f= 3514.6 Hz
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1418 VI. Laboratory Testing The previous static laboratory testing data presented on the ring resulted from supplying voltage to the actuators using a Hewlett-Packard Lab Power Supply and measuring the deflection of the spring with a Keyence Laser Displacement Sensor (LDS). A more in depth system used for testing included the use of a LabVIEW program and a National Instruments USB-6251 Data Acquisition (DAQ) Board for communication with the actuators. This DAQ board allows a range of 0 to +10 V and communicates through USB. As the LDS measures the displacement of the ring and sends the data through the DAQ board, the LabVIEW program corrects for this displacement and supplies a voltage through a TREK Piezo Amplifier to the actuators. The gain on the amplifier is set to 1:100 V. The actuators then expand or contract to the necessary height, seemingly that the stabilization mount never moved. The first round of static testing was performed by activating a single actuator. It was very important to isolate the system and separate the ring mount and laser head from the other devices, which cause noise in the system. For example, the fan in the laptop computer and the piezoelectric amplifier cause small motions in the table, creating an unstable reading on the laser head. Figure 8 shows the set up of the experiment.
Figure 8. Static experimental set-up: isolated mount and separate power supply on the left, computer, amplifier, DAQ and laser head controller on right A static test was performed by deflecting the base of the high frequency stabilization mount and running the LabVIEW program such that the control system corrected for the known displacement. An open loop controller was used to perform a single calculated correction to the system. In this manner, the entire system was exercised including both hardware and software components. The LabVIEW block diagram is shown in Figure 9. The open loop control system allowed for end-to-end testing and communication between all hardware, as well as demonstrating the ability of the LabVIEW program. The initial value was entered manually into the program in the Initial Value on the block diagram. After the base of the vibration mount was given a deflection, the inputs were read from the laser with a DAQ board through the DAQ Assistant vi. The constant 0.0022 is entered into the program to calibrate the sensor. The data is then processed through the system by calculating the difference in the laser voltage readings and multiplying that value by 67.5675, a constant gain that is the ratio from μm to V for the piezoelectric actuators. That value is then added to 0.2 V, which was selected as the starting voltage for the experiment. The starting voltage partially activates the actuators at 20 V, which is equivalent to 2.96 μm, to allow for motion in the positive and negative directions. Finally, the voltage, with a safety limit of 1.4 V, is fed into the DAQ board (140 V from the amplifier to the actuator). The check is done because the maximum voltage allowed per actuator is +150 V. This test was repeated five times for verification purposes and the results are shown in Table 5. The average deflection of the base is 13.88 micrometers, with an average error of 2.66%. The low percent error for this experiment shows that this experiment was a success. The actuators proved to respond appropriately, as well as demonstrated a fully operational LabVIEW program. Table 5. Laser readings for static deflection test results Initial Value Deflection Return % Average % (mm) Value (mm) Value (mm) Difference Difference 3.0254 3.0111 3.0255 0.70% 2.66% 3.0583 3.0444 3.0582 0.72% 3.0278 3.0151 3.0282 3.15% 3.0254 3.0110 3.0248 4.17% 3.0505 3.0373 3.0499 4.55%
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Figure 9. Open loop LabVIEW program for stabilization mount VII. Conclusion The data obtained in these experiments demonstrates the effectiveness of the high frequency stabilization mount prototype. Analytical calculations, computer simulations and static laboratory experiments have all shown the reliability of the system. The modal analysis in ANSYS simulates the various modes and frequencies for the system and gives a range of usable frequencies. Future work for the stabilization mount includes the addition of a Proportional, Integral, Derivative (PID) closed loop controller to eliminate the high frequency dynamic oscillations. The finalized high frequency stabilization mount will be a miniaturized version of this prototype, implementing micro DC-DC amplifiers and a programmed digital controller to produce a small lightweight system that will be operated autonomously onboard a UAS. VIII. Acknowledgements This research was supported in part by Department of Defense contract number FA4861-06-C-C006 “Unmanned Aerial System Remote Sense and Avoid System and Advanced Payload Analysis and Investigation,” and the North Dakota Department of Commerce, “UND Center of Excellence for UAV and Simulation Applications.” The authors would like to also acknowledge the contributions of the Unmanned Aircraft Systems Laboratory team at UND. IX. References [1] Semke, W., Schultz, R., Dvorak, D., Tandem, S., Berseth, B., and Lendway, M., “Utilizing UAV Payload Design by Undergraduate Researchers for Educational and Research Development,” Proc. of 2007 ASME International Mechanical Engineering Congress and Exposition, IMECE2007-43620, November 2007. [2] Lendway, M., Berseth, B., Tandem, S., Schultz, R., and Semke, W., “Integration and Flight of a UniversityDesigned UAV Payload in an Industry-Designed Airframe,” Proc. of the AUVSI, 2007. [3] Semke, W., Stuckel K., Anderson K., Spitsberg R., Kubat B., Mkrtchyan A., Schultz R., “Dynamic Flight th Characteristic Data Capture for Small Unmanned Aircraft,” 26 annual IMAC SEM Conference, Feb ’08. [4] Buisker, M., “Statistically Significant Factors that Affect the Pointing Accuracy of Airborne Remote Sensing Payloads,” M.S. Mechanical Engineering, University of North Dakota, May 2007.
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BookID 214574_ChapID 125_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Crystal Clear SSI for Operational Modal Analysis of Aerospace Vehicles∗ 2 ¨ M. Goursat1 , M. Dohler , L. Mevel2 , and P. Andersen3 1 INRIA, 2 INRIA, 3 Structural
Domaine de Voluceau - Rocquencourt, BP 105, F-78153 Le Chesnay Cedex, France Centre Rennes - Bretagne Atlantique, Campus de Beaulieu, F-35042 Rennes, France
Vibration Solutions A/S, NOVI Science Park, Niels Jerners Vej 10, DK-9220 Aalborg East, Denmark
Abstract In this paper we revisit the problem of the modal analysis of space launchers. We consider the Ariane 5 launcher with its usual equipment during a commercial flight under the natural unknown excitation. The case of space launchers is a typical example of a complex structure with sub-structures strongly and quickly varying in time. This issue becomes especially important in e.g. estimation of damping of aerospace vehicles. The eigenfrequencies are also sliding during the flight but the modeshapes are more stable. Recently, a new implementation of the subspace identification method has been proposed, leading to cleaner and more stable stabilization diagrams. We monitor the behavior of estimated modal parameters by applying this “crystal clear” implementation of the data driven and the covariance driven Stochastic Subspace Identification algorithms. We show the importance of “crystal clear” to monitor successfully frequencies and damping estimates over time in such a non stationary case.
1
INTRODUCTION
In modal analysis of vibrating structures it is usual that operating conditions differ completely from those of laboratory experiments. The first major difference is that under natural loading conditions, excitations cannot be measured and are usually non-stationary; this does not mean that laboratory results are not valid but that in-operation treatment needs different techniques. Subspace-based algorithms are currently used and have been proven efficient for modal parameter estimation (natural frequencies, damping ratios, modeshapes). In this paper, we use output-only covariance-driven and data-driven Stochastic Subspace Identification (SSI) methods for the identification of the modal parameters. These methods assume that the structure is stationary. Even for many non-stationary structures, e.g. aircraft, the stationary assumption can be still assumed because the non-stationarity is due to the load which varies slowly. However, in our case the structure is not stationary. The mass of the launcher is strongly and quickly varying in time. Nevertheless, we want to apply SSI methods to identify the modal parameters. In the case of Ariane we could have some positive points for the method. The output are continuously measured and so it is possible to use sliding windows and to follow the time evolution of a specific eigenfrequency. Moreover modeshapes are more slowly varying than the eigenfrequencies and it is easier (if possible) to compare and follow the modeshapes. Another important point is that Ariane is a complex structure with different sub-structures and the sub-structures are not simultaneously varying: for example, during the initial part of the flight, the boosters are burning and strongly varying but the other parts of the launcher can be considered as stationary parts excited by the boosters. A preceding analysis of the data of Ariane 5 was made in [8] and [9]. The current paper extends the previous research by the analysis with the data-driven SSI algorithm, which is recalled briefly in the first part of this paper together with the covariance-driven SSI procedure. We present some results of the modal parameter identification for Ariane 5 using also the recently developed “crystal clear” implementation [2] which is especially helpful in the presence of non-stationary data and it leads to cleaner and more stable stabilization diagrams. ∗ This work is a revisited version of a work carried out under contract with CNES (Centre National d’Etudes Spatiales) in cooperation with Aerospatiale
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_125, © The Society for Experimental Mechanics, Inc. 2011
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2
IDENTIFICATION PROCEDURE
2.1
Modeling
The mechanical system is supposed to be a stationary linear dynamical system ! ¨ + C Z(t) ˙ + KZ(t) = ν(t) M Z(t) , Y (t) = LZ(t) with • Z: displacements of the degrees of freedom, • M , C, K: mass, damping, stiffness matrices, • t: continuous time, • ν: excitation, • L: observation matrix giving the observation Y . The modal characteristics • μ vibration modes or eigenfrequencies • ψμ modal shapes or eigenvectors are solutions of the following equation: (M μ2 + Cμ + K)Ψμ = 0 , ψμ = LΨμ . We switch to the state space model in discrete time by sampling at the rate 1/δ with Z(kδ) Xk = , Yk = Y (kδ) ˙ Z(kδ) and get
!
Xk+1 Yk
= F Xk + Vk = HXk
with F = exp(Aδ) and H =
where A=
0 −M −1 K
L 0
I −M −1 C
The input noise is assumed to have zero-mean and its covariance is: QV (k) =
" (k+1)δ kδ
˜ Q(s) =
˜ exp(AT s)ds exp(As)Q(s)
0 0
0 M −1 Qν (s)M −T
Qν (s) is the covariance matrix of ν. The modal characteristics (μ, ψμ ) are given by the eigenstructure (λ, Φλ ) of F : eδμ ψμ
= =
λ Δ φλ = HΦλ
In the sequel the dimension of the observed output Y is much smaller than the dimension of the state X.
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2.2
Output-only subspace-based modal analysis
We consider the previously defined discrete time model in state space form: ! Xk+1 = F Xk + Vk+1 Yk = H Xk
(1)
Only knowing the output data Yk at the time instants k = 1, . . . , N , we want to identify the eigenstructure (λ, φλ ) of the system with Stochastic Subspace Identification algorithms. For doing so, we follow two approaches: the covariance driven [4, 1] and data driven approach with the Unweighted Principal Component algorithm [10]. For both approaches we choose p and q as variables with p + 1 ≥ q that indicate the quality of the estimations (a bigger p leads to better estimates) and the maximal system order (≤ qr with the number of sensors r). Normally, we choose p = q − 1, but in the case of measurement noise p = q − 1 + l should be chosen, where l is the order of the noise. We build the data matrices ⎛ ⎞ ⎛ ⎞ .. .. Yq+2 . YN−p ⎟ Yq+1 . YN−p−1 ⎟ ⎜ Yq+1 ⎜ Yq ⎜ ⎟ ⎜ ⎟ .. . ⎜ Yq+2 Yq+3 . YN −p+1 ⎟ Yq .. YN−p−2 ⎟ def ⎜ + − def ⎜ Yq−1 ⎜ ⎟ ⎟ Yp+1 = ⎜ (2) ⎟ , and Yq = ⎜ . ⎟ .. .. .. .. .. .. .. . ⎜ ⎟ ⎜ ⎟ . . . . . . . . ⎝ ⎠ ⎝ ⎠ .. .. Y Y . Y Y Y . Y q+p+1
q+p+2
N
1
N −p−q
2
and, according to the method, a Hankel or a weighted Hankel matrix as follows: • For the covariance driven approach, we build the Hankel matrix ⎛ R1 R2 R3 ⎜ ⎜ R2 R3 R4 ⎜ ⎜ T def + cov R4 R5 Hp+1,q = Yp+1 Yq− = ⎜ ⎜ R3 ⎜ . .. .. ⎜ .. . . ⎝ .. .. . Rp+1 .
... ... ... ...
Rq .. . .. . .. .
⎞ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎠
. . . Rp+q
T where Ri = E Yk Yk−i is the correlation of the output data and E is the expectation operator. The matrix cov Hp+1,q has the factorization property cov Hp+1,q = Op+1 F Cq with the matrix of observability
⎛
Op+1
H ⎜ HF ⎜ =⎜ . ⎝ ..
⎞ ⎟ ⎟ ⎟ ⎠
HF p
and the matrix of controllability Cq . • For the Unweighted Principle Component algorithm of the data driven approach, we build the weighted Hankel matrix1
T T −1 − + data def Hp+1,q = Yp+1 Yq− Yq− Yq− Yq data The matrix Hp+1,q enjoys the factorization property data Hp+1,q = Op+1 Xq
into matrix of observability and Kalman filter state sequence. 1 As Hdata p+1,q is usually a very big matrix and difficult to handle, we continue the calculation in practice with the R part from an RQ-decomposition of the data matrices, see [10] for details. This will lead to the same results as for the system identification only the left part of the decomposition of Hp+1,q is needed.
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In what follows, we skip the superscripts and subscripts of the Hankel or weighted Hankel matrix H, as the following identification procedure is the same for the covariance and data driven approach. Now we want to obtain the eigenstructure of the system (1) from a given matrix H. The observability matrix Op+1 is obtained from a thin SVD of the matrix H and its truncation at the desired model order: H
= =
Op+1
=
U ΔVT Δ1 (U1 U0 ) 0
0 Δ0
V T,
1/2
U1 Δ1 .
(3)
The observation matrix H is then found in the first block-row of the observability matrix Op+1 . The state-transition matrix F is obtained from the shift invariance property of Op+1 , namely ⎛ ⎞ HF ⎜HF 2 ⎟ def ⎜ ⎟ ↑ ↑ Op (H, F ) = Op (H, F ) F, where Op (H, F ) = ⎜ . ⎟ . (4) ⎝ .. ⎠ HF p Of course, for recovering F , it is needed to assume that rank(Op ) = dim F , and thus that the number p + 1 of block-rows in H is large enough. The eigenstructure (λ, φλ ) results from det(F − λ I) = 0, F ϕλ = λ ϕλ , φλ = Hϕλ ,
(5)
where λ ranges over the set of eigenvalues of F . In practice, we increase the truncation order of the SVD from 1 to the maximal system order in Equation (3) and get a stabilization diagram of the obtained modes vs model order. This gives results for successive different but redundant models and we can distinguish the modes that are common to many successive models from the spurious modes. This step gives frequency bands corresponding to the identified natural frequencies. Then we can select such a frequency band and plot the MACs for the modeshapes corresponding to the frequencies of the band. This information indicates whether the modes for all the orders agree on the same mode shape and are hence part of the modal signature. There are many papers on the used identification techniques. A complete description can be found in [4], [5], [6], [7], [10] and the related references. A proof of non-stationary consistency of these subspace methods can be found in [11]. 3 3.1
THE EXPERIMENTAL CASE: ARIANE 5 Ariane 5
Ariane 5 is a launch vehicle under ESA’s responsibility with CNES as prime contractor. Aerospatiale is the industrial architect for the complete vehicle and prime contractor for parts of the launcher. Ariane 5’s lower section consists of a cryogenic central main (EPC) stage fueled by liquid hydrogen and liquid oxygen, plus two solid boosters(EAP). There is an upper stage using storable propellants. The vehicle is also fitted with a bearing structure (Speltra), a fairing and an equipment bay. The structure is equipped with more than 100 sensors. The measurements are of different types: acceleration, constraint, displacement or mechanical vibration. The number of sensors decreases during the flight as the used parts separate from the launcher. For the same reason the length of the records depends on the location of the measurement on the vehicle. The difference with ground modal analysis (such as an aircraft certification) is that the locations of sensors are imposed and chosen for other purposes than modal analysis (mainly for control). Due to mechanical constraints some parts of the vehicle cannot be equipped. Moreover 100 is a low number of sensors for such a complex structure. There are about 30 “interesting” modes for the structure and with only 100 sensors they cannot be all distinguished. Moreover, each mode is observed only by few sensors as many modes are purely local. Hence it is only possible to examine MAC correlations of identified modes when some sensors (e.g. 4) are used for the identification of the modes.
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Figure 1: Effect of frequency sliding on successive spectra
3.2
Signal examination and preprocessing
The output of 103 sensors were available for this study. The goal of the results presented here is not the full modal analysis of Ariane 5 but only to demonstrate the capabilities of the methods. We will use only 6 sensors chosen for some important modes considered difficult to be identified. There are 3 different periods for the flight: the starting phase, the flight with the boosters (EAP flight) and the long phase with the main cryogenic stage (EPC flight). Figure 1 shows the sliding behavior of frequencies during the flight; on the same display we compute 2 averaged spectra. Each spectrum is computed for a block of 20s and the second is computed 20s after the first one. It is clear that the variation of frequencies depends on the modes and on the period of the flight. 4 4.1
RESULTS AND DISCUSSION Identification parameters • Choose the records used for the identification: the number and location of sensors. The best results are obtained with a low number of sensors: 2 to 10; generally 2, 3 or 4 are good choices. The choice of the sensor locations is important as well and a specialist was requested to do the best choice corresponding to the modal parameters we want to identify due to the existence of local modes. This is a crucial point for Ariane and the choices were done by the experts of Aerospatiale and CNES. In most cases, the best results for natural frequencies and damping ratios are obtained with 2 sensors. • Perform the system identification procedure of Section 2.2. The Hankel and weighted Hankel matrices for the covariance-driven and data-driven approach are built and the stabilization diagrams containing the modal parameters computed. • Signature selection: it is automatically done after the choice of a window size and the number of occurrences in the stabilization diagram. We find the natural frequencies located in a frequency window and compare the number of these occurrences to the number of model sizes. The best results are obtained with the “crystal clear” selection where an approximate mean square solution is considered when the system transition matrix is computed (presented in [2]). • Local examination: for every frequency previously selected we plot the result of the identification w.r.t. the state-space order, i.e. the natural frequency, the damping coefficient or the MAC value.
4.2
Some results
All the stabilization diagrams in this section are obtained with the modal analysis toolbox COSMAD [12], [13] which is freely available at www.irisa.fr/i4s/cosmad. For the data-driven approach, we also include some results obtained with the commercial software Artemis (www.svibs.com).
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4.2.1
Covariance driven method
The effect of low-pass-filtering the data and projection stabilization (crystal clear, CC algorithm) are illustrated by Figures 2, 3, 4 and 5.
Figure 2: Stabilization diagram - unfiltered data
Figure 3: Stabilization diagram - unfiltered data - CC algorithm
Figure 4: Stabilization diagram - filtered data
Figure 5: Stabilization diagram - filtered data - CC algorithm
The alignment for the lowest frequencies and the corresponding damping for the first mode are shown on Figures 6 and 7. On Figures 8 and 9 the identification at two different periods shows the evolution of some modes, while some other remain constant. We can also follow the evolution of a mode during the flight period using an automatic monitoring procedure that shifts a window over the data processed by the identification algorithm. Here the CC algorithm was essential for obtaining good results, otherwise the modal parameters did not stabilize sufficiently for the automatic monitoring procedure. The results for the first mode are shown in Figures 10 and 11. Both the natural frequency and the damping increase, while the fluctuation of the damping estimates is much higher than of the frequency estimates.
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4.2.2
Figure 6: Lowest eigenfrequencies - CC algorithm
Figure 7: Damping of first mode - CC algorithm
Figure 8: Eigenfrequencies at t=30s - CC algorithm
Figure 9: Eigenfrequencies at t=150s - CC algorithm
Figure 10: Mode #1 Monitoring with CC algorithm - Evolution of the frequency
Figure 11: Mode #1 Monitoring with CC algorithm - Evolution of the damping
Data driven method
We compare the covariance and data driven methods applied to the same data and using the same tuning for the parameters. The respective stabilization diagrams are on Figure 5 for the covariance driven approach and Figure 12
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for the data driven approach. The corresponding damping evolutions for the first mode are represented on Figures 7 and 13. We also present some results of the modal parameter estimation of the Ariane data processed with the crystal clear UPC algorithm with the commercial software Artemis on Figures 14, 15 and 16. Note that in the first two figures the frequency is plotted on the horizontal axis and the model order on the vertical axis.
Figure 12: Data-driven - Filtered data - CC algorithm
Figure 13: Data-driven - Damping of first mode - CC algorithm
Figure 14: Stabilization diagram for the low modes from Artemis
Figure 15: Stabilization diagram for the high modes from Artemis
Figure 16: Damping of first mode (Artemis)
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4.2.3
Comments
Both the covariance- and the data-driven methods gave similar results for the stabilization diagrams, especially for the estimation of the frequencies. Damping estimates benefit a lot from the “crystal clear” implementation in both methods, even with this very nonstationary case study. In general, the crystal clear procedure (CC) is a major improvement with respect to the basic subspace identification algorithm. It helped to get clear results even in the case of nonstationary data, and a monitoring of the structure during intense changes of the modal parameters as in Figures 10 and 11 was still possible due to the CC procedure. It may be mandatory for an automated monitoring procedure. 5
Conclusions
We have presented some capabilities of automated subspace identification methods for non stationary structures. We succeeded in the identification of a quickly varying structure and could follow the evolution of the system in continuous time with the help of the “crystal clear” algorithm. Both the covariance- and data-driven approach gave satisfying results, while the covariance-driven approach turned out to give more stable damping estimations. With the help of specialists in the processing procedure, the results for Ariane 5 are good. The main problem is not the variation of the structure but the location of sensors. These locations have been chosen for other purposes than modal analysis and are hard constraints.
ACKNOWLEDGEMENTS The authors would thank again the CNES and different partners for their participation. Dominique Langlais, Benoit Ryckelynck (both from Aerospatiale) and Luc Gonidou (from CNES) were the correspondents for the first study. Their mechanical engineer experience and their knowledge of Ariane launchers were essential for many points as selection of sensors or interpretation of results. In the second stage we would thank again Vincent Le Gallo who made, with the help of Luc Gonidou, a complete study for the parameter choices (see [9]). References [1] M. Basseville, A. Benveniste, M. Goursat, L. Hermans, L. Mevel, H. Van der Auweraer, Output-only subspacebased structural identification: from theory to industrial testing practice, ASME Jal Dynamic Systems Measurement and Control, Special Issue on Identification of Mechanical Systems, 123(4), 668–676, 2001. [2] M. Goursat, L. Mevel, Algorithms for Covariance Subspace Identification: a Choice of Effective Implementations., Proc. IMAC XXVII, Florida, USA, 2009. [3] L. Mevel, M. Goursat, M. Basseville, A. Benveniste, Steelquake modes and modeshapes identification from multiple sensor pools, Proc. IMAC XX, Los Angeles, 2002. [4] A. Benveniste and J.-J. Fuchs, Single Sample Modal Identification of a Non-stationary Stochastic Process, IEEE Trans. Aut. Cont., vol. AC-30, pp. 66-74, 1985. [5] P.V. Overschee and B. De Moor, Subspace Algorithms for the Identification of Combined Deterministicstochastic Systems, Automatica, vol. 30, no 1, pp. 75–93, 1994. [6] M. Abdelghani, M. Goursat, and T. Biolchini, One-line Monitoring of Aircraft Structures under Unknown Excitation, Mechanical Systems and Signal Processing, 1999. [7] L. Hermans and H. Van der Auweraer, Modal testing and analysis of structures under operational conditions: industrial applications, Mechanical Systems and Signal Processing, 13, 193–216, 1999. [8] M. Basseville, A. Benveniste, M. Goursat, L. Mevel, Output-only Modal Analysis of Ariane 5 Launcher, Proc. IMAC XIX, Orlando, 2001. [9] V. Le Gallo, M. Goursat, L. Gonidou, Damping characterization and flight identification, Proc. IMAC XXV, Orlando, 2007.
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[10] Overschee, P. V. and Moor, B. D., Subspace Identification for Linear Systems: Theory, Implementation, Applications, Kluwer, 1996. [11] A. Benveniste and L. Mevel, Nonstationary consistency of subspace methods, IEEE Transactions on Automatic Control, vol. 52(6), pp. 974–984, 2007. [12] M. Goursat and L. Mevel, An example of analysis of thermal effects on modal characteristics of a mechanical structure using Scilab, In S.-Y. Qin, B. Hu, S. Li, and C. Gomez, editors, Scilab Research, Development and Applications, pp. 241–256. Tsinghua University Press - Springer, Beijing, China, 2005. [13] M. Goursat and L. Mevel, COSMAD: Identification and diagnosis for mechanical structures with Scilab, Proceedings of the Multi-conference on Systems and Control, International Symposium on Computer-Aided Control Systems Design (CACSD), San Antonio, TX, US, 2008.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Implementation of Multi-Sine Sweep Excitation on a Large-Scale Aircraft Gordon Hoople Kevin Napolitano ATA Engineering, Inc. 11995 El Camino Real, Suite 200 San Diego, California 92130
Abstract This paper presents the first successful application of multi-reference sine sweep testing on a large-scale military aircraft. The results from this testing are compared with traditional testing methods performed on the same aircraft. Multiple reference sine testing is shown to capture equivalent information in a reduced amount of testing time on a full-scale ground vibration test (GVT).
1. Introduction Ground vibration testing is a key component in preflight certification of new and newly modified aircrafts. Advances in sensor technology and computing power have dramatically increased the efficiency of conducting GVTs; however, with tight schedules, aircraft manufacturers are always looking for ways to make further reductions in testing time. Many different excitation options for GVTs are currently available. The most commonly implemented methods are random and sine sweep testing [1]. In “Multiple Sine Sweep Excitation for Ground Vibration Tests,” Napolitano and Linehan put forward a novel approach to accelerate sine sweep testing by performing multiple sine sweeps simultaneously. This new method of testing, when implemented properly, has two major benefits. First, it can dramatically reduce the amount of testing time required during a GVT by performing all sine sweep tests concurrently rather than sequentially. Second, multi-sine testing does not have the low signal-tonoise ratio problems sometimes associated with random testing. This paper describes an implementation of multiple sine sweep testing on the GVT of a largescale aircraft. The results from this testing are then compared with traditional testing methods performed on the same aircraft. Multi-sine testing is shown to capture equivalent information in a reduced amount of testing time on a full-scale GVT.
2. Background ATA Engineering, Inc., has a long history of conducting GVTs on a wide variety of aircraft and is constantly trying to improve test methods. Currently, ATA primarily uses two excitation methods when performing a typical GVT: multiple reference burst random testing and single reference sine sweep testing. Multi-reference burst random testing provides simultaneous excitation over the frequency range of interest and has been shown since the early 1980’s to dramatically reduce test time over T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_126, © The Society for Experimental Mechanics, Inc. 2011
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classical sine dwell techniques. This is because frequency response functions from all references can be collected from one set of test data. In general, with a sufficient number of shakers, good broadband excitation can be applied across the test article. A potential drawback to burst random testing for particularly large aircraft is that the RMS energy input may be low relative to background noise. This may lead to lower quality FRF [1]. Increasing the excitation level, as well as increasing the number of frame averages, helps improve data quality in this case; however, the penalty is increased test time. This is particularly true for large test articles where the frequency range of interest is low and hence the time required for a good set of data may be as much as an hour. One of the major advantages of sine sweep testing over random testing is that when the methods have equivalent peak force, the sine testing produces a higher overall RMS level. This is because at any given instant during sine sweep testing the energy is concentrated at a single frequency, while in random testing it is distributed over multiple frequencies. This is important because it is the increased RMS level, not the peak force, that leads to better signal-to-noise ratio and higher quality FRF. Another benefit to sine testing is that many structures can be excited in a symmetric or antisymmetric manner in order to emphasize particular modal responses. The major drawback of this approach is that only a single reference can be used during any given test because the input signals are correlated. Separate tests are conducted for symmetric and antisymmetric configurations on the wings, engines, tails, etc. On a typical large aircraft, this often requires six to eight different sweeps. Multi-sine testing can be used in different ways to improve data quality when compared to both burst random testing and sine sweep testing. In this paper, two types of multi-sine excitation are described: full multi-sine and symmetric/antisymmetric multi-sine. Full multi-sine is defined as having the same number of independent reference signals as shakers. For the multi-sine case described in this paper, each of the signals passed to each pair of shakers is composed of two sine waves covering the same frequency range. The two sine sweeps are slightly offset in the time domain so that at each instant the structure is being excited at two different frequencies. When testing a structure which has symmetric and antisymmetric modes, differences in phasing of pairs of shakers can be used to simultaneously excite the structure symmetrically and antisymmetrically. Figure 1 shows both time and frequency domain plots for shakers positioned on the wingtips of a test article at a particular instant. Notice that the signal at each shaker is composed of the addition of two frequencies. For the lower frequency these signals are in phase, and for the higher frequency these signals are out of phase. Also note that the instantaneous frequency pairs sent to each pair of shakers are also offset so that no independent input signal frequency matches another. Figure 2 shows this offset in the frequency pairs sent to each pair of shakers at a particular instant. Another way of thinking of this approach is a symmetric sweep being performed with a time delay anti-symmetric sweep.
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The second type of multi-sine described in this paper is ideal for decreasing testing time on test articles which have both symmetric and antisymmetric modal behavior. This type of multi-sine will be referred to as symmetric or antisymmetric multi-sine. Here the number of independent reference signals is equal to half the number of shakers. On a typical aircraft GVT, the test engineer would first perform a symmetric sweep of the wings, then a symmetric sweep of the engines, and then a symmetric sweep of the tails. All of this would then be repeated for the antisymmetric test configuration. With multi-sine testing, the symmetric sweeps for the wings, engines, and tails can be performed simultaneously with slight offsets in the time domain. This means that instead of six individual sweeps, the testing can be reduced to two sweeps – a symmetric multi-sine and an antisymmetric multi-sine. In this case, three uncorrelated signals are passed to the three pairs of shakers, as shown in Figure 3. During postprocessing, FRF can be generated using a single reference (i.e., the wing reference) as is traditionally done when individual sweeps are performed.
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The following sections will describe implementation of these techniques on a test article and will compare the results to traditional testing methods.
3. Testing ATA was contracted by Lockheed Martin Aeronautics Company to perform the ground vibration test on the Advanced Composite Cargo Aircraft (ACCA). To build the ACCA, Lockheed Martin replaced the mid/aft fuselage and vertical tail of a Dornier 328J aircraft with an advanced composite structure.
Figure 4 - Lockheed Martin's advanced composite cargo aircraft.
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For the GVT, the aircraft was excited at six locations: the engines, wing tips, and horizontal stabilizer tips. The aircraft was tested in a simulated free-free boundary condition. For all the test runs, data was acquired using a sampling frequency of 128 Hz. The data was processed with a blocksize of 2048 samples, which corresponds to a frame length of 16 seconds. In order to validate the multi-sine method, first a complete set of traditional data was taken. Burst random excitation was applied using six shakers with uncorrelated inputs. No windows were applied, and a total of thirty frames of data were measured with no overlap. Sine data was collected using traditional sine sweep techniques with two shakers at a time. The same signal, in a symmetric or antisymmetric configuration, was passed to two of the six shakers at a time. The data was processed using a single reference with an overlap of 90% using a Hanning window. The sine sweep began at 50 Hz and swept down to 1 Hz at a rate of 0.2 decades per minute. Both methods of multi-sine excitation described above were also used to excite the aircraft. The first test was a full multi-sine case using six shakers with six independent signals. For each pair of shakers, two independent signals were injected into the shakers simultaneously providing symmetric and antisymmetric excitation. This multi-sine sweep started at 50 Hz and swept down to 1 Hz at a rate of 0.2 decades per minute. During the sweep, a ratio of 1.1 was kept between the frequencies of all six signals in order to ensure independence. Signals for the shakers were created using ATA’s Multi-Sine Sweep Creator utility [1]. The data was processed using all six references with an overlap of 90% using a Hanning window. The second multi-sine test was symmetric/antisymmetric multi-sine testing, which used six shakers and three independent signals. In this case, three symmetric or antisymmetric signals were sent to three pairs of shakers. The shakers on the wings received the first signal, the shakers on the engines the second, and the shakers on the tails the third. All three signals covered the same range in the frequency domain but were offset in the time domain to ensure that they were independent. The sweep began at 50 Hz and swept down to 1 Hz at a rate of 0.2 decades per minute. During the sweep, a ratio of 1.1 was kept between the frequencies of all three signals in order to ensure independence. The data was processed using only one reference at a time for comparison to the equivalent single sine sweep used in traditional testing methods. The data was processed using a Hanning window, and 90% overlap was used for averaging. The differences in testing time between the traditional and multi-sine approach are presented in Table 1. The results show that a complete set of multi-sine testing takes less than half the time of the complete set of traditional testing methods. For tests with requirements for multiple configurations or multiple levels, the time savings offered by the multi-sine approach could have a significant impact on the overall schedule of the test program. In order to understand the time savings further, a few comparisons should be made. The first case is burst random versus full multi-sine. In this case, due to the frequency range of interest, the burst random runs were a relatively short eight minutes. When low frequency, high resolution, data is desired, a burst random run could take up to forty minutes. The multi-sine approach, on the other hand, could achieve equivalent frequency resolution with a shorter ten to twenty minute
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sweep. This time savings is possible because multi-sine data can be overlap processed while burst random data can not. Another comparison to highlight is that the six symmetric and antisymmetric sweeps can be replaced with a symmetric and an antisymmetric multi-sine sweep. The traditional sweeps take 54 minutes, while the multi-sine sweeps take only 18 minutes. This is a factor of three difference, which can quickly impact the test schedule if there are multiple configurations. This discussion has presented the savings in testing time; the next section will compare the results of a traditional testing approach to the multi-sine testing approach. Table 1 - Comparison between testing times for traditional and multi-sine approaches. Traditional Runs Burst Random Symmetric Sine Wings Symmetric Sine Tails Symmetric Sine Engines Antisymmetric Sine Wings Antisymmetric Sine Tails Antisymmetric Sine Engines Total
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4. Presentation of Results This section presents results for the two implementations of multi-sine previously discussed: full multi-sine and symmetric/antisymmetric multi-sine. In each case, the multi-sine data is compared to traditional testing methods and found to be a suitable equivalent. In the interest of simplicity, comparisons for symmetric and antisymmetric sweeps will be based only on data from shakers located at the wingtips. The authors, however, have reviewed the data from the other locations and confirmed that it supports the same conclusions. 4.1. Full Multi-Sine The full multi-sine data collected for this paper can be directly compared to burst random data. Further data processing methods can be used to compare full multi-sine data to other types of data sets, but this is beyond the scope of this paper. For further discussion of this topic see “Order Tracking with Multi-Sine Excitation on a Ground Vibration Test [2].” Figure 5 shows a time history of the excitation force. Notice that while the peak force for the multi-sine and burst random are on the same order of magnitude, the energy input for the sine wave is concentrated at two frequencies while the energy for the burst random data is spread across the entire frequency band. Therefore the multi-sine data provides a significantly higher excitation level across the entire sweep range. Another point to make is that there is no attempt to keep the force level constant as the shakers are part of an open loop control system. As time progresses the full multisine signal decreases in force level due to attenuation characteristics of the shaker at lower frequencies.
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A plot of the complex mode indicator function (CMIF) for both excitation types is shown in Figure 6. The CMIF is one type of summary plot of all the collected FRF and is similar to FRF in that peaks indicate modes of the structure [3]. Figure 6 shows that all but two of the dominant modes of the structure fall below 25 Hz. The increased force levels of the multi-sine testing, particularly at the low frequencies, help to avoid the signal-to-noise problems sometimes associated with burst random. The CMIF plot also shows the impact of the increased force level. Notice that the CMIF for the burst random data, on the left, shows ragged peaks. The full multisine has more clearly defined peaks, and the curve-fitting algorithm is able to solve for potential mode shapes with a finer tolerance for frequency and damping. 10
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As shown in Figure 7, several peaks are present in the burst random data that are not in the full multi-sine data. Further review of other data collected during testing suggested that the peaks below 8 Hz were due to noise and do not indicate modes. Conversely, there is a mode present in the burst random data just below 10 Hz that is somewhat obscured in the multi-sine data. Since the multi-sine data was in general much cleaner than the burst random set, the multi-sine data was used to extract the complete set of mode shapes for this GVT.
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Figure 7 - Burst random and full multi-sine – 5 to 10 Hz only. Peaks which are shown in the burst random data in the 6 to 7 Hz range are actually due to noise which is not present in the multi-sine data.
4.2. Symmetric and Antisymmetric Multi-Sine Symmetric and antisymmetric multi-sine can be compared to a traditional single sine sweep. In this case, the excitation signals are effectively identical, the difference being that in the multisine testing, all three pairs of shakers performed sweeps at the same time, as opposed to individually. Figure 8 shows the time history for the symmetric case on the wings; notice that the two signals are almost identical.
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Before comparing the two sets of sweeps, it is important first to remember that the goal of these sweeps is to excite modes involving a particular part of the test article. In this case, the data presented is from sweeps that were performed on the wings. Therefore, the relevant information obtained from the data is modes involving wing responses. Taking this into account, the relevant data to compare are the frequencies associated with wing modes, rather than all of the frequencies, as was done in the previous section. First consider the symmetric sweep case shown in Figure 9. Mode shapes have been fit for both sets of data, and the primary wing modes have been highlighted in the figure using frequency
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tags. In the right half of the figure, the peaks in the CMIF plots corresponding to the first three wing bending modes are effectively identical. Examining the full frequency range, some difference can be observed at 34 Hz. This frequency corresponds to symmetric wing torsion. In the multi-sine case, one mode dominates in the CMIF: a wing torsion mode. In the single sine case, however, wing torsion and tail torsion can both be seen in the CMIF. When modes were extracted from both of these data sets using AFPoly™, the torsion modes showed excellent agreement. The frequencies were within 0.5% and had a cross-MAC of 98.9%. This indicates that while the CMIF may look slightly different, the same modes are contained within both data sets. Next, consider the antisymmetric sweep case shown in Figure 10. Again, mode shapes were fit for both sets of data, and the primary wing modes were identified in the figures using frequency tags. There is excellent agreement in the CMIF between the tagged peaks, with the exception of the mode at 31.5 Hz. This corresponds to the antisymmetric torsion mode of the wings. Similar to the symmetric case, when this mode was extracted from both data sets, excellent agreement was found between the two torsion modes. The frequencies were within 0.6%, and the MAC was 96.7. This reinforces the conclusion that although the CMIF looks slightly different in this case, the underlying modal information is the same in both data sets, allowing a complete modal extraction to be performed.
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Another observation in the CMIF for the anti-symmetric multi-sine was a more clearly defined mode at 9.5 Hz. This mode actually corresponds to a response from the horizontal stabilizers, and, while it is interesting that it was captured using excitation at the wings, it is preferable to use the horizontal stabilizers excitation to capture this mode. 10
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Figure 9 - Symmetric single sine and symmetric multi-sine. The tagged frequencies indicate wing modes. The figure on the left shows the full range of 0 to 50 Hz, while the figure on the right shows only the first three wing modes below 12 Hz.
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Figure 10 - Antisymmetric single sine and antisymmetric multi-sine. The tagged frequencies indicate wing modes. The figure on the left shows the full range of 0 to 50 Hz, while the figure on the right shows only the first three wing modes below 12 Hz.
5. Conclusions This paper has explained the use of full multi-sine and symmetric/antisymmetric multi-sine on a large-scale military aircraft. Multi-sine has been shown to produce equivalent results compared to traditional testing methods, and all modes of the aircraft were successfully extracted using multi-sine data. The major advantage of multi-sine testing is that it allows the test engineer to dramatically reduce testing time, in this case by a factor of two. Symmetric and antisymmetric data that previously required six separate sine runs can now be acquired in two symmetric/antisymetric multi-sine runs. Burst random data, which can require long acquisition times for high frequency resolution, can instead be replaced with full multi-sine data that can be acquired in half to one third the time. Additionally, there is little cost to implementing these methods; if a target mode is not successfully extracted, then the test engineer can revert to traditional methods. Future work will show that by using data processing techniques, all of the runs discussed in this paper can be extracted from a single full multi-sine data set.
References [1] [2] [3]
Napolitano, K and Linehan, D., “Multiple Sine Sweep Excitation for Ground Vibration Tests,” IMAC XXVII, Orlando, Florida, Feb. 2009. Hoople, G. and Napolitano, N., “Order Tracking With Multi-Sine Excitation on a Ground Vibration Test,” IMAC XXVIII, Jacksonville, Florida, Feb. 2010 Shih, C.Y. and Brown, D.L., “The Complex Mode Indicator Function Approach to Modal Analysis,” Pre-IMAC 8 Symposium, UC-Irvine, 1989.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
An Integrated Experimental and Computational Approach to Analyze Flexible Flapping Wings in Hover
Pin Wu, Erik Sällström, Lawrence Ukeiley, and Peter Ifju University of Florida, Gainesville, FL, 32611 Satish Chimakurthi, Hikaru Aono, Carlos E.S. Cesnik, and Wei Shyy University of Michigan, Ann Arbor, MI, 48109
Abstract Biological flyers exploit wing deformation during flapping flight. There is a substantial need to improve the understanding of the aeroelastic effects associated with the wing deformation to build flapping wing micro air vehicles. This paper presents an effort to develop an integrated approach involving both experimental and computational methods to realize this goal. As the first step, an isotropic flat plate aluminum wing is manufactured and actuated to perform a single degree-of-freedom flapping motion. The wing deformation and airflow around the wing are measured with digital image correlation (DIC) and particle image velocimetry (PIV), respectively. Computational analyses are performed on this wing configuration using a combined nonlinear structural dynamics and Navier-Stokes solution. Reasonable agreement obtained between experimental and computational data in this preliminary effort shows a potential to analyze more complicated flexible flapping wings in future. Introduction There has been a growing interest in the development of flapping wing micro air vehicles (FWMAVs) in the recent past [1-3]. The physics behind flapping wing flight, in particular, the coupled interaction between aerodynamics and wing deformations, needs to be better understood for FWMAV development, which has been stimulated by the long history of natural flight studies. State-of-the-art reviews in this subject are given in references [2] and [3]. Photography and videography studies indicate that most biological flyers undergo orderly deformation in flight [4]. Insects, bats, and birds exploit the coupling between flexible wings and aerodynamic forces, using passive wing deformation to gain aerodynamic advantage [5]. The interaction between unsteady aerodynamics and structural flexibility is, therefore, of considerable importance for FWMAV development [2]. There are numerous publications discussing and analyzing flapping wings [2, 5-9] using both experimental and computational approaches. Some of those efforts that primarily focus on fluid structure interactions are discussed here. Ho et al. [10] used the two-way coupling feature of a commercially available flow solver (CFD-ACE+) and the structural dynamics solver FEMSTRESS to create an aeroelastic solution to analyze membrane flapping wings. They showed that stiffness distribution is a key parameter in determining vortex interaction and thrust production. Heathcote and Gursul [11] have experimentally investigated the effect of chord-wise flexibility on
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thrust generation of airfoils undergoing a plunging motion under various free stream velocities. Direct force measurements showed that the thrust/input-power ratio was found to be greater for flexible airfoils than for the rigid one. Katz and Weihs [12] also showed that chord wise flexibility could increase the propulsive efficiency of a combined pitching/plunging wing. To study the effect of the spanwise flexibility on the thrust and propulsive efficiency of a plunging wing, a water tunnel study was conducted by Heathcote et al [13] on a NACA 0012 uniform wing of aspect ratio 3. They observed that for Strouhal numbers greater than 0.2, a degree of spanwise flexibility was beneficial. In a subsequent effort, Chimakurthi et al [14] and Aono et al. [15, 16] conducted computations to explain the impact of spanwise and chordwise flexibility on aerodynamics with the help of a rigorous analysis of flow structures, pressure distributions, aeroelastic deformation, and phase lag. In the case of spanwise flexible plunging wing structures, it was shown that the effective angle of attack due to prescribed motion and the phase lag of the response are the two key factors that affect the aerodynamic force generation. However, in the case of the chordwise flexible wings, it was shown that wing deformation in the chordwise direction could result in an effective projected area normal to the flight trajectory that supports thrust force generation and a subsequent re-distribution of the aerodynamic loading in the lift and thrust directions. Liu and Bose [17] suggested that propulsive efficiency of oscillating flexible wings can be increased, over the value of an equivalent rigid wing, by careful control of the phase of the spanwise flexibility relative to other motion parameters. It has been observed in both nature and experiments that flapping wing elasticity/flexibility is crucial to flapping flight. Wu et al. [18] investigated structural deformation and airflow around three Zimmerman wings with varying flexibility flapping under hovering conditions. The wings were made from a carbon fiber skeleton and covered with a flexible membrane. Flexibility was varied by changing the number of layers of carbon fiber reinforcing the leading edge and hence it mainly affected the spanwise bending stiffness. The study showed that flapping wing stiffness and mass distribution has significant impact to the aerodynamic performance: certain structural compliance enables flapping wings to produce more thrust. Similarly, fresh hawkmoth (Manduca sexta) wings have been compared with dry wings [19] (laminated with spray paint to recover the weight loss) and found that the more compliant fresh wing has much superior aerodynamic performance with airflow visualized via particle image velocimetry (PIV). Dynamic response is another important aspect: it is a direct benchmark for both the wing inertia (mass) and stiffness (inverted flexibility). This aspect has been studied by measuring the wing deformation caused by inertial loading and aerodynamic loading [20]; the results show that a hawkmoth’s wing deformation at 26 Hz is mainly caused by inertial loads, as the experiments conducted in air and in helium (approx. 15% air density) produce similar deformation pattern, suggesting aerodynamic forces have minimal effect. This may indicate that the passive deformation caused by inertia is utilized to affect aerodynamics. However, as pointed out by the authors, the result may be species specific and many contradictory conclusions have been reported [21, 22]. To further our understanding of flexible flapping wing under hovering conditions, this paper presents an ongoing effort to develop an integrated experimental and computational approach to investigate coupled flow-structure interactions. As the first step, an isotropic aluminum wing is manufactured and actuated to perform a single degree-of-freedom flapping motion (±21º stroke amplitude and 10 Hz flapping frequency). The wing deformation and airflow around the wing are measured using DIC and PIV respectively. Computational analyses are performed on the same wing configuration using a combined nonlinear structural dynamics and Navier-Stokes solution. The main objectives of this paper are to: a) discuss the development of an integrated experimental and computational approach to analyze flapping wing configurations, and b) to show some preliminary comparisons of flow structures and wing deformation between experimental data and computational response. Experimental Setup A) Flapping Mechanism and Tested Wings A single-degree-of-freedom flapping mechanism is designed and built for this study, as shown in Figure 1 (the same mechanism as the one used in reference [18]; but the wings shown in the figure are different from the aluminum wings tested). The design is created based on a Maxon motor system that includes a 15 W brushless DC motor EC16, a 57/13 reduction ratio planetary gear head, a 256 counts-per-turn encoder and an EPOS 24 controller. This system provides precise control of the motor system: the sensor provides position and velocity feedback to the controller that actively regulates the motor. Utilizing the high precision pre-assembled planetary gear head rather than constructing a custom gear transmission is also advantageous. The final output range of the motor shaft is: speed 0 to 45 revolutions per second (RPS) and nominal torque 0 to 21 N•mm.
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The rotation output from the motor is first transformed into a reciprocating motion with a crank-slider mechanism; then a bar linkage mechanism realizes the flapping motion at the wing mount. A detailed schematic description of the flapping kinematics is presented in Figure 1. The geometric relationship between motor rotation (angle T) and flap angle D is expressed in the equations in the figure, where D is the flap angle; T is the motor rotational angle; x is the vertical displacement from the center point when the wings are horizontally positioned. The rest of the parameters are selected so that a ±21 º amplitude is maintained. The experiments are performed at 10 Hz flapping frequency. The isotropic wings tested in this study are made from 0.4 mm thick aluminum sheets. The wing planform is of a 7.65 aspect ratio and has a Zimmerman shape, i.e. is formed by two ellipses which intersect at the quarter-chord point. The wing length is 75 mm and the root chord length is 25 mm. The wings are manufactured with CNC machine to achieve the exact contour and avoid stresses that may cause warping. After machining, the wing surface is primed with flat white coating for the background and speckled with black dots for DIC measurements. Another coating (Rhodamine) is applied for reducing blooming of the laser sheet hitting the wing during PIV measurements. The final weight of a single wing is 1.685±0.005 grams. The flatness is examined with DIC and bounded within -0.05~0.05 mm. The wing is mounted to the mechanism at a 5x5 mm2 square region at the crossing of the leading edge and root. This region of the wing is assumed not to deform during the flapping motion (therefore serving as the reference for calculating the wing deformation and also the boundary condition).
Figure 1. Flapping mechanism and its schematics. B) Deformation Measurement with DIC The kinematics and deformation of the flapping wings are measured with a high speed digital image correlation system. DIC is a well-developed non-contact measurement technique used to capture full-field displacement and deformation of surfaces via stereo-triangulation. A random speckle pattern is applied to the flapping wing, which is then digitized into wing surface coordinates with stereo triangulation. The full-field displacements of the wing during the flapping motion are computed with temporal matching, by minimizing a cross correlation function between discrete regions of speckle patterns on a deformed wing surface and an undeformed one. The DIC system (Correlated Solutions Inc., South Carolina, USA) used in this study consists of two Phantom high speed
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cameras that can store 2800 pictures of 800 pixels by 600 pixels resolution in its flash memory at a maximum rate of 4800 frames per second. The exposure time was 150 ȝs and the frame rate was set to obtain 100 frames per cycle. In the 10 Hz case therefore, the frame rate is 1000 fps. Only the left wing is measured with DIC for the kinematics and deformation. The two DIC cameras are symmetrically placed facing the wings at mid-plane. Field of view on both cameras is adjusted to fit and fill the frame. Then the cameras are calibrated and used to measure the full-field deformation and kinematics of the flapping wing. A reference image at the mid-plane of stroke is taken beforehand. All data are taken in raw format (.tif) until correlated with VIC3D (software by Correlated Solutions Inc.) and output as text files (.txt) for post processing. The data structure contains coordinates of all points on the measurement surface. During post processing, the structural deformation is separated from rigid body kinematics by comparing the undeformed reference data to the deformed wing at the same flap angle. The errors of DIC are induced during the calibration phase and the uncertainty will remain the same after calibration. If a camera pair is carefully calibrated, the system uncertainty is under ±0.1 mm in the current setup (except occasionally the speckle pattern introduces other error). C) Airflow Measurement with PIV Particle image velocimetry, a mature technique used to measure the velocity of particles (assumed to follow the airflow) in the flow of interest [7, 18, 23, 24], is applied to examine the airflow around a flapping wing. The flow field measurement area is defined by the confluence of the position and dimension of a laser sheet, along with the camera’s field of view. Images are captured at each laser flash so that the particle spatial displacement can be extracted by correlating two images taken within a small time interval. One LaVision Imager pro X 4M camera capable of capturing up to 7 image pairs per second with a pixel resolution of 2048x2048 is used. The camera is mounted perpendicular to the wing root, measuring a plane in the wing span direction at 7 Hz frame rate. A laser sheet is generated from a Litron Nano L PIV Pulsed Nd:YAG Laser System (max energy 135 mJ/pulse, 532nm) and is directed at the target from below. The air is seeded by a LaVision Aerosol Generator, with olive oil, an modal radius of approximately 0.25 microns. Davis 7 (software from LaVision) is used to control the PIV system and process the data. The measurements are made in the test section of a sealed open jet wind tunnel. Snapshots of the velocity field are acquired at 3 different chord locations (0.25, 0.5 and 0.75 of the chord length at 2 the root). The measurement area is 121x121 mm , enclosing the wing at the upper right corner. The PIV images are captured at up to 7 Hz, with the snapshots skipped that the system does not have time to process. 2500 images are captured at each location and flapping frequency. The images are processed using a multi-pass algorithm with shrinking interrogation size to produce velocity vectors. The region next to the wing and the area where the laser is shaded by the wing are masked out. The raw images are then processed to find the wing angle. The wing angle is then fit to a sine wave to extract the phase of the wing in each snapshot. The snapshots are then divided into 50 bins, with the bins equally spaced in time within the flapping cycle. Phase averages are then produced by averaging each bin. The wing angle variation within a bin varies between approximately ±0.5º to ±2.5º, and each bin contains on the average 50 snapshots. Computational Model Computational analyses in this work have been performed using a partitioned aeroelastic solution combining a nonlinear structural dynamics solver called UM/NLAMS [25] and a time-accurate Navier-Stokes solver called UM/STREAM [26]. Full details of the aeroelastic solution are available in reference [25]. A summary of the geometric and mechanical properties of the aluminum wing configuration studied in the experiment are included in Table 1. Table 2 provides information about the flow properties (dimensional). In Table 3, the key dimensionless parameters related to either the structure, the flow, or to both for the aluminum wing configuration are furnished. The dimensionless parameters U* and 31 are defined in references. [14-16]. An O-type structured multi-block grid around the Zimmerman wing of aspect ratio 7.65 is used for the computational fluid dynamics (CFD) simulations. Based on a grid sensitivity study (discussed later), the grid configuration has 0.7 million points. The finite element mesh configuration developed in UM/NLAMS has triangular finite elements. A 5 mm x 5 mm square region near the root at the leading edge is constrained in all degrees of freedom (with respect to the global frame) in the structural solver, since the flapping mechanism in the experiment is used to actuate that region on the wing. A total of 480 elements (275 nodes) are used in the finite element discretization.
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To assess the independence of the numerical solution to CFD grid refinement, a grid convergence study is performed and a suitable grid (approximately total 0.7 million cells) is subsequently chosen. A time step of 1.5 x10-3 s and hence approximately 650 time-steps per period of computation are used for the computation. The convergence criterion for the aeroelastic convergence is chosen as a check on the Euclidean norm of the entire solution vector computed in two consecutive fluid-structure subiterations. Table 1 Geometric and mechanical properties of the Zimmerman aluminum flapping wing configuration. Quantity Symbol Value Semi-span at quarter chord b 0.075 m Chord length at wing root croot 0.025 m Structural thickness t 0.4 ×10-3 m Poisson’s ratio ȣ 0.3 Material density ȡalum 2700 kg/m3 Young’s modulus of material Ealum 70.0 GPa Table 2 Flow properties associated with the Zimmerman aluminum flapping wing configuration. Quantity Symbol Value Reference flow velocity Uref 1.0995 m/s (hover) Air density ȡair 1.209 kg m-3 Table 3 Dimensionless parameters associated with the Zimmerman aluminum flapping wing configuration. Quantity Symbol Value Root chord-based Reynolds number Re 2605 Reduced frequency k 0.56 Aspect ratio AR 7.65 Density ratio 2233 U* Scaling parameter – I 3.8×104 Ȇଵ It may be noted that the wing is defined in the X-Z plane wherein, the X axis goes through the wing chord and the Z-axis going through the wing length. Results – Comparison between Experiments and Computations The results of the comparison between the experimental data and simulation output are presented in this section. The comparison is first made between fluid studies and then between structure studies. Figures 2 and 3 show a comparison of the velocity magnitude and vorticity between computation and the experiment for two different points in the flapping cycle (t/T = 0.3 and 0.48, where T is the period of prescribed flap rotation). The flow field in these plots is shown on a slice that is cut at the quarter chord station going through the span. The experimental data could not be obtained in the region near the wing that is outside the laser sheet (appears as a white conelike region above the wing in the experimental velocity magnitude and vorticity contours).
Experiment (Phase averaged)
Computation (6th cycle) Velocity magnitude [m/s]
Computation (Phase averaged)
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Experiment (Phase averaged)
Computation (6th cycle)
Computation (Phase averaged)
X-Vorticity [1/s] Figure 2. Comparison of velocity magnitude and vorticity between experiment and computation for a slice of the wing at the quarter chord going through the span for time instant t/T = 0.3. The sub-plots in the first column correspond to the phase averaged velocity magnitude and X-vorticity contours of the experimental data. The sub-plots in the second column correspond to the velocity magnitude and Xvorticity contours corresponding to the 6th cycle of computation. The sub-plots in the third column correspond to the phase averaged (using the data corresponding to the 3rd, 4th, 5th, and 6th cycles) velocity magnitude and X-vorticity contours of the computational data. (Re= 2605, k= 0.56, Ȇ1= 3.8×104, ȡ*= 2233).
Experiment (Phase averaged)
Computation (6th cycle)
Computation (Phase averaged)
Velocity magnitude [m/s]
Experiment (Phase averaged)
Computation (6th cycle)
Computation (Phase averaged)
X-Vorticity [1/s] Figure 3. Comparison of velocity magnitude and vorticity between experiment and computation for a slice of the wing at the quarter chord going through the span for time instant t/T = 0.48. The sub-plots in the first column correspond to the phase averaged velocity magnitude and X-vorticity contours of the experimental data. The sub-plots in the second column correspond to the velocity magnitude and Xvorticity contours corresponding to the 6th cycle of computation. The sub-plots in the third column
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correspond to the phase averaged (using the data corresponding to the 3rd, 4th, 5th, and 6th cycles) velocity magnitude and X-vorticity contours of the computational data. (Re= 2605, k= 0.56, Ȇ1= 3.8×104, ȡ*= 2233). As seen from the plots, there is a good overall agreement in the flow structure between the computation and the experiment at both points in the flapping cycle. However, the vorticity is weaker and more fragmented in the experiment than in the computation. Also, in general, there is more discrepancy near the wing tip than in the rest of the wing. This is confirmed from the comparison of the velocity distributions between the computation and the experiment shown in Figure 4. Each of those sub-plots is obtained by considering a line of points vertically above and below the wing obtained by intersecting a slice going through the quarter chord all along the span and another slice at either a section near the mid-span or the tip that goes through the entire chord. For example, the sub-plots (A-1) and (B-1) of Figure 4 show such velocity magnitude distributions corresponding to a line of points obtained by intersecting the chordwise slice at mid-span and tip respectively one after the other with the slice going through the span at the quarter chord, both for the time instant t/T = 0.3. Then, the sub-plots (C-1) and (D-1) of Figure 4 correspond to the same line of points but now for time instant t/T = 0.48.
(A-1) – slice at the mid-span
(B-1) – slice at the tip Instantaneous
(A-2) – slice at the mid-span
(B-2) – slice at the tip Phase Averaged
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(C-1) – slice at the mid-span
(D-1) – slice at the tip Instantaneous
\ (C-2) – slice at the mid-span
(D-2) – slice at the tip Phase Averaged
Figure 4. Comparison of velocity magnitude between computation and experiment at two different time instants and for two different slices along the wing span: sub-plots A-1, B-1, C-1, and D-1 correspond to instantaneous velocity magnitude profiles and sub-plots A-2, B-2, C-2, and D-2 correspond to phase averaged velocity magnitude profiles. Sub-plots on the left column correspond to those at the mid-span slice of the wing and those on the right column correspond to a slice near the tip. The errorbars around the phase averaged flow indicate the interval within which 95 % of the instantaneous values are expected to fall, assuming the distribution is Gaussian.
Also included in Fig. 4 are phase averaged velocity profiles for both points in the cycle, t/T = 0.3 and t/T = 0.48, (sub-plots A-2, B-2 correspond to the former time instant and C-2, D-2 correspond to the latter time instant) including the error bars for the experimental data. While there was good qualitative agreement from the contour plots in Figures 2 and 3 here one can see a more quantitative comparison. In general, there are cycle-to-cycle variations in the velocity magnitudes in both computation and experiment. Notwithstanding that, specifically, it appears that the spatial locations associated with the rise in velocities due to the wing motions do not match too well between the experiments and computations however the maximum amplitudes of the velocity magnitudes do agree quite well. In the experimental data, the magnitudes of variance near the tip region are generally larger than those near the mid-span. Figure 5 shows the lift coefficient computed from numerical data on the wing as a function of non-dimensional time. Figure 6 shows the iso-surfaces of the vorticity magnitude (the color
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corresponds to the spanwise vorticity) corresponding to two different time instants labeled as (a) and (b) in Figure 5. Three-dimensional vortex generation is seen in Figure 6 at both time instants wherein the counter-rotating vortices at the leading and the trailing edge interact with the tip vortex during the wing motion. In particular, the vortices generated during a previous stroke (indicated as “PV” in the figure) are captured by the wing and interact with the vortices generated during a current stroke indicated as “CV”.
Figure 5. Lift coefficient on the wing as a function of normalized time (time is normalized with respect to a period of flap rotation), DS- downstroke, US – upstroke
Figure 6. Iso-contours of vorticity magnitude on the aluminum wing and Z-vorticity contours at two stations along the wing length at two different time instants: sub-plots (i) and (ii) correspond to the isocontours of the vorticity magnitude (color indicates the magnitude of the Z-vorticity) for time instants (a) and (b) of Fig. 5 respectively. Sub-plots (iii) and (iv) correspond to the Z-vorticity contours at two stations along the wing length (as indicated) for time instants (a) and (b) of Fig. 5 respectively. (CV – Vortex generated in the current stroke, PV – vortex generated in the previous stroke, blue color indicates clockwise vorticity from the viewpoint of an observer looking into the plane of the plot). Figure 7 shows the comparison of the normalized vertical displacement (with respect to the chord length at the wing root) at a point on the wing tip between the experiment and the computation. As shown in the figure, there is a decent overall agreement in both amplitude and phase between the computational response and the
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experimental data. The tip deformation obtained is only around 3 % of wing chord which means that the selected wing configuration is not compliant enough to deform significantly. Future studies will be focused on more flexible cases.
Figure 7. Aluminium wing response due to flapping excitation (normalized with respect to chord length at the root). Conclusions This paper has presented an integrated approach involving both experimental and computational methods having a potential to examine flexible flapping wing configurations. An aluminum wing is prescribed with single degree-offreedom flap rotation at 10 Hz frequency and +/- 21º amplitude and both flow velocities and deformations are measured in the experiment using digital image correlation and digital particle image velocimetry techniques respectively. Preliminary comparison of flow velocities and wing deformation between the computational and the experimental data showed a decent agreement. This work is part of an ongoing study of the analyses of flexible flapping wing structures and the effect of flexibility on aerodynamics will be reported in future. Acknowledgements This work is supported by the Air Forced Office of Scientific Research under MURI program 69726. The authors would also like to thank Lunxu Xie, Dr. Tony Schmitz and his research group for helping with the manufacture of the wings. References [1] Shyy, W., Berg, M. and Ljungqvist, D., “Flapping and Flexible Wings for Biological and Micro Air Vehicles,” Progress in Aerospace Sciences, 35, 155-205, 1999. [2] Shyy, W., Lian, Y., Tang, J., Viieru, D. and Liu, H. Aerodynamics of Low Reynolds Number Flyers, Cambridge University Press, NY, 2008. [3] Mueller, T. J. Fixed and Flapping Wing Aerodynamics for Micro Air Vehicle Applications, AIAA Inc., VA, 2001. [4] Wootton, J. “Support and Deformability in Insect Wings,” Journal of Zoology, 193, 447-468, 1981. [5] Weis-Fogh, T., “Quick Estimates of Flight Fitness in Hovering Animals, Including Novel Mechanisms for Lift Production,” The Journal of Experimental Biology, 59, 169-230, 1973. [6] Ellington, C. P., Van Den Berg, C., Willmott, A. P. and Thomas, A. L. R., “Leading-edge Vortices in Insect Flight,” Nature, 384, 626-630, 1996
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[7] Dickinson, M. H., Lehmann, F-O., and Sane, S. P., “Wing Rotation and the Aerodynamic Basis of Insect Flight”, Science, 284, 1954-1960, 1999. [8] Wang, Z. J., “Dissecting Insect Flight,” Annual Reviews of Fluid Mechanics, 37, 183-210, 2005. [9] Liu, H. and Aono, H., “Size Effect on Insect Hovering Aerodynamics: an Integrative Computational Study,” Bioinspiration & Biomimetics, 4, 1-13, 2009. [10] Ho, S., Nassef, H., Pornsinsirirak, N., Tai, Y-C, Ho, C-M., “Unsteady Aerodynamics and Flow Control for Flapping Wing Flyers,” Progress in Aerospace Sciences, 39, 635-681, 2003. [11] Heathcote, S. and Gursul, I., “Flexible Flapping Airfoil Propulsion at Low Reynolds Numbers,” AIAA Journal, 45, 1066-1079, 2007. [12] Katz, J. and Weihs, D., “Hydrodynamic Propulsion by Large Amplitude Oscillation of an Airfoil with Chordwise Flexibility,” Journal of Fluid Mechanics, 88, 485-497, 1978. [13] Heathcote, S., Wang, Z. and Gursul, I., “Effect of Spanwise Flexibility on Flapping Wing Propulsion,” Journal of Fluids and Structures, 24, 183-199, 2008. [14] Chimakurthi, S. K., Tang, J., Palacios, R., Cesnik, C.E.S. and Shyy, W., “Computational Aeroelasticity Framework for Analyzing Flapping Wing Micro Air Vehicles,” AIAA Journal, 47, 1865-1878. [15] Aono, H, Chimakurthi, S. K., Cesnik, C.E.S., Liu, H. and Shyy, W., “Computational Modeling of Spanwise Flexibility Effects on Flapping Wing Aerodynamics,” AIAA-2009-1270. [16] Aono, H., Tang, J., Chimakurthi, S.K., Cesnik, C.E.S., Liu, H., and Shyy, W. “Spanwise and Chordwise Flexibility Effects in Plunging Wings”, under review [17] Liu, P. and Bose, N., “Propulsive Performance from Oscillating Propulsors with Spanwise Flexibility,” Proceedings of The Royal Society A. Mathematical Physical & Engineering Sciences, 453, 1763-1770, 1997. [18] Wu, P. Ifju, P., Stanford, B, Sällström, E., Ukeiley, L., Love, R. and Lind, R., “A Multidisplinary Experimental Study of Flapping Wing Aeroelasticity in Thrust Production,” AIAA 2009-2413. [19] Mountcastle, A., and Daniel T., “Aerodynamic and functional consequences of wing compliance,” Exp. Fluids. 46, 873-882, 2009. [20]Combes, S., and Daniel, T., “Flexural Into thin air: contributions of aerodynamic and inertial-elastic forces to wing bending in the hawkmoth Manduca sexta,” The Journal of Experimental Biology, 206, 2999-3006, 2003. [21] Sun, M. and Tang, J., “Lift and power requirements of hovering flight in Drosophila virilis,” The Journal of Experimental Biology, 205, 2413-2427, 2002. [22] Wakeling, J. M. and Ellington, C. P., “Dragonfly flight. III. Lift and power requirements,” The Journal of Experimental Biology, 200, 583-600, 1997. [23] Bomphrey, R. J., Lawson, N. J., Harding, N. J., Taylor, G. K. and Thomas, A. L. R., “The Aerodynamics of Manduca Sexta: Digital Particle Image Velocimetry Analysis of the Leading-Edge Vortex,” The Journal of Experimental Biology, 208, 1079-1094, 2005. [24] Warrick, D. R., Tobalske, B. W. and Powers, D. R., “Aerodynamics of the Hovering Hummingbird,” Nature, 435, 1094-1097, 2005. [25] Chimakurthi, S. K., Standford, B., Cesnik, C.E.S. and Shyy, W., “Flapping Wing CFD/CSD Aeroelastic Formulation Based on a Co-rotational Shell Finite Element,” AIAA 2009-2412. [26] Shyy, W., Udaykumr, H. S., Rao, M. H. and Smith, R. W., Computational Fluid Dynamics with Moving Boundaries. Dover. 1996.
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BookID 214574_ChapID 128_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Random Decrement Signal Processing of Modal Impact Test Data
Rune Brincker, Professor Anders Brandt, Associate professor Department of Industrial and Civil Engineering, University of Southern Denmark, Niels Bohrs Allé 1, DK-5230 Odense M, Denmark
NOMENCLATURE
t, t k :
Continous, discrete time
X (t ), Y (t ) : x(t ), y (t ) : h(t ) :
Stochastic Processes Signals
Z:
Cyclic frequency
X (Z ), Y (Z ) :
Signal Fourier transforms
G XX (Z ), G XX (Z ) :
Spectral densities
H 1 (Z ), H 2 (Z ) : J (Z ) :
Transfer functions
D XX (W ), D XY (W ) : Dˆ XX (W ), Dˆ XY (W ) :
Random Decrement functions
Impulse response function
Coherence
Random Decrement function estimates
ABSTRACT When performing modal impact tests on mechanical systems many practioners utilize hardware based digital signal processors to rapidly estimate average spectral properties from a set of repeated impact and response time traces. However, if the complete set of time traces is saved as is commonly done when performing impact tests on civil structures then more generalized analysis and triggering conditions can be used to process the dataset. One such more general triggering condition can be referred to as random decrement signal processing. In this paper it is explained and illustrated how random decrement signal processing can be used to accept double hit impact and overlapping responses and it is shown that in many cases, the estimated transfer function are estimated with higher accuracy then using traditional triggering/averaging techniques. .
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_128, © The Society for Experimental Mechanics, Inc. 2011
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1454 1. INTRODUCTION The way most vibration data acquisition systems are designed is based on the premises in the 1970’s when the first FFT analyzers became available. One restriction in those days was the price of memory, and thus the way the data processing was implemented was to reduce data as soon as possible after acquisition. The result became the frequency block averaging that we use today. The process waits for a trigger event and, when this is fulfilled, acquires a block of samples into the buffer, which is then sent off to the FFT processor as soon as the buffer is full. Once the FFT process is completed, the data is sent to the averaging process where each frequency value of the latest FFT results is averaged into auto and cross spectra, the averaging performed over N data segments. Usually, for impact testing, there is an interrupt after the FFT process, so that, prior to including the new FFT results in the averaging process, the user can decide to include the new frequency results in the averaging process, or, if it was a bad impact – for instance a double hit - the user can decide to discard the data and make a new impact. Using the classical technique, the user has to make at least two choices: 1. select a trigger level, and 2. decide to include or discard data. When these choices has been made, traditionally the spectral densities are estimated by Welsh averaging
Gˆ XX (Z ) Gˆ XY (Z ) Gˆ YX (Z ) Gˆ YY (Z )
N
1 N
¦
1 N
¦ X k* (Z )Yk (Z )
1 N 1 N
X k* (Z )X k (Z )
(1)
k 1 N
k 1 N
¦ Yk* (Z ) X k (Z )
k 1 N
¦ Yk* (Z )Yk (Z ) k 1
the Frequency Response Function (FRF) estimates Hˆ 1 (Z ) and Hˆ 2 (Z ) are estimated from
Hˆ 1 (Z )
(2)
Gˆ XY (Z ) Gˆ (Z ) XX
Hˆ 2 (Z )
Gˆ YY (Z ) Gˆ (Z ) YX
and the coherence is found as
J 2 (Z )
Hˆ 1 (Z ) Hˆ (Z ) 2
Gˆ XY (Z )Gˆ YX (Z ) Gˆ (Z )Gˆ (Z ) XX
(3)
YY
Following this good and well tested practice, the user is often faced with some classical problems: 1. he might not have the best information extraction, 2. In case of bad coherence, there is no chance to re-analyze the data, only option is do the test once again (better hopefully), 3. You do not know what went wrong in the first
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1455 test, it is trial and error and learning by experience. One way to overcome some of these problems is to import all data into the computer, so that all data are available for sub sequent analysis, and then use a more flexible tool for the analysis. Such a flexible tool could be the Random Decrement (RD) technique.
2. THE RANDOM DECREMENT ALGORITHM Some initial considerations about using the RD technique for modal data processing have been given in Rasmussen and Brincker [1] and in Brincker and Rasmussen [2]. An introduction to the general RD technique is given in Rasmussen [3]. The Random decrement function is defined as the conditional expectation of a stochastic Process X (t )
D XX (W )
E>X (t k W ) T ( X (t k ))@
(4)
and from a time series the corresponding estimate is found as the conditional mean
Dˆ XX (W )
(5)
1 N ¦ x(tk W ) T ( x(tk )) Nk 1
where the triggering condition
T ( x(t k )) is given by for instance
T ( x(t k )) : x(t k )
(6)
a
T ( x(t k )) : x(t k ) ! a This defines an auto RD function where the averaging and the triggering is performed on the same time series. corresponding cross RD functions can be defined as
E>Y (t k W ) T ( X (t k ))@
DYX (W ) D XY (W )
E>X (t k W ) T (Y (t k ))@
Dˆ YX Dˆ XY
N
1 N
¦ y(t k W ) T ( x(t k ))
1 N
¦ x(t k W ) T ( y(t k ))
(7)
k 1 N
k 1
3. RANDOM DECREMENT ON IMPACT TESTING SIGNALS If we assume the classical linear input-output relation between X (t ) and Y (t ) t
Y (t )
³ h(t K ) X (K )dK
f
then it is easy to show that the following relations exist between the RD functions
(8)
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³ h(t K ) D XX (K )dK
DYX (W )
(9)
f
W
³ h(t K ) D XY (K )dK
DYY (W )
f
Now taking the Fourier transform of the RD functions defines the Fourier transform pairs
D XX (W ) l Z XX (Z )
(10)
D XY (W ) l Z XY (Z ) DYX (W ) l Z YX (Z ) DYY (W ) l Z YY (Z ) and the Fourier transform of Eq. (9) defines the corresponding FRF estimates
Zˆ XY (Z ) Zˆ XX (Z ) Zˆ (Z )
Hˆ 1 (Z ) Hˆ 2 (Z )
(11)
YY
ZˆYX (Z )
The analog to the classical coherence function is then finally defined as
c(Z )
Hˆ 1 (Z ) Hˆ (Z ) 2
Zˆ XY (Z ) Zˆ YX (Z ) Zˆ (Z ) Zˆ (Z ) XX
(12)
YY
4. COMPARING RD WITH TRADITIONAL PROCEDURE One of the problems comparing the quality of the RD estimate to the traditional estimate is that we like to consider the quality as defined by the coherence, and this function does not mean the same for the two cases. In both cases it is a ratio between two estimates for the FRF function, but for the traditional technique, the coherence is calculated as a statistics over the considered data segments, which ensures that the number calculated by Eq. (3) always will be between zero and one. However, for the RD estimate, this not the case. The RD analog given by Eq. (12) defines a number that can be both positive and larger than one - and smaller than zero. Thus in order to be able to compare with the traditional coherence, it is natural to make a transform of the RD coherence that for values of c(Z ) close to one, but smaller than one, is the identity transform, for values of
c(Z ) ! 1 , the transform is approximately 2 c(Z ) , and finally for values close to zero, and smaller than zero, the transform is always positive and close to zero. Such transformation can be simply defined as 2 J RD (Z )
exp( 1 c(Z ) )
(13)
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1457 Modal data 15
Force
10 5 0 -5
0
0.5
1
1.5
2
2.5
3
3.5
4 5
x 10
Response
200 100 0 -100 -200
0
0.5
1
1.5
2
2.5
3
3.5
4 5
x 10
Figure 1. Original modal test data.
Random Decrement functions 6
Dxx
4 2 0 -2
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
40
Dyx
20 0 -20 -40
ˆ (W ) and Dˆ (W ) Figure 2. RD functions D XX YX The results of comparing can be summarized as follows. If a good test is performed, i.e. for a case where the coherence is close to one using the traditional testing and estimation technique, no big difference is seen between the classical and the estimated FRF’s and the coherences. If test is performed, that has some drawbacks, some limited improvement can be made by using the RD technique, such case is considered in
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4
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0.8 0.6 0.4 0.2 0
Figure 4. Top plot: Estimated FRF’s, Hˆ 1 (Z ) (blue) and Hˆ 2 (Z ) (red) (the two function falling nearly on top of each other), bottom plot: Traditional coherence using a “best choice” estimation (magenta) and coherence by RD estimation (blue).
Figures 1-4. Figure 1 shows the original data with ten impacts, and a rather noisy force signal in between the impulses. Further the force signal shows several double hits. Figure 2 and 3 shows the RD functions, and finally Figure 4 shows the resulting FRF’s as estimated by Eq. (11) and the RD coherence estimated by Eq. (12) and Eq. (13) compared to traditional coherence. As it appears from the coherence plot, the RD
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1459 coherence is better than traditional coherence only in the low frequency region, whereas the RD coherence has some amplified drop-downs at some of the zeros of the FRF.
5. CONCLUSIONS The RD technique has been introduced and used on real modal data. An analog to the traditional coherence function has been introduced, however, in order to make sure that the coherence is defined only over the values from zero to one, a transformation to ensure this property has been proposed. The so defined procedure for estimation of FRF’s seem to work as well as the traditional technique for tests of good quality, whereas some improvements has been seen on data with double hits and noise.
6. REFERENCES [1] Rasmussen, J.C. and Brincker, R.: Estimation of Frequency Response Functions by Random Decrement. th Proc 14 International Modal analysis Conference, Dearborn, Michigan, USA, Feb. 12-15, 1996, pp. 246252. th [2] Brincker, R. and Rasmussen, J.C.: Random Decrement Based FRF Estimation. Proc 15 International Modal analysis Conference, Orlando, Florida, USA, Feb. 3-6, 1997, pp. 1571-1576. [3] Rasmussen, J.C. Modal Analysis Based on the Random Decrement Technique. Ph.D. Thesis, Aalborg University, Department of Building Technology and Structural Engineering, 1997.
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BookID 214574_ChapID 129_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Using Impulse Response Functions to Evaluate Baseball Bats David L. Formenti BlackHawk Technology, Inc. Santa Cruz, CA
David Ottman Hillerich & Bradsby Co. Loomis, CA
ABSTRACT In this paper, we demonstrate a new approach to evaluating the dynamic behavior of baseball bats. Using this approach we can compare the ball striking one spot on a bat versus another, and also compare the performance of one bat design versus another. We can quantify a ball striking the “sweet spot” on a bat versus the “sting” felt at the handle when the ball strikes the wrong spot. This new approach uses IRFs (Impulse Response Functions), which simulate the impact of a ball striking a bat. The IRFs are synthesized using an experimentally derived modal model of the bat. The modal data is obtained by a standard roving impact test of the bat. Two different quantitative measures are used for comparing IRFs. One measure is called the SCC (Shape Correlation Coefficient). It is a numerical measure of the co-linearity of two deflection shapes. It is the same as the FRAC (Frequency Response Assurance Criterion) calculation, but we apply it to the time domain IRFs as well as frequency domain FRFs. The second numerical measure is called the SPD (Shape Percent Difference). The SPD is a numerical measure of the difference between two deflection shapes. It not only indicates when two shapes are different, but quantifies the magnitude of their difference.
Mark H. Richardson Vibrant Technology, Inc Scotts Valley, CA
After a modal model was obtained for each bat, its synthesized FRFs were compared with the original FRF test data using SCC and SPD calculations. These calculations were done at each frequency sample to compare the experimental and synthesized FRFs. Additionally, both sets of FRFs were Inverse FFT’d and their corresponding IRFs also compared using SCC and SPD calculations at each time sample. These comparisons validated the accuracy of the modal models. Finally, the IRFs of the different bats were compared using SCC and SPD calculations. These results quantified not only the similarity or difference of the bat IRFs, but they also showed which bats had a higher level of vibration at the handle due to an impulsive force on the barrel. Deflection Shape A deflection shape is defined as the deflection of two or more points on a structure. Stated differently, a deflection shape is the deflection of one point relative to all others. Deflection is a vector quantity, meaning that each of its components has both location and direction associated with it. Deflection measured at a point in a specific direction is called a DOF (Degree of Freedom) [2].
The IRFs of several different baseball bats are compared using both the SCC and SPD calculations over all time samples. These measures show graphically how similar or different the impulse responses of different bats are. INTRODUCTION This research was conducted to develop new methods for comparing the performance of baseball bats. The approach taken involved the following steps; 1) Perform a roving impact test on each bat to obtain a calibrated set of FRFs. 2) Curve fit the FRFs to obtain experimental mode shapes. 3) Scale the mode shapes to obtain a modal model. 4) Synthesize acceleration, velocity, or displacement FRFs using the modal model. 5) Inverse FFT the FRFs to obtain a set of IRFs. 6) Compare the impulse responses of the bats at the handle due to an impulsive force on the barrel.
Figure 1. Baseball Bat Showing Test Points A deflection shape can be defined from any vibration data, either at a moment in time, or at a specific frequency. Different types of time domain data, e.g. random, impulsive, or sinusoidal, or different frequency domain functions [3], e.g. Linear spectra (FFTs), Auto & Cross spectra, FRFs, Transmissibility’s, or ODS FRFs can be used to define an ODS.
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_129, © The Society for Experimental Mechanics, Inc. 2011
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Data Acquisition Five different bats were tested using the roving impact hammer method. A (fixed) reference accelerometer was attached to each bat in approximately the same position on the barrel. Then each bat was impacted with an instrumented hammer at 1-inch intervals along the length of the bat from one end to the other. FRFs were then calculated from the impulse force and accelerometer response signals. Each FRF had 2100 uniform frequency samples, over a span from DC (0Hz) to 2998.6Hz. A typical FRF measurement is shown in Figure 2.
Figure 2. Typical FRF Measurement
Two of the bats were 31 inches long, and a total of 31 FRFs were measured on them. Three of the bats were 29 inches long, and a total of 29 FRFs were measured on them. Each set of FRFs was curve fit to obtain the experimental modes of the bat. The modal frequency & damping of the bats are shown in Figure 3. These results clearly show that the resonances of baseball bats can be quite different from one another. Modal frequencies range from a low of 75.6 Hz to a high of 2958 Hz. Likewise, modal damping decay coefficients range from a low of 0.47 Hz to a high of 39 Hz. Typical mode shapes from one of the bats are shown in Figure 4. Two different numerical methods were used to compare two sets of IRFs. (These same calculations can also be done on two sets of FRFs.) One set of IRFs is called the Baseline IRFs and the other is called the Comparison IRFs. Each set of IRFs contains a deflection shape at each sampled time value. To compare two sets of IRFs, their deflection shapes are compared at each time sample, using two different methods. Figure 3. Modal Frequencies & Damping
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One method is called the SCC (Shape Correlation Coefficient), and the other is the SPD (Shape Percent Difference). Both of these calculations yield a percentage value. The SCC measures the co-linearity of the two deflection shapes at each time sample. The SPD measures the percent difference between the deflection shape of the Baseline IRFs and the deflection shape of the Comparison IRFs at each time sample. SCC (Shape Correlation Coefficient) A deflection shape is in general, a complex vector with two or more components, each component having a magnitude & phase. In this application, each component of the deflection shape is obtained from an IRF at a specific time sample. The SCC measures the similarity between two complex vectors. When this coefficient is used to compare two mode shapes, it is called a MAC (Modal Assurance Criterion) [1]. The SCC is defined as;
SCC
DSC $ DS*B DSC DSB
where: DSB
DSC * B
DS
Baseline deflection shape Comparison deflection shape complex conjugate of DS B
indicates the magnitude squared $ indicates the DOT product between two vectors
The SCC is a normalized DOT product between two complex vectors. It has values between 0 and 1. A value of 1 indicates that the two deflection shapes are the same. As a “rule of thumb”, an SCC value greater than 0.90 indicates that two shapes are similar. A value less than 0.90 indicates that two shapes are different. The SCC provides a single numerical measure of the similarity of two deflection shapes. It measures whether or not two vectors are co-linear, or lie along the same line. If two deflection shapes are co-linear but have different magnitudes, the SCC will still have a value of 1. Therefore, a measure of the difference in the magnitudes between two deflection shapes is required. SPD (Shape Percent Difference) A direct measure of the difference between two deflection shapes is the SPD (Shape Percent Difference).
SPD
DSC DSB DSB
where: DSB
DSC
Baseline deflection shape Comparison deflection shape
indicates the magnitude of the vector Figure 4. Typical Bat Mode Shapes
If DSC DSB then the SPD is negative
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The SPD measures the percentage difference between the two shapes relative to the Baseline deflection shape. A value of 0 indicates no difference, and a value of 1 is a 100% difference between the two shapes. To summarize, if two deflection shapes are the same, their SCC will be at or near 1, and their SPD will be at or near 0. As the two shapes become different from one another, the SCC will decrease toward 0, and the SPD will increase or decrease depending on which shape, Comparison or the Baseline shape, has a greater magnitude.
At the first cursor position (275.71Hz) the SCC value is “0.99” and the SPD value is “0.11”. The additional cursor positions show the frequencies of the other 5 modes in the model. It is evident that the SCC is near 1.0, and the SPD is near 0.0 at all of these frequencies.
Mode Shape Interpolation In order to compare deflection shapes between all sets of IRFs, they all have to have a common set of DOFs. Two of the bats were tested at 31 points spaced 1 inch apart, and the other three bats were tested at 29 points spaced 1 inch apart. To obtain five sets of mode shapes with common DOFs, the mode shapes with 29 DOFs were interpolated so that they contained 31 evenly spaced DOFs. With each modal model having 31 DOFs, they could then be used to synthesize FRFs (and obtain IRFs) with the same number of DOFs. The mode shapes with 29 DOFs were interpolated into 31 DOFs by using geometric interpolation. Geometric interpolation uses a weighted summation of the mode shape components at 29 evenly spaced DOFs to calculate new mode shape components at 31 evenly spaced DOFs. A typical 29 DOF mode shape and its interpolated 31 DOF mode shape are shown in Figure 5.
Figure 6A. Two Synthesized & Experimental FRFs Overlaid
Figure 6B. SCC & SPD for Deflection Shapes from 31 FRFs Figure 5. Mode Shape Interpolated From 29 to 31 DOFs
Synthesized Vs. Experimental FRFs To compare the IRFs of the five bats, FRFs were first synthesized for each bat using its modal model (scaled mode shapes). Then the FRFs were Inverse FFT’d to obtain the IRFs. However, before calculating the IRFs, the synthesized FRFs were compared with the experimental FRFs, both visually and using SCC and SPD. A typical result is shown in Figure 6. Two FRFs are overlaid in Figure 6A. SCC and SPD values comparing all 31 FRFs are displayed in Figure 6B. Figure 6B shows that when the SCC is close to “1”, the SPD is also close to “0”, indicating that all 31 synthesized and experimental FRFs are closely matched.
Synthesized Vs. Experimental IRFs Figure 7 shows the IRF comparisons for the same bat as shown in Figure 6. (The IRFs are compared at each time sample, whereas the FRFs were compared at each frequency.) The IRFs match differently than the FRFs, but the result is the same. In the cursor band (0 to 0.1 sec) shown in Figure 7B, the maximum of the SCC is 0.99, the minimum is 0.19, and the mean is “0.90”. The maximum of the SPD is “-0.14” the minimum is 0.05 and the mean is “-0.0012” indicating that all 31 synthesized and experimental IRFs are closely matched. (For the SPD calculations, the deflection shape from the peak response of the IRFs was used to normalize the shape difference.)
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Comparison of Deflection Shapes Figure 9 shows the SCC and SPD of the deflection shapes of Bat#1 compared with all five Bats. When Bat#1 is compared with itself (on the left), its SCC values (upper graph) are 1, and its SPD values (lower graph) are 0.
Figure 7A. Three Synthesized & Experimental IRFs Overlaid
The IRFs of the other four Bats correlate well with Bat#1 near the beginning of the impulse responses. However, the shapes of the other Bats soon digress from the shapes of Bat#1, indicated by SCC values less than 0.5. Moreover, when the SCC values are near 1, the SPD values are also high (0.5 or 50%), indicating that the deflection shapes of Bat#1 are quite different from the deflection shapes of the other Bats. CONCLUSIONS A new method for comparing the impulse responses of baseball bats was introduced in this paper. It is based on comparing the deflection shapes between two sets of IRFs. IRFs contain the combined response all of modes that are excited, which depends on their mode shapes, frequencies, damping, and the impact and response DOFs. IRFs were synthesized for each bat using experimentally derived modal data. Using a modal model provides the flexibility of synthesizing IRFs between any pair of DOFs of the bat where the mode shapes are defined.
Figure 7B.SCC (top) & SPD (bottom) from 31 IRFs
The FRF and IRF comparisons both confirm that a modal model is sufficiently accurate so that synthesized IRFs can be used for comparing the impulse responses of the different bats. The advantage of using the modal model is that impulse responses can be calculated between any pair of DOFs (impact and response DOFs) where the mode shapes are defined. Hence, an impact force could be simulated at any DOF on the barrel, and the response simulated at any DOF on the handle. The experimental IRFs themselves could also be used for comparisons, but experimental data only contains a limited number of reference (impact) DOFs, usually only one. Comparison of IRFs Figure 8 shows synthesized IRFs of all five bats. Each IRF is the response at 25Z (the bat handle) due to an impulsive force applied at 9Z (on the barrel of the bat). The initial 0.2 seconds of each IRF are shown. Comparing the IRFs makes it clear that the bats respond quite differently. The vibration of Bat#1 (on the left) is completely damped out while Bat#5 (on the right) still has substantial vibration after 0.2 seconds. Furthermore, the IRFs make it clear that the peak responses (in g’s/lb) of the Bats are different. For example, the peak response of Bat#2 is much less than the response of Bat#4.
Two measures for comparing deflection shapes were introduced. The SCC (shape correlation coefficient) quantifies the co-linearity between two shapes, and the SPD (shape percent difference) measures the difference between two shapes. Both of these measures provided clear graphic evidence of the differences between the impulse responses of five different baseball bats. Two other innovations were used in this research. First, geometric interpolation was used to create mode shape components for three of the bats to match the same DOFs of the other two bats. Secondly, SCC and SPD were used to verify that the FRFs and IRFs synthesized from the modal models correlated well with the experimental data. This quantitative approach to comparing the dynamic behavior of structures should be useful in many other applications. Once a modal model is validated, the mode shapes themselves can be integrated or differentiated and then used to synthesize and compare the displacement, velocity, and acceleration responses of structures. REFERENCES 1. R.J. Allemang, D.L. Brown "A Correlation Coefficient for Modal Vector Analysis", Proceedings of the International Modal Analysis Conference 2. M.H. Richardson, “Is It a Mode Shape or an Operating Deflection Shape?” Sound and Vibration magazine, March, 1997. 3. B. Schwarz, M.H. Richardson, “Measurements Required for Displaying Operating Deflection Shapes” Proceedings of IMAC XXII, January 26, 2004.
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Figure 8. Synthesized IRFs (Response at 25Z due to impulsive force at 9Z)
Figure 9. SCC (top) & SPD (bottom) Bat #1 Compared To Five Bats
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
FFT Integration of Time Series using an Overlap-Add Technique
Rune Brincker, Professor Anders Brandt, Associate Professor Department of Industrial and Civil Engineering, University of Southern Denmark, Niels Bohrs Allé 1, DK-5230 Odense M, Denmark
NOMENCLATURE Time t
't T f
Sampling time
B x(t ) y (t ) X(f )
Frequency band width Signal
Xn
Fourier coefficient of signal
Data segment length Frequency
Integrated signal Fourier transform of signal
ABSTRACT Many times in vibration problems it is of importance to be able to integrate signals. Well known cases are Operational Deflection Shapes and earth quake problems where the displacements often need to be estimated from acceleration time series. When digital signals are integrated some classical problems arise; one of these is numerical noise introduced by the inaccurate integration algorithms resulting in large errors in the low frequency region leading to large DC drift. These problems are normally dealt with by using high pass filters that often introduce additional implementation problems like instability and amplitude/phase distorting problems. In this paper an FFT based procedure is introduced. The idea is to perform the integration in the frequency domain dividing the Fourier Transform by 2Sif , and then transforming back to time domain by IFFT. The technique is implemented using an overlap-add finite data segment approach, and the drift problem is solved by forcing the DC value of the frequency domain representation of the integrated signal to zero.
1. INTRODUCTION Classical problems in integration are: instrumentation errors, sampling errors and numerical integration errors, more information about these errors and classical ways to deal with them can be found in Mahdi [X]. In this paper however it is assumed that instrumentation errors are not present, i.e. we are dealing solely with sampling errors and numerical integration errors.
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_130, © The Society for Experimental Mechanics, Inc. 2011
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1468 In this paper the effect of integration is studied by considering a harmonic with the frequency f and the amplitude
X0 x(t )
X 0ei 2Sft
(1.1)
This does not limit our results as long as the considered more general signals can be considered as a linear combination of harmonics as expressed by a Fourier series for a finite data segment
x(t )
f
¦ X n e i 2Sfnt
(1.2)
n f
or as expressed by a Fourier integral for the case of infinite time f
x(t )
³ X ( f )e
i 2Sft
(1.3)
df
f
In both cases the simple harmonic is just a typical term in the Fourier expansion. The basic idea of this paper is to use the well known fact that appears from the three equations above that integration in time domain correspond to dividing by i 2Sf in the frequency domain. This is also known as the integration theorem in Fourier transform theory. The procedure is rather simple, the principle is illustrated in Figure 1. Data segments are taken from the original time series and taken to frequency domain by Discrete Fourier Transform (DFT). In order to minimize leakage and to ensure that overlapping data segments can be added to reconstruct the original time series, a window is multiplied onto the data segment before applying the DFT. In frequency domain, the data is divided by i 2Sf , and finally the so integrated frequency domain data is taken back to time domain by Inverse Discrete Fourier Transform (IDFT). In the following it is explained how the data segments are captured from the time series, and what conditions have to be fulfilled for the applied windows in order to reconstruct the original time series. Some examples of window classes are given. Further it is shown, that even though it looks like the integration theorem can only be theoretically justified in continuous time, it is argued that the integration theorem also holds for discrete time. Forcing the DC value of the integrated frequency domain data to be zero the drift error is removed in each data segment. Further since the integration theorem does not by itself introduce any error, also the well known numerical integration error is removed. Thus for periodic data the proposed algorithm is error free , only basic round-off errors are left and some minor errors due to the small leakage introduced by the applied window. For frequencies that fall in between the allowed frequencies for periodic data (falling between frequency lines
fn
error is introduced.
n'f ; 'f
1 / T where T is the length of the considered data segment), a leakage
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Figure 1. Illustration of the proposed technique for integration of time series, step 1: data segmentation to obtain original time domain data, step 2 to obtain the corresponding data in the frequency domain by DFT, step 3 to integrate the data in frequency domain by dividing by i 2Sf , step 4 to take the integrated data back to time domain by IDFT, step 5 to construct the integrated time series by adding the overlapping integrated data segments together.
2. TIME SERIES SEGMENTATION In this investigation, the time series is subdivided and processed in smaller data segments. It is well known, Brigham [X], that taking that DFT of smaller data segment of time length T instead of using the whole time series will introduced a leakage error. The reason is that the energy of the signal with a frequency content that does not fall exactly on the frequency lines of the data segment defined by the discrete frequencies f n n'f will “leak” to the adjacent frequency lines of the data segment. It is also well known, Brigham, that the leakage can be reduced by application of a suitable “window” that is multiplied onto the data segment before performing the DFT. Also in order to use overlapping data segments, a tapered (soft shouldered) window must be applied. Figure 3 illustrates the undisturbed data taken with a boxcar window, the soft shouldered window, and finally the result of multiplying the captured data with soft shouldered window in order to obtain a data segment with tapered data resulting in reduced average leakage. The data segment and the window has the time length T . It is clear that the applied overlapping windows must add to unity. Many different windows can be adjusted with proper overlap to fulfill this condition. However a simple form are windows with 50 % overlap. A broad class of such windows can be constructed in the following way. Let
g1(W ) be an arbitrary continuous function
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W /T
W /T
W /T Figure 2. Illustration of data segmentation, top: a data segment defined as a smaller part of a time series without changing the sample values of the end points of the data segment, middle: a soft shouldered window (Hanning), bottom: the soft shouldered window multiplied onto the original data segment in order to reduce average leakage and to make sure that overlapping windows add to unity.
W >0; T / 4@ only conditions are that g1 (0) 0 , g1(T / 4) 1 . Now define a new function g 2 (W ) by combining g1(W ) with the mirror around the point (T / 4,1) , thus g 2 (W ) g1 (W ) for W >0; T / 4@ and g 2 (W ) 2 g1 (T / 2 W ) for W >T / 4; T / 2@ . Finally define the window by w(W ) g 2 (W T / 2) / 2 for W > T / 2;0@ and w(W ) 1 g 2 (W ) / 2 for W >0; T / 2@. Two obvious classes to consider for the
defined over
segmentation and windowing process is the Hanning class
g1 (W )
(1 cos(2SW / T ))D
(2.1)
and the polynomial class
g1 (W )
(4W / T )D
(2.2)
where the exponent D can be any positive real number. Plots of some windows from the two classes are shown in Figure 3. In the following only the normal Hanning window is used, thus we are considering the Hanning class for
D 1.
3. INTEGRATION THEOREM In this section we will give the theoretical basis for using the integration theorem in discrete time. It will be argued that the integration theorem is valid also in discrete time.
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Figure 3. Some examples of windows from the two classes of windows defined in Eq. 2.7 and 2.8. Left: Some Hanning class windows. Right: Some polynomial class windows. Note that adjusting the parameter D changes the shape of the window, and that the two window classes are nearly identical.
Now let the two functions x(t ) , X ( f ) form a Fourier transform pair, Brigham [X]
x(t ) l X ( f )
(3.1)
then the well known Fourier differential and integral theorem follows directly from Eq. (1.1), (1.2) and (1.3)
dx(t ) l i 2Sf X ( f ) dt
(3.2)
And thus t
y (t )
³ x(W )dW l X ( f ) / i 2Sf
(3.3)
f
Of course we have to make some proper assumptions about X ( f ) for f o 0 for Eq (3.3) to be meaningful. Now, let us investigate what happens when discrete time is introduced by sampling the signal x (t ) with the sampling interval
't . As it is well known, in this case, the frequency band is limited by the Nyquist frequency
B 1 / 2't
(3.4)
but the spectrum is still continuous. However, since X ( f ) is assumed to be zero outside the band B Eq. (1.3) yields B
x (t )
i 2Sft ³ X ( f )e df
B
And the sampled values are then given by
(3.5)
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x(t k )
³ X ( f )e
i 2Sftk
df ; t k
(3.6)
k't
B
This means that the Fourier transform X ( f ) determines the original function x(t ) completely, since a function is determined if its Fourier transform is known. Therefore the original signal can be integrated as a continuous function, i.e. Eq. (3.3) still holds. For finite data segments, the spectrum becomes discrete, but again, the Fourier coefficients determine the original time function completely, and therefore, the integration theorem (3.3) still holds. For the integration theorem applied to finite data segments, a problems exist at DC (
f
0 ). The problem is
solved by forcing the Fourier coefficient at DC of the integrated signal to be zero. This corresponds to forcing the mean value of the integrated signal to be zero
(3.7)
T
1 y (t )dt T ³0
0
which is the only meaningful solution to the arbitrary constant that is introduced into the integration problem by starting integration at the arbitrary time
4.
t 0.
ERROR ANALYSIS
In this section the errors introduced by the algorithm will be studied. First let us consider the errors introduced by segmentation of the time series. Since any signal can be considered as a linear combination of windowed versions of the same time series, and since the integration is a linear operation, the procedure of integrating each data segment individually using the integration theorem does not by itself introduce any errors. The same is true for discrete time as it appears from the results of the preceding section. However, in practice we are not considering the data segments as we should as a windowed version of the original time series, we are only considering the non-zero part of the windowed data. We do this because we prefer to use the FFT algorithm, that assumes periodic data, and thus, because of this assumption, leakage is introduced. As indicated in the introduction, errors are most easily studied by estimating the effect on single harmonics. Thus studying a harmonic as given by Eq (1.1), the corresponding exact solution for the integrated signal is given by
y (t )
X 0 (i 2Sf ) 1 e i 2Sft
(4.1)
The error on the proposed integration scheme can be studied by estimating the difference between the exact integrated signal as given by Eq (4.1) and the integrated signal yˆ (t ) estimated by the proposed integration scheme. An example of the error on a single data segment is shown in Figure 4. Here the procedure is the simplest possible; a one shot DFT, integration in the frequency domain by division by
i 2Sf n , setting the DC
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215 data points, the
10.5 times the frequency resolution. Thus the
Figure 4. Illustation of the error on a single data segment. Top picture: Blue curve showing the exact integrated solution given by Eq (4.1), red curve showing the signal estimated by the proposed integration scheme (Hanning taperings on the outmost parts (left and right quarter) are used). Middle picture: difference between the two signals, the middle part where the window is unity is indicated by red. Bottom: Zoom on the signal axis to illustrate the error on the middle part of the signal.
frequency is relatively low (for maximum expected leakage effect on the integration), and the frequency is falling just in between two frequency lines (for maximum leakage effect). As it appears from the bottom picture of Figure 4.1, an error is in fact present on the middle part, but the error has no visible frequency content with the same frequency as the considered harmonic illustrating the absence of amplitude error on the integrated signal. The reason of the present errors is that the leakage spreads some energy to all the frequency pins of the frequency domain representation. Thus, the pins close to DC will be polluted by (a very small) leakage, and then, when we divide by
i 2Sf n , the small error components close to DC is heavily amplified, and a low
frequency error is introduced. Here only the DC pin is forced to zero. We can of course choose to force several pins around DC to zero, this will reduce the introduced error, but this corresponds to high pass filtering and is not the subject of this work. The similar case when using several windows are illustrated in Figure 5, for this case the window size is
4096 data points. The average error H is calculated as the root mean square (RMS) of the difference 'y (t )
yˆ (t ) y (t )
divided by the RMS of the exact integrated signal, for this case the average error is found to be or about 0.4 % of the exact signal amplitude.
(4.2)
H
0.0042 ,
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Figure 5. Illustation of the error when using several data segments. Top picture: Blue curve showing the exact integrated solution given by Eq (4.6), red curve showing the signal estimated by the proposed integration scheme (Hanning window). Middle picture: difference between the two signals, the middle part where the window is unity is indicated by red. Bottom: Zoom on the signal axis to illustrate the error on the middle part of the signal.
CONCLUSIONS And FFT based integration has been formulated that in principle is error free. The procedure is based on the integration theorem that exactly perform the integration in the frequency domain. However, leakage errors are introduced due to final data segment size.
REFERENCES [1] Teimouri Sichani, M., Brincker, R.: Investigating efficiency of time domain curve fitters versus filtering for rectification of displacement histories reconstructed from acceleration measurements. Proc. of ISMA2008, Int. Conference on Noise and Vibration Engineering, Sep. 15-17, Leuven, Belgium, 2008. [2] Brigham, E. Oran: Fast Fourier Transform and Its Applications, Prentice hall, 1988.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Requirements for a Long-term Viable, Archive Data Format
Allyn W. Phillips, Research Associate Professor Randall J. Allemang, Professor Department of Mechanical Engineering PO Box 210072, University of Cincinnati, Cincinnati, 45221-0072
Abstract Within the vibration technical community, there is a demand for a long-term viable, open definition file format for the archiving of data and results (not to be confused with a database management structure). For many years, the Universal File Format has been the defacto standard in this area. However, as technology has progressed, the aging nature of this FORTRAN card image based format has become problematic. In order to satisfy the increasing legal requirement of long term record keeping, a flexible archive, not dependent upon any particular hardware or operating system environment, is needed. With a discussion of some of the strengths and weaknesses of existing data formats, this paper focuses upon the identified feature set needed for realistic, longterm reliable recovery of information and successful community adoption. Introduction This paper is a presentation of a work-in-progress. For the purposes of discussion, the existing Universal File Format (UFF) has been taken as the initial starting reference point. Various other formats were reviewed and considered for content and applicability; however, in order to facilitate technical community adoption, the final resulting format has been specified to be open, extensible, and non-proprietary. In addition, the principle of ‘keeping it simple’ has been adhered to. The recognition is that if it gets too complicated, no one will use, support, or adopt it. For this reason, the final resulting format will probably not be perfect for everyone but it should sufficient for everyone's needs, in other words, a 95% solution. This decision is consistent with the consensus of opinion expressed at a meeting of users and vendors held at IMAC in 1998. The focus of that meeting was upon ideas for extending the UFF to address some of its basic deficiencies. In many respects, this project has benefited from and is somewhat of an outgrowth of that activity. What the New Format will NOT Be Before discussing the new format, it is important to avoid initial misconceptions by discussing briefly what the new format is not. The new format is focused upon long term archival of dynamic data; as such, things like the data storage media (hardware) and the vendor specific internal database structures are not being addressed. There is no intention or desire to force any particular hardware or internal database structure upon individual vendors or users. The only goal is to produce a long term, viable, cross platform, open architecture, dynamic data storage format.
DISCLAIMER This work of authorship and those incorporated herein were prepared by Contractor as accounts of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor Contractor, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, use made, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency or Contractor thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency or Contractor thereof.
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_131, © The Society for Experimental Mechanics, Inc. 2011
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1476 It is also important to recognize that the since the content focus is dynamic data, other data content types, such as CAD/CAE, video, pictures, etc., will not be specifically included in the format. It should be noted, however, that although such data will not be specifically identified and targeted for support, nothing in the definition will prevent referencing such information via the meta data records or including it within the data archive container. Background / Motivation / Objective The expected format is intended to fundamentally extend and/or replace the UFF; hence it might be thought of as roughly ‘UFF2ish’, a sort of second generation UFF. The advantage is that the UFF has been relatively stable and effective for over 30 years. While nobody particularly likes it, nonetheless, as a least common denominator, it has in the past basically gotten the job done. What is needed is to address the core UFF weaknesses that have developed over the years as technology has advanced. One of these key UFF weaknesses is in the area of meta data. The desired format must include a naturally extensible meta data capability by providing mechanisms for easy, natural extension as new needs develop, while providing backward compatibility, as much as practical. Another is the aging, eighty ASCII character, FORTRAN card image format. Again, another important point of clarity should be noted: the purpose of the format is primarily archival, not an active database. As a result, the focus of the definition is upon an archive (streamed) format, NOT upon any particular programming language implementation or representation. The long term goal is to encourage adoption by the dynamics community (both vendor and user) as an export and import format by having a set of libraries in both source and executable format on the University of Cincinnati Structural Dynamics Research Laboratory (UC-SDRL) web site for use by the community. The UC-SDRL web site will provide a clearing house for enhancements and bug fixes which can be submitted back to UC-SDRL for incorporation into the reference implementations. Currently, the UC-SDRL web site provides documentation for the existing UFF data structures. Finally, it is the intention that long term there will be a set of software test suites to facilitate compliance and validation checking of implementations. There is no intention to require the community (and vendors in particular) to use the reference implementations in order to achieve compliance. Anyone may develop an optimized version from the specification and validate against the compliance test suite. Historical Abuses of the UFF Over the years, because of misunderstanding of the format definition and because of uncertainty about handling various data, there has arisen several frequent and yet understandable abuses of the UFF which cause the files to be less portable than they might otherwise be and effectively non-transportable between different hardware and software systems or even unreadable and unrecoverable. x Storing critical, non-documentary information in textual ID lines x Exceeding 80 character line lengths x Inconsistent, order dependent units issues x Misunderstanding the format definition x Invalid field data values and formats (C vs. FORTRAN) x White space errors (spaces vs. tabs) x No clear procedure for format error handling Current Project Summary Overall, the project is focused upon the long term archival of dynamic measurement and associated meta data. Performance and size of the archival are not the primary objectives; data integrity and recoverability are the prime objectives. The project is limited to the detection of inadvertent data corruption and extraction of remaining valid data. The problems associated with malicious damage are specifically outside the scope. The issue of refreshing the data, as media storage technology changes, will be required but is also not the concern of this project. [1]
The current plan for the primary data container is to use the ECMA-376 'Open Container' . It is essentially a restricted format, industry standard ZIP file. The contents are envisioned primarily as sets of XML data streams. The strength of this container is that it can hold structurally organized data and retain the structure. It can also contain and store non-format defined (vendor specific) data.
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1477 One of the challenges to this project has been that there are very few data formats that are open and usable without restriction. In the course of the project, three primary candidates were identified for consideration as the [2] [3] format basis: HDF , ASAM-ODS ATF-XML , and a custom developed XML format. Unfortunately, each has a significant weakness. x The weakness of the HDF format is that it is essentially a binary file system embedded in a file. Long term damaged data recovery will be problematic. x The weakness of the ASAM-ODS is that the format prefers external binary files for large data. Using direct file references for these parts makes long term data integrity problematic, potentially placing the entire dataset at risk of complete loss. x The weakness of a straight XML implementation is that the XML standard requires that any conforming parser stop processing upon encountering any error. Because the only historically successful, long term archival format is the traditional book, there is a focus upon ASCII/textual data type formats. Format Development Activity The process of reviewing the three primary archival format candidates noted above proceeded, in part, by reviewing existing available data format options with a specific focus upon applicable features for incorporation into the resultant archival format. Since most data formats are targeted at either data transport or active database manipulation, some of their design decisions are at odds with the long term archival goal of the project; nonetheless, many of their specific data features are still relevant. Some of the positive and negative aspects of these different formats were considered in light of these specific features and how long term implementation might be affected. Consideration was also given to the feature characteristics needed for long term read/recovery viability and industry acceptance. In particular, these two elements favor a format that is principally textual (ASCII) encoded data, which is nominally familiar, is simple to implement and can be mapped relatively straightforward to existing proprietary databases. Of the formats previously reviewed, the HDF format, while providing the potential mapping structure, is essentially a high performance file system embedded in a file. Besides being relatively complicated, corruption of the data container appears to make recovery of the data difficult. The use of a straight XML format has the advantage of being basically verbose textual (ASCII) information, but unfortunately, the XML standard requires that conforming parsers must halt at any parsing error. Additionally, extraction of any data information requires effectively reading the entire file. Although proprietary data formats exist which are fundamentally ASCII/binary data interleaved, such formats cannot be considered because of their proprietary nature. However, one format specification standard, developed in the area of textual document interchange, appears to have significant application to this project. It is the 'Office Open XML Format, ECMA-376, Second Edition, Dec. 2008.' The textual document attributes of headers, footers, cross references, body text, etc. share many conceptual features common to the archiving of dynamic data, that is, data headers, meta data, cross channel references, etc. Hence, the packaging of such data can conceptually be considered a type of dynamic document. The Office Open XML Format and in particular the portion referred to as 'Open Packaging Conventions', includes many of the characteristics of the desired archive data definition. While the specification was primarily developed to support textual documents, the actual specification is general and not specific to such documents. Effectively the definition is a random access container holding primarily textual data. By being based upon familiar industry standards (some being de facto definitions), the format has the potential for easier industry acceptance. Since the data recovery features of the potential format are not focused upon deliberate malicious data manipulation, but upon inadvertent corruption, depending upon the type and degree of corruption, through the use of appropriately tagged prefix meta data (effectively providing redundant container information), the valid uncorrupted data could still be extracted from a damaged archive. Thus, potentially all or most of an archive could be reconstructed in the event of container information corruption.
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1478 Archive Feature Suggestions During the various formal and informal discussions with users and vendors that occurred during this project, many suggestions for desirable features were offered which, while not immediately applicable to the initial project effort, were worth noting for consideration during future work. Many of the suggestions do not affect the principal data per se, but rather focus on the retention of "historical" meta information and the like. Examples of these suggestions and concerns are: x It should be possible to write "noisy" or verbose output (ie. redundant info) with "equivalence" constraint testing capability. (eg. writing multiple measurement vectors from measurement matrix and checking measurement characteristics or constraints. [fmin, deltaf, testid, block length, etc.]) x When preserving data it should be possible to write a "noisy" or verbose output with some form of "backtrace" to the original database fields. (eg. perhaps writing <meas vendorSource="floogle[1]">... data ... where "floogle[1]" may be the original vendor data ID.) x It might be advantageous to reserve all 'vendorXXX' attribute fields for vendor use. x It might also be advantageous to reserve all 'userXXX' attribute fields for end-user use. x In developing the XML data specification, attributes should not provide any data information, but only meta information about the data. x It should be possible to tag or log any hardware or software that has "touched" the data. (ie. retain the data history path.) x It should be possible to document vendor specific or proprietary information within the container using human readable ASCII/XML - *NOT* PDF/DOC/etc. x The 'Open Container' should allow inclusion of other non-format defined information types. (eg. images, sounds, etc.) x The format should have clearly defined behavior as well as content. (ie. specified error handling in the presence of malformed data.) x Inline data should be written in decimal: floating point or bytes. Complex data should be specified as successive pairs of real values. Although additional feedback is expected as the project continues to progress, these types of comments favor the development of an 'XMLized' UFF-like format definition. Additionally, many of these suggestions are inherently supported by the current concept through the synergy of the ECMA 'Open Container' coupled with a predominantly XML data definition. Conclusions The design decision of utilizing the ECMA 'Open Container' with a predominantly XML data definition is relatively firm. The evaluation of the relative strengths of basing the XML data definition on the ASAM-ODS ATF-XML definition or, currently more likely, of developing an XML format that encompasses the familiar UFF characteristics (but without the data stream sequence dependency) remains. The final decision will be made as the project progresses subject to additional technical feedback; the ultimate goal being to have a relatively final format definition and working reference code completed over the next couple of years. References http://www.ecma-international.org [2] http://www.hdfgroup.org [3] http://www.asam.net [1]
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1479 Appendix The following example is presented for conceptual discussion purposes, giving only an impression of the style of data storage. It is not intended to be complete or to represent any particular likely final implementation. container header [binary information]
container data record header [binary information]
container data record content <measurement type="FRF" storage="3D" blocklen="512"> badw8...[encoded binary data]...se64Aa5
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Modal Testing of Complex Hardened Structures Janet C. Wolfson*, Jason R. Foley, and Lashaun M. Watkins *
Air Force Research Laboratory AFRL/RWMF; 306 W. Eglin Blvd., Bldg. 432; Eglin AFB, FL 32542-5430, [email protected]
Alain L. Beliveau Applied Research Associates, Inc.
Preston C. Gillespie Jacobs Engineering
ABSTRACT A new testing method is being developed by the Air Force Research Lab to excite a desired multi-dimensional response in a structure using tuned resonances. The structure consists of a Aluminum plate and a perpendicular shelf. The test article is excited through the use of either an impact hammer or pyrotechnics (e.g., a pyrotechnic plunger) which in turn inputs a specific frequency profile. The dynamic response of the plate and shelf spans the entire spectrum from low (10 Hz) to high (10 kHz) frequency as well different peak amplitudes (i.e., accelerations). They are captured by a variety of instrumentation methods including modal accelerometers, laser vibrometers, and a digital image correlation system. The location of the shelf as well as its material and stiffness properties is modified to reproduce the design objective (frequency response functions with the desired amplitude and phase spectrum). These modifications are achieved through interpretation of modal data. INTRODUCTION The Air Force Research Lab (AFRL) conducts research in a wide variety of energy regimes. This research is designed to evaluate aspects of a test article over a variety of scales from components to systems and sub-scale to full-scale. One area that AFRL is specifically interested in is multi-axially exciting a system over the entire frequency spectrum from low (10 Hz) to high (10 kHz) as well as different amplitudes (i.e. – accelerations). The ranges desired will be demonstrated through the use of a Shock Response Spectrum (SRS). The desire is to develop a scientific, repeatable, field experiment that can reproduce the desired forces in order to determine the failure mechanisms in the systems under test. The test fixture, currently under design, consists of an aluminum plate with a shelf on the back where the item under test will be placed. The front of the plate will be excited through the use of pyrotechnics (e.g. a pyrotechnic plunger or detonation cord) which in turn will input a specific force and frequency profile. This paper will provide a brief overview of similar test fixtures used in industry, discuss the data from one pyrotechnic test, and compare that data with three LS-DYNA simulations. It will conclude with a discussion of how AFRL plans to develop this test apparatus. BACKGROUND In 2001 AFRL contracted with Wyle Laboratories to perform a pyroshock test on a system component. The results from that test were intriguing as they were able to impart low frequency energy over a significant duration on all three axes. This was the impetuous to develop our own unique test apparatus, where we would expect to impart the low energy into our test article, but also evaluate the ability to excite the high frequencies as well. In 2006 Ensign-Bickford Aerospace & Defense Company (EBA&D) presented a similar type of test article [1]. AFRL is currently trying to compare the data gathered from these tests using computer simulations in their effort to develop the Multi-Axial Pyrotechnic Plate (MAPP) test apparatus. The current design of the test fixture consists of an 8’ x 4’ aluminum plate approximately 1” thick and supported in 2 locations along the top edge of the plate; which simulates free-free end conditions. A support shelf, or “bookshelf” is located on the back of the plate and is where the item under test is located. It is assumed that the shelf and its supports are welded to the plate.
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_132, © The Society for Experimental Mechanics, Inc. 2011
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APPROACH Pyroshock is the desired method to excite the MAPP test apparatus because it differs from other types of mechanical shock. Compared to mechanical shock, there is very little rigid-body motion of a structure in response to pyroshock. The acceleration time-history of a pyroshock, measured on the structure, is oscillatory and approximates a combination of decayed sinusoidal accelerations with very short duration as shown in Figure 1 [2]. When a test article is very close or in contact with the explosive, it is considered near-field. In these cases the Pyroshock acceleration time-history consists of a high-frequency, high-amplitude shock that may have transients of microseconds or less. This near-field energy is distributed over a wide range of frequencies and is typically not dominated by a few selected frequencies. The energy deposition time for a pyrotechnic event is very small and does not strongly excite the rigid body modes of the structure. The resulting stress waves, from the explosives, propagate through the test article and high-frequency energy is gradually attenuated due to various material and structural damping mechanisms. That high-frequency energy is then transferred or coupled into the lower frequency modes of the structure. It is through these modes that AFRL hopes to tune the structure and the “bookshelf” to excite specific modes. 2
x 10
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Figure 1 Pyroshock Time History TEST SET-UP AFRL is developing a test apparatus called the Multi-Axis Pyrotechnic Plate (MAPP) that consists of an aluminum plate with a “bookshelf” type support on the back. The current design utilizes a 8’ x 4’ x 1” plate that is supported at two locations along the top of the plate to simulate a free-free-free-free boundary condition. The bookshelf on the back is attached to the plate for 12” along one edge and protrudes out another 12”. The shelf itself is 1” thick and made out of aluminum. Stiffening supports are placed along both edges of the plate and are also constructed of 1” thick aluminum. The “bookshelf” will be welded to the Aluminum plate at the desired location, which is currently under investigation. The location, stiffness, and material properties of the shelf will determine what type of multi-axial accelerations the item under test will be exposed to. Schematic drawings of the MAPP set-up and the test bunker are shown in Figure 2 (a) and (b), respectively. In order to determine the properties and location of the shelf on the MAPP system the desired forces and frequencies applied to the system under test needs to be determined. In the complex environment that AFRL is interested in there is a methodology of developing test requirements using a Shock Response Spectrum (SRS). The shock response spectrum, or SRS, has been proposed as a tool for evaluating the damage potential in a given acceleration time history. The SRS is defined using an array of 1-D spring-mass systems, each with a spring constant tuned to a different resonant frequency (Z = ¥k/m). The maximum acceleration by an oscillator
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Test Item Shelf
Plate a) MAPP test apparatus
b) MAPP test bunker set-up
Figure 2: Schematic drawing of the MAPP test article and arena when coupled to a rigid base moving with the specified acceleration time history defines the “positive” or “negative” SRS depending on the direction of the shock. Further details on the SRS can be found in comprehensive reviews, e.g., Irvine [3]; other spectral analyses can be found in Scavuzzo and Pusey [4]. The SRS is calculated in this paper using the improved filter bank method developed by Smallwood [5, 6]. The positive and negative maximum SRS gives the maximum acceleration of the 1-D spring mass in the respective directions due to the acceleration time history. An example SRS of impact test data is shown below in Figure 3(a). The SRS provides a measure of the effect of the pyroshock on a simple mechanical model with a single degree of freedom. Generally, a measured acceleration time-history is applied to the model and the maximum acceleration response is calculated. An ensemble of maximum absolute-value accelerations responses is calculated for various natural frequencies of the model. Since near-field pyroshock usually has broad-band frequency content its SRS exhibits a more complex shape. [2] Figure 3 compares SRS’s determined from two different types of impacts. The one on the left [Figure 3(a)] is from a pyroshock event and [Figure 3(b)] is a combination of SRS’s from a variety of traditional impact tests. The differences in the shape of the SRS is apparent. In the pyroshock data the slope of the SRS increases, then plateaus, after which it continues to increase. The SRS of the impact data follows a more traditional shape and has an increasing slope until it levels off in the high frequency range.
a) Pyroshock SRS Example
b) Impact SRS Example
Figure 3: Shock Response Spectra The desired testing requirements for the MAPP system were determined by evaluating the results from a variety of impact tests. The acceleration profiles of the different axes (vertical, horizontal and axial) were evaluated
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separately and SRS’s were created. For the development of the testing requirements a desired envelope was established as shown by the black lines in Figure 3(b) and Figure 4. In order to further demarcate our testing goals energy requirements at specific frequencies were identified. They are shown highlighted in pink in the following figures. Figure 3(b) shows the vertical requirement, Figure 4(a) depicts the horizontal requirements, and Figure 4(b) shows the axial requirements. These will be used to evaluate the effectiveness of the MAPP system and fine tune the inputs. Specifically the input energy (pyroshock) and the location and stiffness of the shelf located on the back of the plate.
a) Horizontal Requirement b) Axial Requirement Figure 4: Desired Shock Response Spectra PYROSHOCK DATA Data was gathered from one pyroshock event on a test set-up similar to the MAPP that is currently under design. A Shock Response Spectrum (SRS) of the data was created and shown in Figure 5(a). The SRS shows a typical pyroshock response as it is rising at approximately 10dB/decade. [2]. The plot depicts both the positive and negative values for the SRS. The autopower spectra of the data were calculated and one of the autopowerss will be shown, and described, below.
a)
SRS from Pyroshock Data b) SRS from Computational Analysis Figure 5: Shock Response Spectra
COMPUTATIONAL SIMULATIONS
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Computational analyses were performed on the Multi-Axis Pyrotechnic Plate (MAPP) test set-up. The purpose of these calculations is to evaluate the effect of the “bookshelf” location. Three locations were evaluated. The first configuration located the shelf in the upper right hand corner of the plate [Figure 6(a)]. The second configuration located the shelf in the middle of the plate on the right hand side [Figure 6(b)]. The final configuration placed the “bookshelf” in the center of the plate [Figure 6(c)]. These configurations were meshed in Truegrid [7] using solid brick elements. The analysis was performed in LSDYNA [8] where the plate and shelf were modeled as a single body using a Plastic-Kinematic material model. An explicit calculation was performed on each of the configurations. A triaxial acceleration trace was recorded at the center node of the shelf for all analyses. A triangular impulse was applied to the face of the plate opposite of the shelf. A peak pressure of 1,000 psi was applied at 0.1 milliseconds; by 0.2 milliseconds; the pressure had returned to zero. The results from these analyses are discussed below. A modal analysis of the three configurations was performed using SolidWorks [9]. That analysis showed that the location of the “bookshelf” had a very small impact on the modal behavior of the plate. The mode frequencies were typically within 5% of each other. The first five mode range from 21.51 Hz to 94.917 Hz.
a) Plate Layout Configuration 1
The SRS’s were calculated for each of the test cases. Figure 3(b) shows the SRS from Test 2. The other tests had very similar responses. The autopowers were also calculated for the data, some of which will be discussed below. ANALYSIS AND DISCUSSION An analysis of the accelerometer traces were performed for the three different configurations shown to the right (Figure 6). Linear spectra, autopower spectra, and SRS’s were calculated from these traces and compared to the experimental data. A comparison of the SRS’s of the experimental and computational data is shown in Figure 5(b). They both exhibit similar shapes; however, the SRS from the experimental data is slightly more complex. This is to be expected from near-field pyroshock events. While the computational SRS does not show that same complex shape it has similar overall characteristics which shows that the plate is seeing an impulsive load and that that computational method being employed is a good guide for fine-tuning the design of the “bookshelf” and plate.
b) Plate Layout Configuration 2
Linear spectra were calculated and autopwer spectra were computed from the computational and analytical c) Plate Layout Configuration 3 data as well. Acceleration traces were analytically determined in all three principal axis: the axial Figure 6: Plate Configurations for Computational direction (up and down on the shelf), the out-of-plane Analysis forces (perpendicular to the plate), and the in-plane forces (parallel to the plate). For the computational data, the linear spectra and autopower spectra did not change
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much from one configuration to another for the axial and in-plane directions. The out-of plane forces of configuration 3 were dramatically different when compared to configurations 1 and 2 as shown in Figure 7. By placing the shelf in the center of the plate it removed the effects of the modal behavior of the bending of the plate, thereby limiting the power that the test article would be subjected to. By designing the MAPP system to have a “bookshelf” on the edges of the plate the out-of-plane forces applied to the test article are dramatically increased.
Figure 7: Computed Power Spectral Density of Out-of-Plane forces for 3 Plate Configurations One of the desired outcomes of the computational analysis was to compare the effectiveness of the computer simulations to data gathered from a pyroshock experiment. This comparison was performed by evaluation the PSD’s of the three configurations and the experimental data in the axial direction. The computational analyses had similar PSD’s for all three configurations as depicted in Figure 8. The PSD of the experimental data in the axial direction is also shown on that plot. That figure shows that the computational analysis is over-predicting the acceleration induced by the experimental pyroshock event 20kHz. At that frequency the experimental data has a severe increase in acceleration while the analytical data continues to decrease. This jump in frequency could be attributed to weaknesses in the data acquisition system or in sensor resonance. Since we do not have any additional data on either of those aspects no definite conclusion can be made on the effectiveness of the calculation at those high frequencies. However, we have discovered that the computational capabilities are limited to less than 10kHz, after which the results become un-reliable [10] FUTURE WORK The design of the MAPP system will be modified on the initial computational analysis. Evaluations of the plate size, thickness, material, and shelf location will be determined though a sensitivity study of the system. Various inputs will be applied and the desired acceleration outputs will be determined. The actual test set-up will then be built and tested within the next year. The test set-up will be improved upon as additional testing requirements are determined.
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Figure 8: Power Spectral Density of Axial Forces for Computational Analysis and Experimental Results SUMMARY The Air Force Research Lab is currently designing a new test apparatus to impart a specific shock level (amplitude and frequency) that has been exhibited under impact tests. Computational analysis were performed to compare the results with a near-field pyroshock test. A possible system design was evaluated with three different configurations. The Power Spectral Density in the axial direction was shown to over-estimate the forces from the pyroshock event. Further analytical studies will be performed to reach desired levels specified through the use of a Shock Response Spectrum. The Multi-Axis Pyrotechnic Plate will be built in the near future and tests using live explosives will be performed. ACKNOWLEDGEMENTS J. W. would like to acknowledge research funding from Mr. Danny Hayles and the Defense Threat Reduction Agency. REFERENCES 1. Keon, S.P., Pyrotechnic Shock Testing: Real Test Lab Experiences at EBA&D. 2006: Spacecraft and Launch Vehicle Dynamic Environments Workshop. 2. Harris, C.M. and A.G. Piersol, eds. Harris' Shock and Vibration Handbook. Fifth ed. 2002, McGraw-Hill. 3. Irvine, T., An Introduction to the Shock Response Spectrum. 2002. 4. Scavuzzo, R.J. and H.C. Pusey, Principles and Technizues of Shock Data Anlaysis. 2nd Edition ed. 1995, Arlington, VA: SAVIAC. 5. Smallwood, D.O., The Shock Spectrum at Low Frequencies. Shock and Vibration Bulletin, 1986. 56(No. 1, Appendix A): p. 9. 6. Smallwood, D.O. Improved recursive formula for calculating shock response spectra. in Proceedings of 51st Symposium on Shock and Vibration. 1981. San Diego, CA: SAVIAC. 7. TrueGrid Users Manual. 2009: XYZ Scientific Applications. 8. LS-DYNA Users Manual. 2009: LSTC. 9. Solid Works User Manual. 2009: Dassault Systems SolidWorks Corp. 10. Foley, J.R., et al. Wideband Characterization of the Shock and Vibration Response of Impact-Loaded Structures. in SEM IMAC XXVII. 2009. Orlando, FL: SEM.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Dynamic Force Characterization for an Industrial Process Using Response Measurements Brad Pridham1, Bujar Morava2,Bruno Purnode3, Steve Mighton4 1
2
Specialist, Project Director Rowan Williams Davies and Irwin (RWDI) Inc. 650 Woodlawn Road West Guelph, Ontario N1K 1B8 Email: [email protected] [email protected] 3,4
Owens Corning Science & Technology Center Granville, Ohio 43023 Email: [email protected] [email protected]
ABSTRACT Characterization of dynamic forces generated by industrial processes is an important part of the design of the process support structure and foundation. Forces generated by reciprocating industrial machinery are often characterized using sinusoidal functions; however, more complex processes, such as those involving fluidstructure interactions, may not be easily approximated by simplified periodic functions. When possible, operational measurements of the acceleration and displacement response of the supporting structure can be used in conjunction with dynamic Finite Element Modeling (FEM) to estimate force spectra associated with the process. In this paper a case study on dynamic force characterization for an industrial process is presented. The study included static and dynamic response measurements of an existing process installation. A FEM of the system was developed and used in conjunction with the response measurements to estimate the force spectra for the process. The force spectra were then used to generate random time histories of the forces for use in the design of the supporting structure and foundation for a future, similar process installation. 1
INTRODUCTION
Most manufacturing and materials processing plants contain equipment and processes that generate vibration. These vibrations are a concern because they can cause fatigue of connections and components, malfunction of the equipment, human discomfort, and in extreme cases, failure of connections or the support structure. Most of these problems can be prevented by including the dynamic loads generated by the system in the design of the supporting structure. This can be a challenge when there are no dynamic loads available for use by the structural designer. Many machines and processes generate periodic forces that can be characterized using sinusoidal functions [1]. However, when the dynamic loads are a random process, their characterization can be derived from stochastic models or operational tests of the system. A numerical model of the loading process can be difficult (or impossible) to develop, costly to the design team, and may possess a high degree of uncertainty. A more accurate characterization of dynamic forces can be achieved using dynamic test data from a prototype installation. This requires careful planning of the dynamic tests to ensure that the right information is obtained for
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_133, © The Society for Experimental Mechanics, Inc. 2011
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1490 force characterization, and that several approaches to the characterization are used for verification/validation of the results. This paper presents the methodology and results from a force characterization study conducted for an industrial process. The objective of the study was to provide estimates of the power spectral densities of random forces generated by the process for use in the design of its supporting structural system. Results from dynamic tests of a prototype installation were used to develop and validate a Finite Element Model (FEM) of the structural support. The dynamic loads were then estimated by solving the inverse problem using the operational response data and the modal properties from the FEM. 2
DESCRIPTION OF THE SYSTEM
The system examined in this study is the tank assembly shown in Figure 1.
Figure 1: Tank assembly examined in this study. The assembly consists of a rigid tank (shown in blue) supported on steel framework. Exhaust ductwork extends vertically from the tank and is supported on steel framework connected to the main building structure (photo top right). The top of the tank is also tied to a steel crossbeam via angle sections (photo bottom right). The base framework is bolted to the slab-on-grade floor and the tank is bolted to a steel plate that is welded to the steel framing. The tank is very stiff and for the dynamic force characterization was considered to behave as a rigid body. A network of piping is connected to the tank via flexible couplings. The contribution of these connections to the stiffness of the system was considered negligible. During operation of the system a dense fluid is boiled inside of the tank and hydrodynamic pressure fluctuations occur on the tank wall. These fluctuating forces are transmitted to the supporting framework, and during specific points in the process significant lateral shaking of the entire system is observed. The design team expressed
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1491 concerns regarding the dynamic displacements of the system and was interested in obtaining estimates of the dynamic forces from the process for use in the structural design of a new support structure. A dynamic testing program was proposed to investigate the behavior of the system during operation. Results from the tests were then be used to estimate the dynamic forces generated by the process. 3
DYNAMIC TESTING
3.1 Test Program Dynamic testing consisted of pull tests (step relaxation) and operational tests. Descriptions of the tests are as follows: •
Pull Tests (step relaxation): During these tests a measured pulling force was applied at the base of the tank and the force-displacement characteristics of the system were recorded. Once a pulling force of 1000 – 1500 lbs was reached, the load was released for measurement of the free vibration response. The purpose of these tests was to obtain estimates of the lateral stiffness of the system and frequency characteristics of the empty melter during free vibration decay. Photos of the setup for these tests are shown in Figure 2.
Figure 2: Photos of loading assembly for pull tests. •
Operational Tests: During these tests, motion of the tank was measured under normal operation of the system. The purpose of these tests was to measure typical vibration amplitudes (accelerations and displacements) associated with normal operation as well as the frequency characteristics of the motion.
Figure 3 shows the sensor locations during the tests. Motions were measured in the horizontal X-Y plane. The displacements of the system were tracked by lasers mounted to the floor, approximately 6 inches above the base of the tank at the center line of each side. Accelerations were measured at these same locations in both the X and Y directions. These locations were selected to correspond approximately with the center of mass of the molten fluid. A total of 25 data sets of varying length (depending on the test) were obtained during the operation of the system and subsequently analyzed to determine peak response levels of the system, and the energy content of the system response. 3.2 Summary of Measurement Results Figure 4 shows the measured force time history and displacements from one of the pull tests. During each of the pull tests, a gradually increasing pulling force was applied at the base of the melter (on the top of the steel supporting frame), until a maximum force of approximately 1500 lbs was reached. The load and deflection
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Figure 3: Measurement set up during the dynamic testing (X axis assigned to the long axis of the tank).
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Figure 4: Load and deflection measurements from one of the pull tests. data from the pull tests were used to estimate the lateral effective stiffness of the system (i.e., the applied pulling force was decomposed into its X and Y components, and then divided by the measured displacements in the X and Y directions to arrive at approximate values for the lateral effective stiffness). A series of operational tests were performed with varying fluid depths and process configurations. Due to the randomness of the excitation, the operators were uncertain which operating conditions caused the maximum dynamic response. Therefore, a variety of tests had to be performed within a limited time span until the team was satisfied that worst-case responses had been observed. The peak lateral deflections of the system measured during the operational tests were 0.0554 in (1.41 mm) in the Y direction and 0.0284 in (0.72 mm) in the X direction.
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1493 The computed effective stiffness values were applied to the measured maximum deflections to derive equivalent static forces causing the peak dynamic deflections. Using this approach, estimates of the equivalent static forces were computed as follows: equivalent x
max δxoperationa l = ⋅ F cos θ max δx pull −test
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Where, θ was the angle between the applied load and the X axis in the X-Y plane (see Figure 3) and F was the measured maximum applied force (1500 lbs). These preliminary estimates of the peak lateral loads were provided to the team to be used as initial values for design. The primary motivation for the pull test measurements was to estimate equivalent static forces that could be cross-checked against the results from the dynamic force characterization procedure. The spectral data from operating tests indicate that the operational acceleration response of the system is dominated by 4 modes of vibration having frequencies in the range of 3 – 20 Hz. The majority of the energy in the response was associated with the highest mode. The modal damping ratios of the first four modes were obtained from the free decay responses of the system using the Eigensystem Realization Algorithm. These estimates are summarized below. 4
NUMERICAL MODELING
A linear dynamic model of the melter tank and support structure was developed for characterization of the dynamic forces generated by the melting process using mass data, structural drawings and information on the as-built condition of the structural connections provided by the design team. . The model was then compared with the measurement data and subsequently adjusted until the dynamics of the system correlated well with the observed dynamic behavior. The results from spectral analysis of all of the measurement data were used to appropriately adjust the initially assumed model boundary conditions and mass distribution until a reasonable agreement between the dynamics of the model and the experimental observations was obtained. The amount of energy dissipation (vibration damping) present in the system was estimated using system identification techniques applied to the measured free decay responses. Multiple identification sets were analyzed, resulting in several damping estimates for the first four modes of the system. The mean value of the estimates for each mode was used for specification of modal damping values in the numerical model. The measured operational frequencies, identified free decay damping ratios, and finite element model frequencies are listed in Table 1. The FEM mode shapes are shown in Figure 5.
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8%
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Table 1: Measured and modeled vibration frequencies and identified modal damping ratios.
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Mode 1
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Mode 4 Figure 5: First four mode shapes of the FEM.
As stated previously, the dominant mode in the response was at a frequency of 17 Hz. This corresponds to the second order sway mode of the system in the Y direction. The large contribution from this mode in the response is attributed to the fact that the peak mode deflection occurs near the center of mass of the dense fluid, at the base of the tank. Applied dynamic forces from the molten fluid are exciting this mode at this location, increasing its contribution to the total dynamic response. It should be noted that the descriptions of system motion provided in Table 1 were confirmed by additional measurements along the height of the tank during operation. Details of these measurements are not presented in this paper. To confirm that a reasonable estimate of the lateral stiffness of the system was obtained from the numerical model, static loads equivalent to the mean X and mean Y forces measured from one of the pull test data were applied to the model. The computed displacements were compared to the measured mean displacements recorded by the lasers. The results from the comparison are listed in Table 2.
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Mean Measured Displacement
Computed Displacement from FEM
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1340 lb
0.0105 in
0.0122 in
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477 lb
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0.0074 in
Table 2: Analytically predicted and measured static displacement from a pull test.
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1495 The computed displacements show satisfactory agreement with the measured mean deflections. This result confirmed that the lateral stiffness of the model reasonably represented the lateral stiffness of the prototype installation. 5
DYNAMIC FORCE CHARACTERIZATION
5.1 Methodology Following validation of the model, a procedure was implemented to characterize the dynamic forces associated with the principal X and Y axes of motion. Two approaches were used: one based on the measured acceleration data and one based on the measured displacement data. The steps in the procedure were as follows: 1. The relationship between an applied dynamic force and predicted response (acceleration or displacement) was established numerically using the model for the X and Y axes of motion. 2. The relationship established in Step 1 above was inverted and applied to the measured accelerations/displacements from the operational tests, resulting in force spectra for the X and Y axes of motion. 3.
The force spectra established in Step 2 above were used to generate independent, random realizations of force time histories compatible with the spectra.
4. The generated X and Y pair of time series were applied to the model (either simultaneously or independently) at the assumed center of mass location of the molten fluid. Predicted acceleration and displacement responses were extracted from the model at locations corresponding approximately with the measurement locations. 5. The predicted acceleration/displacement responses were compared with the measured data for validation of the generated force time histories. This procedure was applied to the operational tests associated with the maximum measured displacements. The band width of the analysis was limited to the frequency range in which the dominant motions were observed (1 – 20 Hz). Step 1 of the force characterization procedure involved computation of the accelerance (acceleration per unit force) and receptance (displacement per unit force) of the model [2]. These relationships were established by applying band-limited white noise to model nodes corresponding approximately to the measurement locations from the dynamic tests. The response at these locations was then computed and the auto-spectra of acceleration or displacement were divided by the auto-spectrum of the input. Force time histories were generated from the force spectra using the methodology originally presented by Shinozuka [3]. The spectra were established in step 2 by multiplying the apparent mass (inverse of accelerance) or dynamic stiffness (inverse of receptance) by operational respective responses measured during the tests. The resulting spectra represent the force characteristics of the system at the measurement points. These points were selected a priori to coincide approximately with the center of mass of the fluid in the tank, when at rest. These locations represent approximately the points at which the peak sloshing loads were expected to occur. A fundamental assumption of the characterization procedure is that the forces generated by the process can be approximated by linear dynamics. If this assumption is valid, the force characterization procedure results in dynamic loads that are independent of the support configuration tested. The estimated force spectra can then be applied to the structural design of new support frame for the system.
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1496 5.2 Force Characterization Using Measured Accelerations Figure 6 shows the modeled and measured acceleration response spectra based on a single realization of the estimated X and Y force time histories. The modeled and measured data for accelerometers 1 and 4 match very well for the case when the loads were applied separately in two separate simulations. However, there are visible discrepancies between the modeled and measured spectra for accelerometers 2 and 3. The modeled results indicate greater acceleration in the 9 – 11 Hz band than were recorded during the tests. This motion is associated with the third mode of the model, which was estimated to be a torsional mode of vibration (see Figure 6). Figure 7 shows two realizations of the force time histories generated using the acceleration approach. Peak dynamic forces were estimated to be approximately 500 lbs in the X direction and 1000 lbs in the Y direction (for a single realization). These values were found to be much lower than the equivalent static force values estimated from the pull tests. The discrepancies observed between the modeled and measured responses using the apparent mass can be attributed to discrepancies between the true and modeled mass distribution of the system (considered approximately uniform in the model), the modeled center of mass location (load application point), and the specified damping level. Inaccuracies in the assumed mass distribution manifest themselves in the estimated mode shapes (dynamic deflections) for a given mode of vibration. Model estimates of the mode shapes differing from the real system will result in differences between the measured and modeled responses. 5.3 Force Characterization Using Measured Displacements Figure 8 shows the measured and model displacement spectra based on a single realization of the estimated X and Y force time histories. In both cases the modeled spectra show good agreement with the measurement data, particularly in the low frequency regions where the response energy is greatest.
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Figure 9: Sample force realizations (displacement-based approach). Figure 10 shows the measured displacement time series from lasers 1 and 2, together with the model predictions from a single realization of the X and Y forces applied simultaneously. From this figure, a satisfactory agreement between the modeled and measured displacement time series can be observed, indicating that the generated force time histories based on the displacement approach provide a reasonable characterization of the dynamic force generated by the melting process. Figure 11 shows the force spectra estimated using the displacementbased approach.
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Figure 11: Estimated force spectra (displacement-based approach). A check of the generated forcing function magnitudes was performed to further verify the consistency between the modeled and measured results. The peak amplitudes of the generated time histories in Figure 9 were multiplied by the experimental lateral stiffness values estimated from the pull tests. The resulting deflection could then be compared with the peak deflections observed during the operational tests. Mathematically: max δˆoperating − test =
max FGenerated max Fpull − test
max ⋅ δ pull − test
where δ are the displacements and F the force amplitudes. These estimated values can be compared with the modeled and/or measured peak displacements. The estimates of the peak displacement were in satisfactory agreement with the peak amplitudes of the time series shown in Figure 10, confirming that the generated force amplitudes are consistent with the load-deflection relationship obtained from the operating tests, and that the lateral stiffness characteristics of the system have been captured by the model. 6
CONCLUDING REMARKS
A force characterization procedure was developed for practical application to industrial processes. The procedure uses vibration test data (operational responses and step relaxation) and linear numerical models of the test assembly to estimate dynamic structural loads from the process. For linear or approximately linear loading process the estimation procedure results in dynamic loads that are independent of the test configuration. These loads can be applied to the design of new structural supports for the process. The technique was applied to a melter tank assembly, in which dynamic loads are generated by boiling of a molten fluid. The dynamic force characterization results were verified using equivalent static load estimates. Results from the case study indicate peak dynamic loads on the order of several thousand pounds for the process considered. The results from the study can be used for the design of a new structural support for the system.
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REFERENCES
[1] Handbook of Noise and Vibration Control Wiley Interscience, John Wiley and Sons, Ed. M. Crocker, 2007. [2] Theoretical and Experimental Modal Analysis, Ed. N.M. M. Maia and J.M.M. Silva, Research Studies Press, 1997. [3] Shinozuka, M., Stochastic fields and their digital simulation, in Stochastic Methods in Structural Dynamics, Ed. G.I. Schueller and M. Shinozuka, Martinus Nijhoff Publishers, 1987.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Operational Modal Analysis of a rotating tyre subject to cleat excitation Peter Kindt(1), Arnaldo delli Carri(2), Bart Peeters(2), Herman Van der Auweraer(2), Paul Sas(1), Wim Desmet(1) (1)
Department of Mechanical Engineering, K.U.Leuven, Belgium LMS International, Interleuvenlaan 68, B-3001 Leuven, Belgium, [email protected]
(2)
ABSTRACT Structure-borne tyre/road noise is an important component of the perceived noise annoyance of passenger cars. More in particular, it was observed that crossing road surface discontinuities (e.g. concrete road surface joints, railroad crossing, potholes, …) causes a significant increase in instantaneous exterior noise level. In addition, it has an adverse effect on the interior vehicle NVH in the sense that the passengers experience high-amplitude transient noise and vibrations. Therefore, an extensive research programme was established at the Department of Mechanical Engineering, K.U.Leuven, to study structure-borne tyre/road noise due to road surface discontinuities. As part of the research activities, an original test setup for impact tyre/road noise was developed so that rolling tyre vibrations, radiated noise and dynamic spindle forces could be measured at different rolling speeds. The test setup is based on the tyre-on-tyre principle and a cleat is used to reproduce a road surface discontinuity. This paper concentrates on the data processing techniques used to experimentally obtain the modes of a rolling tyre. Since the forces introduced by the cleat cannot me measured, Operational Modal Analysis was selected as processing technique. A major challenge is the requirement to obtain spatial information on the tire from a single-point measurement device. Therefore, a dedicated triggering and time-domain averaging procedure was elaborated. The purpose of averaging is obviously to reduce random noise whereas triggering is required to be able to correlate different tyre locations that have not been measured at the same time (a singlepoint Laser Doppler Vibrometer was used). 1 INTRODUCTION The increase of the road traffic density over the past decades resulted in a growing noise burden for most inhabitants of urban areas [1][2][3]. Nowadays, there is a high awareness among policymakers of the problems that traffic noise causes to the society. Therefore, road traffic is subjected to ever tightening noise limits. The three main sources of vehicle noise are: power unit noise, aerodynamic noise and tyre/road noise. Tyre/road noise refers to the noise that is generated by the interaction between the rolling tyre and the road surface. For modern vehicles, the tyre/road noise becomes more important than the power unit noise for driving speeds above approximately 40 km/h. The aerodynamic noise is small for normal driving speeds. Thus, tyre/road noise has become the dominant vehicle noise source for most driving conditions. Although tyre/road noise has been extensively studied for decades, still some of the noise generating phenomena are not yet fully understood and the generation of tyre/road noise for certain tyre-road configurations has never been studied in detail. For instance, the noise caused by passing a road surface discontinuity, such as joints in concrete road surface, railroad crossings, bridge joints, cobbled roads, etc. , has hardly been studied. The interaction between the tyre and a road discontinuity causes a transient noise that reaches significant peak levels and that is perceived as highly annoying. This noise causes serious discomfort, particularly in cities where a large number of these discontinuities are found. Reduction of tyre/road noise requires an integrated approach which comprises both low noise road surfaces and low noise tyres. Moreover, all design aspects – such as durability, wet grip performance, rolling resistance – have to be considered in the development of both new tyres and road surfaces. Therefore, a full understanding of all noise generating phenomena is essential.
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Similar to the noise exterior to the vehicle, the tyre/road interaction also contributes to the noise inside the passenger compartment [4]. In addition, the driver of a vehicle experiences vibrations at the seat and steering wheel. Depending on the amplitude and frequency content, these vibrations can reduce the driver’s comfort significantly. For most road surfaces, the interaction between the tyre and the road surface is a major source of vibrations that are transmitted through the suspension towards the vehicle body. The harshness of a vehicle expresses the subjective perception of transient vibrations and noise. Crossing road surface discontinuities causes transient vehicle interior noise and vibrations that can reach significant peak levels. Passengers perceive this kind of excitation as annoying, which results in a considerable reduction of the comfort for vehicle occupants. Improving the Noise, Vibration and Harshness (NVH) characteristics of a vehicle requires a thorough understanding of the different noise sources, vibration sources and transmission paths of structural and acoustic energy in the vehicle. Over the last two decades, the development times for vehicles have decreased significantly due to the introduction of advanced numerical and experimental methods. This evolution, combined with the increasing comfort requirements for new cars, has lead to a demand for more accurate tyre models for vehicle NVH simulations. Therefore, an extensive research programme was established at the Department of Mechanical Engineering, K.U.Leuven, to study structure-borne tyre/road noise due to road surface discontinuities [3]. As part of the research activities, an original test setup for impact tyre/road noise was developed so that rolling tyre vibrations, radiated noise and dynamic spindle forces could be measured at different rolling speeds. This paper concentrates on the data processing techniques used to experimentally obtain the modes of a rolling tyre. 2 TEST SETUP The rotation is known to have an influence on the dynamic behaviour of a tyre [5]. However, it is practically infeasible to measure the excitation forces on the rolling tyre caused by the surface texture. Thus, a classical modal analysis is not applicable to characterize the dynamic behaviour of a rolling tyre. Therefore, an operational modal analysis (OMA) will be used to identify the dynamic behaviour of the rolling tyre out of measured responses only. A novel test setup, which is based on the tyre-on-tyre contact, was developed in order to simulate a tyre rolling on a flat road surface [6][7]. The studied tyre is of size 205/55R16 without tread pattern. Two identical tyres that are statically loaded against each other both deform as if they are loaded against a flat road surface. Figure 1 illustrates the tyre-on-tyre contact between two identical tyres.
Figure 1: Static deformation of two identical tyres loaded against each other. Dotted line represents the tangent line to both deformed tyres in the middle of the contact patch.
The tangent line to the tyre in the middle of the contact patch is equal for both tyres and is perpendicular to the line that connects the two tyre spindles. A road surface discontinuity is simulated by guiding an aluminium cleat through the contact area of the two tyres. The cleat is mounted on the driven tyre (Figure 2). The cleat is attached to the steel wheel by means of four pre-tensioned rubber springs. The pre-tensioned flexible fixture keeps the
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cleat also connected to the tyre surface when the cleat approaches the contact area where the belt is deflected radially. A cleat which is rigidly connected to the wheel will not follow the intended trajectory through the contact area. As the cleat passes through the contact area of the two tyres, the cleat is indenting both tyres (Figure 3). This deformation is similar to the one of a tyre rolling over a cleat with half the size of the cleat used in the test setup. The test tyre is mounted on a multiaxial wheel hub dynamometer with built-in encoder (Figure 4). The piezoelectric dynamometer measures the three spindle forces and the three spindle moments. The x, y and z direction correspond to the longitudinal, lateral and vertical tyre direction, respectively.
(b)
(a) Figure 2: (a) Test setup with two tyres mounted; (b) Cleat fixation.
Figure 3: Circular cleat passing through the contact patch between the two identical tyres.
Figure 4: Multiaxial wheel hub dynamometer with built-in encoder.
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Figure 5 (a) shows the setup to measure the radial component of the rolling tyre treadband vibrations. A Laser Doppler Vibrometer (LDV) is used for this purpose. A plane mirror is used to reflect the laser beam of the LDV such that the laser hits the tyre tread surface perpendicularly. The mirror is mounted on a support which can be considered rigid in the frequency range of interest. The mirror support can be positioned at different locations around the tyre, allowing a circumferential measurement resolution of 10 degrees. Since the mirror can be fixed at different positions and angles on the support, measurements can be performed over the entire width of the treadband.
Figure 5: (a) Rolling tyre vibration measurement by means of a LDV (dotted line shows the path of the laser beam); (b) Measurement points on the tyre cross-section; (c) Tyre measurement grid.
Figure 5 (b) shows the measurement points on the tyre cross-section. Vibrations are measured in two points on the treadband and in one point on the sidewall. The points are chosen such that the different cross-sectional modes can be identified. The laser vibrometer is equipped with an indicator that shows the intensity of the scattered light received by the vibrometer. When the laser beam is aligned perpendicularly to the tyre surface, a maximum amount of light is scattered back in the direction of the laser beam. The direction of the laser beam is therefore adjusted such that the intensity of the received scattered light is maximized. This assures that the laser
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beam is aligned perpendicularly to the tyre surface. This approach is still valid when the laser beam is deflected by a mirror since the light reflected from the tyre surface is also deflected by the mirror. Vibration measurements in the contact area of the tyre are impossible since the laser beam has no access to this region of the tyre. In this setup, measurements on the tread can be performed at circumferential angles between 30 degrees and 330 degrees (Figure 5 (c)). The sidewall rolling tyre vibrations are measured directly without deflecting the laser beam. All the vibration measurements are performed with respect to a fixed reference frame in this setup. 3 REFERENCE ANALYSIS RESULTS This section will review the analysis results as presented in [3] and obtained through rather non-classical data (pre-) processing. These results will be considered as reference results throughout this paper and will be compared to other, sometimes more classical, Operational Modal Analysis (OMA) processing results in next sections. Since it is difficult to measure all responses simultaneously, a sequential measurement will be performed in which the responses are measured separately. Sequential measurements can only yield information about the complete vibration pattern if the phase relation between the different responses is maintained. This can be achieved by measuring simultaneously the response point and reference point vibration. This requires at least two laser Doppler vibrometers. However, if the excitation is perfectly repetitive it is possible to use a time reference instead. The acquisition of the individual responses has to start at the same time instant relative to the excitation. Here, the vertical component of the spindle force Fz (Figure 4) is used to synchronize the different response measurements since the spindle force and tyre surface vibrations are always measured simultaneously in this setup. Therefore, the time reference is obtained from the force reference. In Section 4, the force reference will be used directly to obtain correct phase relationships between acquisition runs. Figure 6 shows an example of the vertical spindle force due to four subsequent cleat passages. Here, the time instant at which the spindle force is maximum (indicated by an arrow) is used as the time reference. The time reference is then used to synchronize the reference response and all the other responses such that the auto- and cross-power spectral density functions between these signals can be calculated. A sufficiently high sampling rate is required to determine accurately the time instant of the maximum spindle force. The same time reference is used to calculate the time averaged responses as described below. Alternatively, the auto-correlation function could have been used to identify the time intervals between each rotation.
Figure 6: Maxima of vertical spindle forces used as time reference between different cleat impacts.
The synchronized, time-averaged responses are further processed in the classical OMA way as implemented in LMS Test.Lab [8], as if they were measured simultaneously in a single run. In the presented analysis, two responses on the tread are used as reference (Figure 5). However, the references are not considered
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simultaneously during the modal parameter estimation. Two different modal parameter estimations will be performed in which a single reference is considered. Certain resonances will be identified in both analyses while others modes are only identified in one analysis, dependent on the reference location. As is shown in [3], the two considered references provide a complete identification of all the excited rolling tyre modes. 4 THROUGHPUT-DATA OPERATIONAL MODAL ANALYSIS In this section a method will be investigated to conduct an Operational Modal Analysis using throughput-data only. Throughput-data is raw-collected data as good as it comes from a measurement system, without averages in a single run or synchronization between different runs. In the presented application, the basic idea is to compute cross-spectra between the spindle forces and the tyre vibration responses for each measurement run. It should be noted that only spindle (reaction) forces are available and not the contact forces between the obstacle and the tyre. By computing these cross-spectra, no triggering or synchronization is needed between the different runs, since the relative phase between two response quantities is independent from the start time of the acquisition: G FR ( f )
G FR ( f ) M ( f )
Fy ( f ) R x* ( f )
(1)
4.1 Time-Data Figure 7 shows the time signal of the vertical spindle force and the vibration response of point (a, 250) on the tyre surface for the entire measurement duration and a zoom around 1 particular cleat impact. The total acquisition time is 13 s, at a sampling frequency of 3200 Hz. Every measurement run counts 32 passages of the cleat in the contact-zone and the driving wheel has a rotation speed of 150 rpm. 0.11
580.00
0.11
-250.00
Real
Time Fy:+Y Time slick:80_250:+X
N Real
(m/s)
F B
Real
Time Fy:+Y Time slick:80_250:+X
N Real
F B
-0.16 -250.00 0.00
s
13.00
(m/s)
580.00
-0.16 5.13
s
5.60
Figure 7: Throughput-data for the vertical spindle force and the vibration response at point (a, 250). (Left) Entire measurement duration; (Right) zoom around 1 cleat impact.
4.2 Pre-processing The so-called correlogram was used as cross-spectrum estimate. The tyre surface velocity time histories (as measured by the LDV) and the vertical spindle force time histories (as measured by the wheel hub dynamometer) were processed into cross-correlations. The vertical force has been selected as a reference since this component was the most significant from the 6 spindle force/moments. For those computations, 640 time lags and no windowing have been used. Afterwards, so called “half cross-spectra” were obtained by applying a single DFT to the positive time lags of the cross-correlations. This procedure is repeated for each of the measurement runs. Each run contains a different tyre surface response point (Figure 5), but the spindle forces as well. Figure 8 (Left) shows some of the calculated cross-spectra. More information about this particular pre-processing, which is very suited for OMA, can be found in [9][10]. Figure 8 (Right) shows the spindle force power spectra for all measurement runs. A tyre rolling over a discontinuity in the road surface will partially envelope this discontinuity, causing the spindle force to be
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significantly smoother than the geometry of the road discontinuity. Despite this enveloping, a road discontinuity causes an unequal distribution of the excitation energy in the frequency domain [11][3]. At distinct frequencies, there is almost no excitation of the tyre. These frequencies are determined by the rolling speed, the length of the contact patch and the geometry of the road discontinuity. These frequencies of low excitation are clearly visible in the spectra of Figure 8 (Right). The different spectra show the same trend, but there are small variations in amplitude which are caused by variations in the tyre temperature between the different measurement runs. This time-invariance can provoke several errors in modal parameters identification. -40.00
slick:180_250:+X/Fy:+Y slick:180_240:+X/Fy:+Y slick:180_230:+X/Fy:+Y slick:180_240:+X/Fy:+Y
40.00
N2 dB
Hp dB
CrossPow er CrossPow er CrossPow er CrossPow er
-80.00 ° Phase
180.00
-180.00
-10.00 0.00
Hz
500.00
0.00
Hz
AutoPow er AutoPow er AutoPow er AutoPow er AutoPow er AutoPow er AutoPow er AutoPow er AutoPow er AutoPow er AutoPow er AutoPow er AutoPow er AutoPow er AutoPow er AutoPow er AutoPow er AutoPow er AutoPow er AutoPow er AutoPow er AutoPow er AutoPow er AutoPow er A t P
Fy:+Y Fy:+Y Fy:+Y Fy:+Y Fy:+Y Fy:+Y Fy:+Y Fy:+Y Fy:+Y Fy:+Y Fy:+Y Fy:+Y Fy:+Y Fy:+Y Fy:+Y Fy:+Y Fy:+Y Fy:+Y Fy:+Y Fy:+Y Fy:+Y Fy:+Y Fy:+Y Fy:+Y F +Y
150.00
Figure 8: (Left) Some of the computed cross-spectra. (Right) Spindle force power-spectra for all runs.
4.3 PolyMAX parameter extraction In order to extract the modal parameters, all runs are analyzed simultaneously using the Operational PolyMAX method [10][12]. Although in the analysis only the cross spectra between tyre surface velocity and spindle force are used, it is assumed in such a global approach that the operational forces are the same for each run. These operational forces introduced by the cleat could not be measured, but at least when considering the spindle reaction forces, it is observed that these are not exactly the same for each run (Figure 8 – Right). The frequency range for the estimation is set from 0 to 445 Hz and the modal order is 45. As can be seen in Figure 9, this method yields quite some global estimates for the poles.
Figure 9: Operational PolyMAX stabilization diagram.
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In Figure 10, the mode set identified here is compared to a mode set identified according to the pre-processing discussed in Section 3. Although both mode sets do not match perfectly, a good agreement is found between both processing ways. In general, the mode shapes look very plausible and clean.
Figure 10: Matching of poles and mode shapes.
4.4 Modal model validation The Modal Assurance Criterion (MAC) was used to validate the modal model. The MAC is a mathematical tool to compare two vectors: assuming that {X} and {Y} are two vectors with the same number of elements, MAC is defined as: MAC XY
{Y }
{ X }*t {Y }
*t
2
{Y } { X }*t { X }
(2)
If the modal assurance criterion is unity, then both vectors are perfectly identical within a scale factor. If the modal assurance criterion is zero, no linear relation exists between both vectors and the estimated modal scale factor has no meaning. Here, the modal assurance criterion is used as a tool to compare different sets of estimated mode shapes or to investigate the validity of the estimated modes within one set (also referred to as the AutoMAC). Figure 11 (Left) shows the AutoMAC, which indicates how much a single mode is self-independent. Ideally, the off-diagonal elements should be low, indicating that the sensor number and locations were well selected to distinguish the mode shapes from each other. Figure 11 (Right) shows the MAC between the classical throughput OMA (this section) and the reference analysis using synchronized, time-averaged signals (Section 3). Especially the lower modes agree very well. Some typical mode shapes are represented in Figure 12.
Figure 11: (Left) AutoMAC for throughput-data Operational Modal Analysis. MAC between classical throughput OMA and the reference analysis using synchronized, time-averaged signals.
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Figure 12: Typical rotating tyre mode shapes obtained by applying operational PolyMAX to cleat excitation data.
5 SCALED CROSS-SPECTRA OPERATIONAL MODAL ANALYSIS Figure 8 reveals that the spindle force spectra are not equal for the different runs, which violates the basic assumption of a global analysis. In an attempt to overcome this problem, every response cross-spectra can be weighted relative to its respective force auto-spectrum. Under the assumption of system linearity, doubling the force also doubles the response. Thus, dividing by the force yields: 2 Fy
(i )
2 FY 2 Fy
(i)
2RX
(i ) (i)
*
R X Fy
(i )
(i )
*
*
(i )
(i )
*
FY Fy
(3)
Figure 13 shows some of the obtained scaled cross-spectra. The PolyMAX method applied to this set of crossspectra yields very good estimates of the system poles, which are nearly identical to the poles obtained by the reference Operational Modal Analysis (see Figure 14). Despite this very good estimation of the system poles, the
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estimated mode shapes look less clean than in the unscaled case (Section 4), whereas rather the opposite was expected. It is speculated that the vertical spindle force (which is used to scale the responses) may not be a good indication of the real forces that are injected into the system by the cleat. -50.00
((m/s)/N) dB
CrossPow er slick:80_40:+X/Fy:+Y CrossPow er slick:80_50:+X/Fy:+Y CrossPow er slick:80_130:+X/Fy:+Y
-120.00 0.00
Hz
500.00
Figure 13: some scaled cross-spectra.
Figure 14: PolyMAX method for scaled cross-spectra set.
6 SPINDLE FORCE EXPERIMENTAL MODAL ANALYSIS Another approach has been investigated in which the vertical spindle force is used as an input force for a classical experimental modal analysis. This approach was inspired by the approach of previous section in which the tyre responses are scaled by the spindle forces. This is exactly what happens in a classical Frequency Response Function (FRF) estimation assuming that the spindle forces are representative for the real forces. Unfortunately, again the estimated mode shapes are not very clean, which indicate that the vertical spindle forces may not be a good indication of the real forces that are injected into the system by the cleat. The coherence function between vertical spindle forces and tyre surface velocities are shown in Figure 15. It can be seen that above the 160 Hz, the responses are not so well correlated to the chosen input force.
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/ Amplitude
1.00
Coherence slick:80_240:+X/Fy:+Y Coherence slick:80_250:+X/Fy:+Y Coherence slick:80_230:+X/Fy:+Y
0.07 0.00
Hz
165.65 300.00
Figure 15: Coherence functions between vertical spindle forces and tyre surface velocities.
7 CONCLUSIONS This paper investigated several possibilities to process data from a rotating tyre with the aim to extract the modal parameters in operational conditions. Following data processing methods can be distinguished: 1. Synchronized, time-averaged responses > auto- and cross spectra using response as reference > OMA: Section 3 and [3]; 2. Synchronized, time-averaged responses > single-block DFT of each response (without references) > OMA: not discussed here, but also valid approach; 3. Raw time responses > auto- and cross spectra using vertical spindle force measured in each run as reference > OMA: Section 4; 4. Raw time responses > auto- and cross spectra using vertical spindle force measured in each run as reference > scale cross spectra by auto spectra of spindle forces > OMA: Section 5; 5. Raw time response > H1 estimator considering vertical spindle force as reference > EMA: Section 6. Essentially, the last two approaches (4 and 5) are identical, except for the fact that in the OMA processing as presented in [8][10], the correlogram approach is used to calculate auto- and cross-spectra, whereas in the traditional H1 FRF estimate, the periodogram approach is used. The scaling by the reaction force was an attempt to compensate for force differences that may exist between runs and also for reducing the non-system-related dips in the response spectra due to the specifics of the cleat excitation. This phenomenon seems to have some resemblance with the well-known “double impact” in impact testing (Figure 7). Despite the good hypothesis, these scaling approaches did not yield satisfactory results, leading to the speculation that the spindle forces may not be entirely representative for the (unmeasurable) forces injected into the tyre by the passing cleat. The unscaled approaches (1, 3, and also 2 although this method is not discussed in the paper) yielded highquality operational modal parameters with realistic frequency and damping estimates and very clean mode shapes. The advantage of method 3 is that no special pre-processing is required to yield synchronized, timeaveraged responses between different measurement runs. REFERENCES [1] U. Sandberg and J.A. Ejsmont. Tyre/road Noise Reference Book. Informex Ejsmont & Sandberg Handelsbolag, Harg, SE-59040 Kisa, Sweden, 2002. [2] K.R. Stassen, P. Collier, and R. Torfs. Environmental burden of disease due to transportation noise in Flanders (Belgium). Transportation Research Part D, 13:355–358, 2008. [3] P Kindt. Structure-borne tyre/road noise due to road surface discontinuities. PhD thesis, Katholieke Universiteit Leuven, Belgium, 2009.
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[4] [5] [6] [7] [8] [9] [10] [11] [12]
C.J. Gagliano, M. Tondra, B. Fouts, and T. Geluk. Development of a experimentally derived tire and road surface model for vehicle interior noise prediction, SAE Paper 2009-01-0068 in Proceedings of the SAE World Congress & Exhibition, Detroit, MI, USA, April 2009. Y.J. Kim and J.S. Bolton. Effects of rotation on the dynamics of a circular cylindrical shell with application to tire vibration. Journal of Sound and Vibration, 275:605–621, 2004. P. Kindt, F. De Coninck, P. Sas, and W. Desmet. Analysis of tire/road noise caused by road impact excitations, SAE paper 2007-01-2248 in Proceedings of the 2007 SAE Noise and Vibration Conference. P. Kindt, D. Berckmans, F. De Coninck, P. Sas, and W. Desmet. Experimental analysis of the structureborne tyre/road noise due to road discontinuities, Mechanical Systems and Signal Processing, 23(8):25572574, 2009. LMS International. LMS Test.Lab Structures, Leuven, Belgium, www.lmsintl.com, 2009. L. Hermans, H. Van der Auweraer and P. Guillaume. A frequency-domain maximum likelihood approach for the extraction of modal parameters from output-only data, In Proceedings of ISMA23, the International Conference on Noise and Vibration Engineering, 367-376, Leuven, Belgium, 16-18 September 1998. B. Peeters, H. Van der Auweraer, F. Vanhollebeke, and P. Guillaume. Operational modal analysis for estimating the dynamic properties of a stadium structure during a football game, Shock and Vibration, 14(4):283-303, 2007. P. Bandel and C. Monguzzi. Simulation model of the dynamic behavior of a tire running over an obstacle. Tire Science and Technology, TSTCA, 16(2):62-77, 1988. B. Peeters, H. Van der Auweraer, P. Guillaume, and J. Leuridan. The PolyMAX frequency-domain method: a new standard for modal parameter estimation? Shock and Vibration, 11:395-409, 2004.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Modal Identification of Flexible Structures with Applications in Robotic Manipulators* by Zamri Abdul Rahman Ahmad Azlan Mat Isa Faculty of Mechanical Engineering, MARA University of Technology, 40450 Shah Alam, Selangor, Malaysia email: [email protected] ABSTARCT: Control of flexible structures for many applications requires an accurate value of their parametric properties such as mass and stiffness. These properties may be derived from natural frequencies and damping ratios. Structures with high flexibility in their configurations such as those with both angular displacement about a transverse direction and torsional displacements about an axial direction pose extreme challenge in determining their dynamics parameters analytically. This paper discusses the determination of modal parameters of a flexible structure with fixed-free end conditions using experimental modal analysis. The accuracy of the results is then verified with the results obtained analytically and also through output-based modal analysis. These results will later be applied to obtain the state variables required for the control of such flexible structure as robot manipulators. Keywords Flexible structures, mode shapes, natural frequency, damping ratio, experimental and operational modal analysis 1.0 INTRODUCTION Flexible structure has gained considerable attention over the past 20 years owing to its high strength to weight ratio, especially in aerospace industries, robotic manipulators, hard disk drives, and micro-mechanical systems [1]. Rigid and flexible structural components exhibit different dynamic characteristics. In the case of rigid manipulators, the state variables which consist of joint angles and their velocities can be easily measured using encoders, tachometers, potentiometers, and so on. However, for the flexible manipulators, the state variables also include elastic deformations and their velocities due to flexibility [2]. Modeled as a distributed-parameter system, a flexible structure has an infinite number of degree of freedom and hence an infinite number of natural frequencies [3]. Control of such structures offers real difficulty especially when both bending and torsional modes of vibration present since these modes occur at relatively many lower natural frequencies. Various approaches have been applied to successfully control and suppress the vibration of flexible structures [1, 2, 4, 5]. Recently, active vibration control strategies have been employed due to rapid advancement in smart materials imbedded as sensors and actuators to such flexible structures [4]. Yoshikawa and friends utilized visual sensor to measure distributed state variables of flexible manipulators [2]. However, there has been a very little attempt on the use of experimental modal analysis to adequately characterize the dynamics properties of a flexible beam with applications to robot manipulators. *This work is partly supported by Malaysian FRGS 2008
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_135, © The Society for Experimental Mechanics, Inc. 2011
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This paper presents the identification of the dynamic properties such as natural frequencies and damping ratios of the flexible beam with fixed-free end conditions using classical experimental modal analysis. The accuracy of the classical modal analysis is then compared with the results obtained through analytical method and through operational modal analysis technique. 2.0 FLEXIBLE BEAM MODELING AND OPERATIONAL MODAL ANALYSIS When only bending modes are assumed, the classical-beam or Euler-Bernoulli beam model may be sufficed to describe the behavior of a flexible manipulator. However, when it is coupled with a torsional mode, the Timoshenko beam model must be applied. This section provides a brief theoretical analysis and mathematical modeling of a single flexible manipulator and operational modal analysis techniques applied to vibration of a flexible beam using an Euler-Bernoulli model. Special consideration is given to the end condition with fixed-free ends. The classical experimental modal analysis technique is well known and will not be discussed here. Its detailed discussion can be found many literatures, for instance [6] and.[7] 2.1 Mathematical Model We consider a single-link flexible manipulator as shown in Figure 1. Although the original set-up of the physical system consists of a hub which may rotate about the z axis with a flexible beam (which is essentially having a pinned-free boundary condition), we consider it here as a fixed-free end condition (i.e. a cantilever beam) since a G clamp is used to tightly secure the hub to the base. z
y m(x), EI(x), A(x)
h
y t
w(x,t)
x
Cross-section A(x)
x
Figure 1 A simple single-flexible beam in transverse vibration Assuming no external force applied to the beam, and that EI(x) and A(x) are constant, the free vibration of the system is then reduced to[3] 4 w 2 w( x, t ) 2 w w( x, t ) +c =0 wt 2 wx 4
where c =
EI UA
(1)
Since the equation contains four spatial derivatives and two time derivatives, hence, it requires four boundary conditions and two initial conditions. These boundary conditions can be determined by examining the deflection w(x,t), the slope of the deflection ww(x,t)/wx, the bending moment EIw2w(x,t)/w2x, and the shear force w[EIw2w(x,t)/w2x]/wx at each end of the beam. For clamped-free end conditions, we have
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EI
w 2w wx 2 x
w ª w2wº « EI » wx ¬ wx 2 ¼ x
=0 L
=0
w(0) = 0
and
L
ww wx x
=0
(2)
0
While satisfying these four boundary conditions, the solution of equation (1) must also satisfy (x,0) = two initial conditions, namely, the initial deflection w(x,0) = w0(x) and initial velocity w
w 0 ( x ) at the initial time t = 0. For motion to occur, w0(x) and w 0 ( x) cannot both be zero. Using a separation-of-variables method, the solution to flexible displacement of beam can be written in the form w(x,t) = Y(x) T(t)
(3)
Substituting equation (3) into (1) and after arrangement yields c2
Y (iv ) ( x) T(t ) =– = Z2 Y ( x) T (t )
(4)
Consequently, we obtain the temporal equation:
T(t ) + Z2 T(t) = 0
(5)
and the spatial equation: 2
Y
( iv )
§Z · ( x ) – ¨ ¸ Y(x) = 0 ©c¹
(6)
The solution to both equations (5) and (6) can be found in many available textbooks on vibration of continuous structures. For instance, Meirovitch [8] derived the natural modes of vibration to be of the form:
Yr (x) = Br[(sin ErL – sinh ErL)(sin Erx – sinh Erx) + (cos ErL + cosh ErL)(cos Erx – cosh Erx)]
r = 1, 2,…
(7)
Where Br = C1/(sin ErL – sinh ErL) with C1 0 for a nontrivial solution. The natural frequencies are found from the corresponding characteristic equation to be
Zr = (rS)2
EI mL4
r = 1, 2,…
(8)
2.2 Operational Modal Analysis 2.2.1
Frequency Domain Decomposition (FDD)
The frequency domain decomposition method (FDD) is an extension of the Basic Frequency Domain method (BFD) or commonly known Peak-Picking method. The technique estimates the modal parameters directly from signal processing data calculations. It utilizes the property that
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the mode shapes can be estimated from the calculated spectral density for the condition of random noise input or stochastic input applied to lightly damped structure where the modes are well separated[9]. Unlike the classical technique where the power spectral density (PSD) matrix is directly and easily estimated via Fast Fourier Transformation (FFT), in the FDD, it is not directly processed, but decomposed using the singular value decomposition (SVD) at each spectral line where the PSD matrix is decomposed into auto spectral density functions consist of single degree of freedom systems. The modes are simply picked by locating the peaks in the SVD plots. The accuracy of the estimated natural frequency depends on the FFT resolution with no modal damping is calculated[10]. 2.2.2
Enhanced Frequency Domain Decomposition (EFDD)
The enhanced frequency domain decomposition method (EFDD) is just an extension to the FDD. While giving an improve estimate of both the mode shapes and the natural frequencies, it also provides modal damping. In EFDD, the SDOF Power Spectral Density function which is identified around a resonance peak is transformed back to time domain using Inverse Discrete Fourier Transform (IDFT). The SDOF function is predicted using the shape previously determined using FDD which is used as reference vector in the correlation analysis based on Modal Assurance Criteria (MAC). This value is calculated between the FDD vector and a single vector for each frequency line[11]. 3.0 EXPERIMENTAL METHODS The experimental procedure was carried out to obtain the mobility of a single-flexible manipulator in the form of a frequency response functions (FRF) using both classical and operational modal testing. Only the type of fixed-free beam end conditions was investigated. Two flexible beams, namely flexible beam #1 and flexible beam #2 were considered in this investigation. Some of their dimensional and other properties are given in Table 1. Table 1 Dimensional and other properties of flexible beams Flexible beam #1 Flexible beam #2 430 mm 440 mm Link Length, L Link Thickness, t 1 mm 1.5 mm Link Width, h 20.8 mm 32.8 mm 7800 kg/m3 7800 kg/m3 Link Density, U Young’s Modulus, E 210 GPa 210 GPa 3.1 Experimental Modal Analysis (EMA) 3.1.1
Experimental Modal Testing
In this test, the excitation is exerted to the test structure by applying an impulse force from an impact hammer (type 8206-002) in a single Z-direction, see Figure 2. The force was applied at a single location, i.e. point 3 in order to minimize the resulting vibration while a single roving accelerometer capturing the response signal at 5 locations along the structure. The FRFs at different positions of the accelerometers are stored for further post processing using ME’Scope to extract dynamic properties such as natural frequencies and damping ratio, and also to simulate the vibration modes of a flexible beam. All measurements utilized Bruel&Kaer PULSE Frontend
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Type 3560D Analyzer data acquisition system and responses are measured using B&K accelerometers type 4508. For all measurements, coherence over 0.98 is considered for a reasonable accuracy since the structure is a simple one. Y 1 Z
3 43 cm
Impact Hammer
5 8.6 cm 2
X
4
6 Flexible manipulator
FFT Analyzer/ PULSE System
Roving Accelerometer
Figure 2 EMA – A roving hammer and measurement points 3.1.2
Experimental Modal Identification
The modal identification was performed using the curve fitting and direct parameter estimation techniques in the frequency range of zero to 800 Hz from which natural frequencies, mode shapes and damping ratios are determined. A typical FRF obtained is shown in Figure 3.
Figure 3 FRF with phase and magnitude plots 3.2 Operational Modal Analysis (OMA) 3.2.1
Operational Modal Testing
To assess the relevant structural vibrations of a flexible beam in operating condition, the operational modal analysis technique is employed where, in this case, the beam was randomly excited along the axial direction in the z-direction. Data were recorded using 2 uniaxial accelerometers with the accelerometer mounted at the free end or node 2 as a reference, Figure 4. This point was selected since it exhibited most information about the vibration and in actual system, may act as a payload. All measurements utilized the Bruel&Kaer PULSE Frontend Type 3560D Analyzer data acquisition system. A total of 4 measurement degrees of freedom were taken in order to obtain the overall modes of vibration.
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Y 1 3 4
43 cm 5
Z X
Random excitation Roving Accelerometer
6 8.6 cm
Flexible manipulator
FFT Analyzer/ PULSE System
2 Reference Accelerometer
Figure 4 OMA – Measurement points and a reference accelerometer The measurements were repeated several times in order to obtain an accurate observation of the structural properties of the flexible beam and also to exhibit the effectiveness of the OMA technique. Figure 5 shows a typical time record of the history of the random excitation signal.
Figure 5 Typical time record of random excitation signal in OMA 3.2.2
Operational Modal Identification
To identify the operational modes of vibration governing the response of the flexible beam, two different operational modal analysis techniques were applied, namely the FDD and EFDD. The time data were calculated using both techniques where the spectral density matrix was calculated utilizing a 2048 point FFT. A typical FRF captured is shown in Figure 6.
Figure 6 FRF of a flexible beam using FDD method 4.0 RESULTS AND DISCUSSION Selected results from the experimental studies were tabulated and comparisons were presented. The results for both flexible beams #1 and #2 using EMA were compared to the results obtained via OMA, and both results were then compared with the theoretical values.
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As can be seen in Figure 6, there are also clearly several modes of vibration which do not correspond to the transverse vibration. This may perhaps represent the torsional modes of vibration picked up by both EMA and OMA techniques. Table 2 and Table 3 compare the first five modal natural frequencies for flexible beam #1 and #2, respectively. The operational modal identification of mode shapes and the natural frequencies utilizing the FDD and EFDD yielded very much consistent results with very small differences in the natural frequencies, at least for the first three modes. These are clearly shown with percentage error with respect to theoretical results. For flexible beam #2 (which is less flexible), the results are very much comparable up to the fifth mode of vibration. Table 2: Natural Frequencies of Flexible Beam #1 Natural Frequency (Hz) MODE OMA THEORY EMA % Err FDD % Err EFDD 1 3.03 3.07 1.32 3.0 -0.99 3.661 2 19.0 20.6 8.42 19.5 2.63 19.5 3 53.18 54.0 1.54 54.5 2.48 55.06 4 104.2 110.0 5.57 108.5 4.13 102.3 5 172.2 196.0 13.82 179 3.95 167.4 Table 3: Natural Frequencies for Flexible Beam #2 Natural Frequency (Hz) MODE OMA THEORY EMA % Err % E rr FDD EFDD 1 5.53 6.14 11.03 5.5 -0.54 5.199 2 34.7 36.9 6.34 35.5 2.31 33.0 3 97.07 105.0 8.17 100 3.02 96.07 4 190.2 208.0 9.36 194 2.00 188.2 5 314.4 337.0 7.19 320.5 1.94 311
% Err
20.83 2.63 3.54 -1.82 -2.79
% Err
-5.99 -4.90 -1.03 -1.05 -1.08
Table 4 gives the values of the damping ratio for both flexible beam #1 and #2 computed using both EMA and OMA techniques. These values may be attributed to the fixed end attachment which is not sufficiently rigid. Table 4: Modal Damping for Flexible Beam #1 and #2 Damping Ratio (%) MODE Beam #1 Beam #2 EMA OMA EMA OMA 1 1.03 12.46 2.8 7.687 2 3.1 11.2 0.954 2.423 3 4.36 5.568 0.388 1.044 4 3.77 3.768 0.291 0.6792 5 3.48 2.423 0.186 0.6013
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5.0 CONCLUSION In this experimental study, we presented a modal parameter estimation of flexible manipulator using experimental modal analysis (EMA) method. We verified the validity of the method with theoretical results and also by means of output-based modal analysis (OMA) technique. The intended purpose of this study is to eventually obtain the state estimation of state variables required to control the vibration of flexible manipulators. The results showed that both EMA and OMA techniques can adequately be used to extract modal parameters of a flexible beam with fixed-free end condition. The techniques are not only able to identify bending but also torsional modes of vibration, though only a unidirectional accelerometer is used. Hence, the data obtained can be used to design appropriate control scheme to suppress vibration of flexible structural systems especially at lower modes. 6.0 ACKNOWLEDGEMENTS The authors would like to thank to all individuals who are directly or indirectly involve in making this project a success, particularly Mr Mohd Fauzi, for the good cooperation with respect to all data acquisition issues. 7.0 REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
Hu, H., et al. Modeling and Vibration Control of a Flexible Structure Using Linearized Piezoceramic Actuators, in International Conference on Intelligent Mechatronics and Automation. August, 2004. Chengdu, China. Yoshikawa, T., A. Ohta, and K. Kanaoka. State Estimation and Parameter Identification of Flexible Manipulators Based on Visual Sensor and Virtual Joint Model. in IEEE International Conference on Robotics and Automation. May, 2001. Seoul, Korea. Inman, D.J., Engineering Vibration. 2nd ed. 2001, New Jersey: Prentice-Hall, Inc. Barboni, R., et al., Optimal Placement of PZT Actuators for the Control of Beam Dynamics. Smart Materials and Structures, 2000. 9: p. 110-120. Barbosa, E.G. and L.C.S. Goes, Modelling and Identification of Flexible Structure Using Bond Graphs Applied on FLEXCAM Quanser System. ABCM Symposium Series in Mechatronics, 2008. 3: p. 129-138. Ewin, D.J., Modal Testing: Theory, Practice and Application. 2nd Edition ed. 2000: Research Studies Press, England. Allemang, R.D., Vibrations: Experimental Modal Analysis, ed. UC-SDRL-CN-20-263663/664. 1999. Meirovitch, L., Elements of Vibration Analysis. 2 ed. 1986, Singapore: McGraw-Hill, Inc. Møller, N.R.B., H. Herlufsen, P. Andersen. Modal Testing Of Mechanical Structures Subject To Operational Excitation Forces. in Proceedings of The 19th International Modal Analysis Conference (IMAC). 2001. Kissimmee, Florida,. Brincker, R.L.Z., P. Andersen. Output-Only Modal Anlysis By Frequency Domain Decomposition. in Proceedings of The ISMA25 Noise And Vibration Engineering. 2000. Leuven, Belgium. Jacobsen, N.J., P. Andersen, R. Brincker. Eliminating the Influence of Harmonic Components in Operational Modal Analysis. in Proceedings of The 25th International Modal Analysis Conference (IMAC) 2007. Orlando, Florida
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Calibration of Very-Low-Frequency Accelerometers A challenging task
Dr.-Ing. Holger Nicklich, General Manager Michael Mende, Physicist both SPEKTRA Schwingungstechnik und Akustik GmbH Dresden, Gostritzer Straße 61, 01217 Dresden, Germany
ABSTRACT The calibration of accelerometers in the very low frequency range below 1 Hz is a special challenge. Since in this frequency range the maximum acceleration that can be provided by a calibration vibration exciter is limited by its maximum stroke, the acceleration and also the electrical output of the accelerometer decreases rapidly with the frequency (12 dB/ octave). Thus the lower the frequency the bigger the problems with noise generated by the shaker bearings, electrical noise or subsonic noise coming from the laboratory floor. As a consequence the use of an excellent air bearing long-stroke vibration exciter mounted on a heavy rigid table that is well isolated from environmental vibrations is mandatory. Also an appropriate reference standard and algorithms for signal processing (Timing Problems, DC-coupling vs. High-pass filter, sine-approximation) must be chosen carefully for good calibration results. Also mechanical problems due to heavy sensors to be calibrated like seismometers, can cause trouble. This paper gives an overview over the requirements that a very-low-frequency calibration system has to meet in order to reach best measurement uncertainties.
Sensors for the Very Low Frequency Range Many applications like the measurement of building vibrations, the measurement seismic activities and earthquakes etc. require sensors suitable to measure acceleration or velocity in the extreme low frequency range below 1 Hz. Such sensors are available in different sizes and using different sensor technologies. Small, lightweight accelerometers may have a weight of 10 gram and a size of 25mm x 25mm x 10mm or even less. The other extreme are seismic sensors that may have a weight of some kilogram and a size of 120 mm x 120 mm x 80 mm or even more. The different sensor technologies used to build such sensors require different signal conditioners like constant current or constant voltage power supplies or charge amplifiers.
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_136, © The Society for Experimental Mechanics, Inc. 2011
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Fig.1 Example for a Seismometer weight 3 kg, diameter120mm Thus a calibration system for the very low frequency range should be capable to handle all these extreme requirements regarding weight and size of the device under test (DUT) as well as the different type of sensor signals. The vibration exciter should not only be able to bear a payload of some kilogram but also to move with low transverse motion according to the limits defined in ISO 16063-21, if a heavy DUT is mounted. Furthermore it must be possible to mount a big sensor with a 100 mm diameter or more stable and safe on the shaker armature.
Very Low Frequency means Very Low Acceleration A special challenge in the very low frequency range is the low acceleration that can be provided even by long stroke exciters. This is due to the fact that every vibration exciter has a limited stroke. (1) (2) (3) As can be seen in equitation (2) above the relation between the velocity amplitude v0 of a vibration exciter having a maximum stroke x0 and the frequency ω is v0 ~ x0 ω. That means the maximum velocity amplitude decreases with 6 dB / octave if the vibration exciter is operated with maximum stroke. For the acceleration amplitude a0 the relationship is even worse since it decreases proportional to the square of the frequency or 12 dB / octave (see fig. 2). Or in other words, the maximum acceleration at 0.1 Hz is only 0.01% of the acceleration that can be provided at 10 Hz by a certain exciter. To give an example how dramatic this decrease is, let’s have a look on a typical exciter for the low frequency range down to 1 Hz. Such exciters have typically a maximum stroke of about 10 mm peak-peak. So at 1 Hz the maximum acceleration that can be provided by this shaker would be only 20 mgn which may hardly be sufficient to calibrate very sensitive sensors. But at 0.1 Hz only 0.2 mgn could be provided which is almost nothing. To overcome this ‘lack of acceleration problem’ at very low frequencies, more stroke is needed. A long stroke calibration shaker like the APS 113AB which is usually used up to a stroke of 100 mm peak to peak for calibration purposes, can already provide a peak acceleration of 0.2 gn at 1 Hz. But at 0.1 Hz the peak acceleration is still only 0.002 gn (2 mgn). The acceleration amplitude scales linear with the stroke but it scales quadratically with the frequency. Thus to provide accelerations that are usually used in the medium frequency range, the maximum stroke of the vibration exciter would have to be in the range of some meters. It is obvious that such vibration exciters would need quite special mechanical solutions like an expensive linear motor drive that may also cause other mechanical problems. Such an exciter could not be practically used in a normal laboratory. Thus in practice compromise solutions like long stroke calibration vibration exciters with 100 mm stroke will be used which need to cope with the low accelerations.
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12 dB / octave
Fig. 2 Performance chart of a APS113 AB long stroke calibration exciter showing the 12 dB / octave decrease of the max. acceleration at low frequencies
Fighting Noise and Transverse Motion A Long Calibration Shaker needs to be designed carefully As shown in the section before even a special calibration vibration exciter for use in the very low frequency range will provide only low acceleration at lowest frequencies. Thus any kind of mechanical noise coming from the guidance of the coil and armature or other parts of the exciter must be avoided in order to achieve the best signal to noise ratio of the mechanical signal (movement). Since the long stroke does not allow using a spring guidance of the armature, one may think of using a ball bearing guidance of armature and coil instead. But ball bearings cause a lot of noise that can not only be heard but also measured with an acceleration sensor. Especially in the turning points of the movements stick-slip effects cause a lot of distortion (see fig.3).
Fig. 3 Stick-slip effects from ball bearings at the turning points of the movement increase distortion and noise in the mechanical signal of the exciter Air bearings reduce this distortion significantly. The total harmonic distortion of an air bearing vibration exciter like the APS129 is in the range of less than 2% (THD at 1 Hz and 110 mm p-p). Another source of distortion can be the rubber bands commonly used to hold the armature of the vibration exciter in a definite position. Due to the long stroke in the low frequency range such bands are extremely stretched and behave like nonlinear springs. The nonlinear behavior can cause a lot of harmonic distortion that may influence the measurement. This shortcoming can be avoided by using an electronic position controller that measures the current displacement of the armature by means of a position sensor and adds a DC voltage to the AC voltage from the vibration controller (see fig. 4). Thus the center position of the armature is controlled by the zero position controller and the characteristics of the controller can be programmed in such a way that it acts like a very soft rubber avoiding additional harmonic distortion.
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Fig.4 Working principle of an electronic zero position controller Another item that has to be carefully taken into account in the design of a vibration exciter is transverse motion. In fact transverse vibration is an unwanted but inevitable feature of all vibration exciters. It emerges from the fact that up to now, no one has succeeded yet in finding a design by which the movable armature of the vibration exciter is guided such that it can move with one degree of freedom only. But as already mentioned above the long stroke calibration vibration exciters used in the very low frequency range must also be able to carry high payloads since some seismic sensor can have a weight of some kilogram. Air bearings with tight gaps may be capable to carry such loads. But especially if the vibration exciter is operated in horizontal direction with a heavy DUT mounted directly on the armature, the huge overhang load tends to make the exciter movement instable and causes transverse motion. This transverse motion is not that much a problem at the lowest frequencies but in a range above some 10 Hz depending on the DUT mass. In order to avoid too high transverse motion the maximum payload for calibration operation should be limited to a certain value. If operated in vertical direction the payload can be well centered on the armature and the DUT applies a force in direction of the armature movement rather than perpendicular to the armature movement. Thus the payload limit can be several times higher in vertical operation than it can be allowed in horizontal operation without the risk of a worse measurement uncertainty. For example the APS113AB integrated in a SPEKTRA CS18 VLF calibration system allows a three times higher payload in vertical direction than operated in horizontal direction if the required measurement uncertainty shall be the same in both operation modes (900 gram in horizontal direction, 3 kg in vertical direction). But also a limitation of the upper calibration frequency depending on the DUT mass should be taken into account. Since especially some heavy triaxial seismic sensors require a calibration of two axes in horizontal direction perpendicular to the gravity field, vibration exciters for the calibration of such DUT’s need an improved design. A possible solution for such an improved design is an additional payload table separated from the electrodynamic drive and guided by its own air bearing (see fig. 5). Payload table and drive are mounted on heavy metal plate and can be adjusted very precisely in their relative position in order to reduce unwanted transverse motion. Since drive and payload table are mechanical coupled by a slender connecting rod also transverse motion coming directly from the armature of the drive is efficiently suppressed (see example measurement in fig. 6.).
Fig. 5 APS129 – Example for a calibration exciter capable
1525 to calibrate heavy DUT’s in horizontal direction with low transverse motion
Fig. 6 Measured transverse motion of an APS 129 vibration exciter with mounted dummy load The measurement results above show that the measured transverse motion stays over the whole interesting frequency range up to 200 Hz below the limits allowed by standard ISO 16063-21 (< 6% up to 20 Hz and < 10 % up to 200 Hz). But it can also be seen that due to the mass on top of the payload table (above the point where the drive force is applied to the table) the vertical transverse motion is increasing more than the horizontal transverse motion at higher frequencies. Transverse motion is depending on the mounted mass. In the low frequency range the measurement ends at 10 Hz. This is due to the fact that the triaxial sensors commonly used for such measurements allow only a limited measurement uncertainty in the very low frequency range. Thus the maximum transverse motion at lower frequencies was estimated from the mechanical parameters of the air bearings. Knowing the maximum gap of the air bearings and assuming that the operation exciter will be operated properly without damaging the bearings, the maximum tilting movement of the payload table can be calculated from the geometry of the bearings. Setting this movement in relation to regular the linear motion of the table the maximum relative transverse motion can be calculated. Furthermore it is assumed that the maximum transverse motion will appear if the exciter is operated at maximum stroke. Taking all that into account the calculated maximum transverse motion possible in the very low frequency range can be achieved. As can be seen in Fig. 7 it is far below the limits allowed by the ISO standard.
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Fig. 7 Calculated maximum transverse motion in the very low frequency range
Vibration Isolation Another source of unwanted vibration and noise is the table on which the vibration exciter is mounted. Since the natural frequencies of common ceilings in buildings are commonly in the range between 5 Hz and 50 Hz, they overlap with a part of the frequency range of a low frequency vibration calibration system. Thus to setup a calibration system on the basement floor is recommend. Since the vibration exciter will be used to drive some kilogram of mass, from a theoretical point of view it should be attached to a compact block. The weight (mass) of the block should be at least 2000 times the mass that is moved in the calibration process. If we assume that armature and calibration object have a maximum total weight (mass) of 1 kg, the result would be a block weighting 2 tons. Furthermore experience showed that the block should be set in the ground and completely decoupled from building (see Fig. 8). If this should not be possible and the calibration system has to be set up in an upper floor, the vibration exciter should be spring-mounted on a heavy-weight block. This spring-mass system shall be so designed that its natural frequency lies sufficiently far below the lowermost calibration frequency.
Fig. 8 Example of a concrete table with a flat granite slab on top set up in a hole in the basement floor decoupled from the building
Signal Processing As shown above even a long stroke calibration vibration exciter can only provide some mgn in the very low frequency range at 0.2 Hz or even less. Thus even a high sensitive reference accelerometer with a nominal sensitivity of 1000 mV / g will deliver an output signal of just some mV. An accurate measurement of such low and noisy signals is also a challenge. Normally the complete amplifier chain is made up of a high-pass filter, the cut-off frequency of which is switchable, a switchable amplifier and a low-pass (anti-aliasing) filter, the cut-off frequency of which is switchable,
1527 too. Its overall characteristics are dependent, therefore, on frequency, selected gain and setting of cut-off frequencies. Since these characteristics can be determined in an electrical calibration process of the system, the influence of the amplifier chain can be easily corrected. Using a high-pass filter at the beginning of the measurement chain has some pro and cons at very low frequencies. The advantage of such a filter is that trouble with a possible drift of the DC offset (e.g. due to thermal effects) of a sensor can be avoided. On the other hand the low input signal may be even more attenuated if the cut-off frequency is in the range of the lowest calibration frequencies. After an AD conversion of the input signal behind the amplifier filter chain, a further digital signal processing is applied. If the phase information can be disregarded, the data can be supplied to a digital root-meansquare calculator operating by the conventional principle (squaring, summation, extraction of root), maybe supplemented by a digital narrow-band filter and a band-stop filter. If the angle of phase difference is further needed, the sequence of samples must be processed using sine approximation according to ISO 16063-11. In this manner even very low signal voltages can be processed with small uncertainty.
Influence of the Operator In the end even the best designed calibration system is operated by a human being who can make mistakes. In fact in practice it turned out that many bad calibration results are not caused by a failure of the calibration system but by a mistake of the operator. In the field of low frequency vibration calibration especially the cable routing is a source of many mistakes. Due to the long stroke of the vibration exciter in the very low frequency range, an accurate routing is mandatory. Too short cables or thick and inflexible cables that influence the movement of the shaker can influence the measurement results significantly. Also too long cables hitting the table surface or cables, routed in a way that they are sharply bent due to the armature movement cause a lot of trouble. Thus in a reasonable measurement uncertainty budget of a very low frequency vibration calibration system the influence of the cable routing is one of the most significant parts especially at lowest frequencies.
Conclusions Accurate calibration in the very low frequency range below 1 Hz is a challenge due to the low available acceleration and the necessary long strokes of the vibration exciters. Modern calibration systems are capable to provide an excellent mechanical performance combined with precise electrical measurement of the sensor signals and subsequent digital signal processing. Thus a secondary calibration system like SPEKTRAs CS18 VLF specialized for the operation in this frequency range, can perform calibration at frequencies below 0.4 Hz with a still good measurement uncertainty of just 2.5%. Using a primary calibration system like the CS18 P VLF with a laser vibrometer as primary reference standard, the uncertainty decreases even to an excellent value of 1%. However, while the technical problems in this extreme frequency range can be solved, the operation of the system needs special care. Every mistake in cable routing or sensor mounting can have a huge influence on the calibration results. So operators should have a good technical education and sufficient experience in calibration to avoid the mistakes.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Comparative Analysis of Triaxial Shock Accelerometer Output Jacob C. Dodson, Lt. Lashaun Watkins, Dr. Jason R. Foley* Air Force Research Laboratory * AFRL/RWMF; 306 W. Eglin Blvd., Bldg. 432; Eglin AFB, FL 32542-5430, [email protected]
Dr. Alain Beliveau Applied Research Associates, Valparasio, FL NOMENCLATURE Number of data points Sampling frequency measurement in the ensemble Fourier transform of Energy spectral density of Averaged energy spectral density Spectral Energy Ratio of Spectral Energy ABSTRACT Shock accelerometer internal and mounting dynamics are analyized and the contribution to the sensor output is evaluated. This includes an analysis of uniaxial and triaxial accelerometer cross-talk (cross axis sensitivity effects), the filtering characteristics of polysulfide films, and the influence of triaxial block transient dynamic response on the shock accelerometer output. It is shown that the polysulfide acts as a lowpass filter and dissipates energy in the frequency range of sensor resonance. Features in the data, such as energy spectral density, cross axis sensitivity, and mode shapes of the triaxial block are highlighted. INTRODUCTION Shock accelerometers are essential for measuring the response of systems that undergo large mechanical shock, i.e., high amplitude impulsive loadings with short recovery time. These accelerometers are used to capture the dynamics of the objects under test however, both uniaxial and triaxial accelerometer outputs contain other dynamics as well. A typical triaxial accelerometer array is made up of three uniaxial accelerometers orthogonally mounted on a substructure such as a block. The output of such shock accelerometers consists of contributions from off-axis dynamic response, sensor resonance, mounting block dynamics, shock filtering materials, and environmental noise. Understanding sensor and mounting dynamics is vital to interpreting the recorded data of inertial sensors. When using shock accelerometers in applications where large amplitudes and broadband excitation is common [1] the accelerometer and mounting dynamics influence and can dominate the measurement of the dynamic structure. This paper focuses on the mounting and sensor dynamics that may affect the output of shock accelerometers in harsh broadband excitation environments. The three specific areas examined are the mitigating properties of polysulfide layers, the dynamics of triaxial blocks, and cross-axis sensitivity of the shock accelerometers. POLYSULFIDE FILTERING The dynamics of several mechanical isolators that contain polysulfide have been examined by Bateman, Brown and Nusser, however, the effect of just the polysulfide layers has not been analyzed in much detail [2, 3]. Winfree and Kang have examined shock mitigation through multiple different metallic layers and have also experimentally shown that the elastomer polysulfide reduces the spectral energy ratio in the frequency range of sensor resonance using a pendulum impact test [4]. In the method presented polysulfide mechanical filtering effects were evaluated through a test series using a direct impact Hopkinson bar, several different flyaways, and multiple layers of polysulfide. The polysulfide layers are evaluated to see how well they act as a high frequency filter, one that does not affect the low frequencies (below 100kHz), but filters the energy in the region of sensor resonance (700 – 900 kHz). T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_137, © The Society for Experimental Mechanics, Inc. 2011
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1530 Experiment The Hopkinson pressure bar is an experimental apparatus used to input reproducible controlled pressure waves, or elastic stress waves into a specimen located at the end of the bar. The direct-impact Hopkinson bar used in this study is located at the Shock Dynamics Laboratory of the Air Force Research Laboratory Munitions Directorate at Eglin AFB, FL. The bar used in the setup, shown in Figure 1 below, is 2 m long and 3.81 cm in 3 diameter and is ASTM Grade 5 (6% Al, 4%V) Ti alloy (ρ = 4.43 g/cm ,E = 114 GPa , ν = 0.33).
Figure 1: AFRL Hopkinson Pressure Bar setup The bar is used to transmit the pressure wave to the “flyaway” which is a 1.12 cm thick cylindrical disk of the same material and diameter of the bar. Accelerometers are mounted on the flyaway which is held in place by a vacuum chuck. The flyaway is released from the end of the bar upon the arrival of the first longitudinal stress wave which creates a single impulse on the flyaway and attached sensors. Three different flyaways were used: one that is flat with no pocket, the second with a pocket where the accelerometer under test was mounted and no polysulfide layers were used, and the third has a pocket such that a shock accelerometer was compressed between layers of polysulfide on either side of the sensor, all flyaways are shown in figure 2. Two piezoresistive Endevco model 7270A-60k shock accelerometers (60 kgn full scale, 700 kHz bandwidth) are mounted on each flyaway [5]. Data from both accelerometers is recorded on a Gagescope PC card at 125 MSa/sec. The output from the accelerometers are preamplified and conditioned by ADA-400A differential pre-amplifiers at full bandwidth (1Mhz). A set of four tests was completed, and for each test an ensemble of five measurements were recorded. The first three tests each used a different flyaway mounting for the accelerometers. The third test which used the flyaway with the compression mount for the accelerometer under test has one layer of polysulfide on both the top and bottom of the sensor, illustrated in Figure 2 (c), and the fourth test uses the same flyaway, but has 2 layers of polysulfide on both top and bottom of the sensor, which is illustrated in Figure 2 (d). Analysis The periodogram, also known as the power spectral density or the energy spectral density, is the spectral energy density of the
measurement of a signal
and is calculated by (1)
where is the Fourier transform of , is the sampling frequency, and the recorded signal [6]. The averaged energy spectral density is defined as
is the number of samples in
(2) where M is the size of the ensemble for each test. The signal energy as
over a certain frequency range is defined
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Figure 2: The three flyaways and four setups used in testing. (a) is the flyaway with no pocket or polysulfide, (b) is the flyaway with an accelerometer mounted in a pocket, but no polysulfide, (c) is the flyaway with the compression mount with one layer of polysulfide on either side of the pocketed accelerometer, and (d) is the same flyaway as (c) but with two layers of polysulfide on either side of the accelerometer. (3) and for the discrete signals equation (3) can be written as .
(4)
The spectral energy of both accelerometers over the range of the resonance frequencies on each flyaway was calculated for a series of 5 measurements. According to the datasheet the shock accelerometers are linear up to 100kHz [5], after which the sensor dynamics exhibit non-linear resonance. The averaged energy spectral density for each test is shown in Figure 3. Each sensor has two natural frequencies between 700kHz and 900kHz and can be seen in Figure 4. The resonance frequencies shift a small amount with the mounting on the different flyaways. The resonant frequencies can be seen in the averaged energy spectral densities for the frequency range of 700900kHz in Figure 4. The spectral energy ratio is given by (5) where is the spectral energy of the accelerometer under test, in the pocket or sandwiched between polysulfide layers, and is the spectral energy of the reference accelerometer [4]. To compare the filtering effect of the polysulfide layers the ratios of both the linear region (0-100kHz) and the frequency range where sensor resonance occurs (700 – 900 kHz) were calculated and are given in Table 1.
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Figure 3: Averaged Energy Spectral Densities
Figure 4: Averaged Energy Spectral densities in the frequency range of accelerometer resonance.
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0 - 100khz
700 - 900kHz
Side-by-Side
1.25
0.41
Pocket
1.01
0.64
1 Polysulfide layer
1.00
0.28
2 Polysulfide layers
1.00
0.26
)
An increase in the spectral energy ratio means that there is an increase in relative energy in the frequency range of the accelerometer under test. A decrease in the spectral energy ratio indicates a decrease in energy in the frequency bin of the accelerometer under test, or a filtering of the energy in that frequency range. The spectral energy ratios in the linear range of the accelerometers are all around 1, so the same amount of energy propagates to both accelerometers in all setups for the frequency range of 0 – 100kHz. It can also be seen that in the range of sensor resonance the pocketed flyaway increases the energy received by the accelerometer under test, while the polysulfide layers do attenuate the energy received by the filtered sensor. The two layers of polysulfide do increase attenuation, but only by an additional 5%. The polysulfide acts as a low-pass filter; while the cut-off frequency is unknown it is shown that the frequency range of resonance is filtered. TRIAXIAL MOUNTING BLOCKS The Munitions Directorate Shock Dynamics Laboratory uses titanium triaxial mounting blocks in measuring harsh multi-axial environments. The mounting blocks allow three uniaxial shock accelerometers to act as a triaxial sensor. Two types of triaxial mounting blocks are examined. The first is built for undamped shock accelerometers and includes the pockets for compression mounting with polysulfide layers, shown in Figure 5 (a). The second is intended to mount damped shock accelerometers can be seen in Figure 5 (b). Using a solid modeling program Solidworks COSMOS the natural frequencies and corresponding mode shapes of the solid models of the two triaxial blocks were computed. It was assumed that there is no relative motion between the bottom of the block and the mounting surface so in the simulation the bottom elements were fixed.
Figure 5: Titanium triaxial mounting blocks for (a) undamped shock accelerometers and (b) for damped accelerometers. The mode shapes for the first mounting block can be found in Figure 6 and the natural frequencies range from 45 kHz to 62.8 kHz, while the mode shapes for the second mounting block can be found in Figure 7 and the natural nd frequencies range from 115 kHz to 144kHz. It can be seen from the geometry of the blocks that the 2 triaxial
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Figure 6: Mode Shapes at the first five natural frequencies of the mounting block for undamped shock accelerometers. The mode shapes are for the corresponding natural frequencies (a) 45 kHz, (b) 50 kHz, (c) 52.8 kHz, (d) 61.3 kHz, and (e) 62.8 kHz.
Figure 7: Mode Shapes at the first five natural frequencies of the mounting block for damped shock accelerometers. The mode shapes are for the corresponding natural frequencies (a) 115 kHz, (b) 118 kHz, (c) 119 kHz, (d) 140 kHz, and (e) 144 kHz.
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1535 block is more rigid and the natural frequencies are expected to be higher. The mode shapes of the first triaxial block fall within the linear frequency range of the undamped shock accelerometer. It should be noted that the stiffness of the blocks will increase when accelerometers are mounted on the blocks, and when the covers to the pocketed block are put on. The increase in stiffness will increase the natural frequencies. Also the mode shapes shown of the second triaxial block may contribute to sensor case bending which will affect the measurement. While no definite conclusions may be drawn from this analysis, the natural frequencies and mode shapes provide frequency ranges to monitor when using the triaxial blocks for measurement. Future work will evaluate how much these modes affect the accelerometer measurement. While some of these modes are in the sensor’s linear frequency range, accurate measurements at that high of frequency range are difficult to measure accurately and the contribution may be difficult to see. FUTURE WORK: CROSS-AXIS SENSITIVITY To further characterize the dynamics of shock accelerometers, the cross axis sensitivity will be examined. There are a few existing measurement techniques of cross axis sensitivity in shock accelerometers. Bateman and Brown examined the cross-axis sensitivity of shock accelerometers using a beryllium Hopkinson bar technique [7, 8]. Sill and Seller more recently developed a transverse sensitivity measurement technique for accelerometers using planar orbital motion [9]. The method proposed expands the capability of the titanium Hopkinson bar located at the Shock Dynamics Laboratory. Two new flyaways have been designed that will allow the measurement of cross axis acceleration with one accelerometer and the in axis acceleration with a reference accelerometer. The two cross axis flyaways can be seen in Figure 8. The calculations of the cross axis sensitivity take into account the transverse motion of the flyaway as well as the radial acceleration due to Possion’s ratio of titanium. The cross axis sensitivity of damped and undamped shock accelerometers will be analyzed and compared. These experiments are planned and will be completed shortly.
Figure 8: Two cross axis flyaway designs (a) has only one cross axis accelerometer and (b) has two accelerometers to be mounted transverse to the acceleration. SUMMARY The results from a test series examining mounting and filter dynamics of shock accelerometers are presented. Polysulfide layers measurably attenuate the energy received by the filtered sensor, but the additional layers of polysulfide do not increase the attenuation linearly. The resonant frequencies of the mounting blocks may affect the accelerometers output, but the frequencies where the resonance occurs are above the easily measurable frequency range of the sensors. The sensor and mounting dynamics can greatly affect the sensor output and must be acknowledged and taken into consideration when analyzing shock accelerometer data.
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1536 AKNOWLEDGEMENTS J. D. would like to acknowledge support from the Department of Defense SMART (Science, Mathematics, And Research Transformation) scholarship program. The authors also wish to thank the Air Force Office of Scientific Research (PM: Dr. David Stargel) for supporting this project. Opinions, interpretations, conclusions and recommendations are those of the authors and are not necessarily endorsed by the United States Air Force. REFERENCES
[1]
[2]
[3] [4] [5] [6] [7]
[8]
[9]
Foley, J. R., Dodson, J. C., Schmidt, M., Gillespie, P., Dick, A., Idesman, A. and Inman, D. J., "Wideband Characterization of the Shock and Vibration Response of Impact Loaded Structures," in SEM IMAC XXVII, Orlando, FL, 2009 Bateman, V. I., Brown, F. A. and Nusser, M. A.,"High Shock, High Frequency Characteristics of a Mechanical Isolator for a Piezoresistive Accelerometer, the ENDEVCO 7270AM6", Sandia National Laboratory Report SAND2000-1528, 2000 Bateman, V. I., R.G., B., Brown, F. A. and Davie, N. T., "Evaluation of Uniaxial and Triaxial Shock Isolation Techniques for a Piezoresistive Accelerometer," in 61st Shock and Vibration Symposium, 1990 Winfree, N. A. and Kang, J. H., "Resonance Prevention of Accelerometers Using Multiple-Layer Rigid Filters," in 79th Shock and Vibration Symposium, Orlando, FL, 2008 "Model 7270A Accelerometer Data Sheet", Endevco Corporation, 2005 Hegge, B. J. and Masselink, G., "Spectral Analysis of Geomorphic Time Series: Auto-Spectrum", Earth Surface Processes and Landforms, Vol 21 No 11, pp 1021-1040, 1996 Bateman, V. I. and Brown, F. A.,"The Use of a Beryllium Hopkinson Bar to Characterize In-Axis and CrossAxis Accelerometer Response in Shock Environments", Sandia National Laboratory Report SAND97-2862, 1999 Bateman, V. I., Brown, F. A. and Davie, N. T., "Use of a Beryllium Hopkinson Bar to Characterize a Piezoresistive Accelerometer in Shock Enviroments", Journal of the Institute of Environmental Sciences, Vol 39 No 6 Nov/Dec, pp 33-39, 1996 Sill, R. D. and Seller, E. J., "Accelerometer Transverse Sensitivity Measurement Using Planar Orbital Motion," in 77th Shock and Vibration Symposium, Monterey, CA, 2006
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Strain Sensors for High Field Pulse Magnets
Christian Martinez1, Yan Zheng2, Daniel Easton3, Kevin Farinholt4, Gyuhae Park4 1
Dept. of Mechanical Engineering, Rice University, Houston, TX 77005 Dept. Civil Engineering, Georgia Institute of Technology, Atlanta, GA, 30332 3 Atomic Weapons Establishment plc, UK 4 Engineering Institute, Los Alamos, National Laboratory, Los Alamos, NM 87545 2
ABSTRACT In this paper we present an investigation into several strain sensing technologies that are being considered to monitor mechanical deformation within the steel reinforcement shells used in high field pulsed magnets. Such systems generally operate at cryogenic temperatures to mitigate heating issues that are inherent in the coils of nondestructive, high field pulsed magnets. The objective of this preliminary study is to characterize the performance of various strain sensing technologies at liquid nitrogen temperatures (-196oC). Four sensor types are considered in this investigation: fiber Bragg gratings (FBG), resistive foil strain gauges (RFSG), piezoelectric polymers (PVDF), and piezoceramics (PZT). Three operational conditions are considered for each sensor: bond integrity, sensitivity as a function of temperature, and thermal cycling effects. Several experiments were conducted as part of this study, investigating adhesion with various substrate materials (stainless steel, aluminum, and carbon fiber), sensitivity to static (FBG and RFSG) and dynamic (RFSG, PVDF and PZT) load conditions, and sensor diagnostics using PZT sensors. This work has been conducted in collaboration with the National High Magnetic Field Laboratory (NHMFL), and the results of this study will be used to identify the set of sensing technologies that would be best suited for integration within high field pulsed magnets at the NHMFL facility. 1. INTRODUCTION High field, multi-pulse magnets are very useful in materials and physics related research. These systems produce magnetic fields of 60-100 Tesla for periods of 15 ms or longer. Such fields are larger than those provided by continuous field magnets, yet without the single-shot limitations of high field, explosively driven destructive magnets. Continuous field magnets are capable of sustaining fields of 45 Tesla, whereas the recently developed 100T multi-shot system at the National High Magnetic Field Laboratory (NHMFL) at Los Alamos National Laboratory (LANL) has reach a record field of 90Tesla, and is currently being conditioned toward the operational goal of 100 Tesla. The 100T multi-shot magnet is composed of a two stage design. The outer stage of this system is composed of a series of six nested, resistive, solenoid coils that are driven by a 1,400 MW generator to provide platform fields of 40 Tesla. This outer stage surrounds an ‘insert’ composed of eleven superconducting coils that are powered by a 1.4 MJ, 18kV capacitor bank to produce a nominal field of 60 Tesla [1]. The duration of this pulsed field is 15ms, and is shown in Figure 1. A schematic of how the insert and outer stage are configured for the 100T TM system is shown in Figure 2. Each coil in the outer stage of this system is fit inside a Nitronic-40 stainless steel reinforcement shell that provides structural support due to the large radial forces that develop during each pulse cycle. Such forces often stress these reinforcement shells to their material limits with strains approaching 1% in the center coils. In addition to the severe electrical and mechanical
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_138, © The Society for Experimental Mechanics, Inc. 2011
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1538 *CourtesyofNHMFL Pulsedfieldprovidedbythe capacitorbankdriveninsert
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Figure 1: Magnetic field profile generated by the 100 Tesla multi-pulse magnet.
environments produced in these large field magnets, the system must also operate at cryogenic temperatures to address heating issues and promote the superconducting properties of the insert coils.
Insert
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1.1 Background In July 2000, the 60T long-pulse magnet at NHMFL underwent a catastrophic failure that destroyed the entire magnet. This failure occurred after performing only 914 pulses of the expected 10,000 pulse lifespan. The entire Figure 2: Schematic of the shell and coil magnet was destroyed in the resulting explosion, taking components of the 100 Tesla multi-pulse less than 5ms to release an estimated 8-80 GW of power in magnet. the process [2]. After extensive forensic analysis, it was determined that the cause of this destructive failure was the improper heat treatment of the reinforcement shells surrounding coils six and seven of the magnet (the 60T is composed of a series of nine nested, resistive, solenoid coils). As scientists and engineers worked to identify the cause of this failure and to develop safeguards to prevent future accidents, they also identified the need for integrated sensing capabilities to monitor the mechanical response of the magnet during operation. A series of subsequent magnets were designed and built to address the use of new design principles to improve reliability and avoid future failures. First, a 65 T magnet was created to introduce the concept of two, nested solenoid coils connected electronically in series. Next, a 75 T added a new Cu-Nb conductor to provide uniform distribution of the insert coil’s mid-plane stress and conductor heating. Subsequently, an 80 T was pushed to failure in which the phenomenon of axial electromagnetic buckling was discovered and winding the metal shells was found to alleviate this stress [3]. Presently, magnet operators utilize the electrical properties of the magnet coils to monitor residual strain within the magnet following each pulse. A near failure in the 100T magnet was recently avoided due to these measurements; however an official health monitoring approach has yet to be developed for these systems. The large magnetic field affects most traditional strain sensors, and the large temperature fluctuations cause adhesion and material fatigue issues. As this poses a significant design challenge for many sensing technologies, this paper focuses on temperature effects on fiber Bragg gratings (FBG), resistive foil strain gauges (RFSG), piezoelectric ceramics (PZT), and piezoelectric polymers (PVDF). A series of static and dynamic experiments were designed to investigate sensor bonding and response at liquid nitrogen temperatures (77 K). Several substrates are considered in this investigation, as well as methods for interrogating the bonding condition to delineate between sensor failure and structural failure.
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1.2 Project Description The purpose of the project is to investigate strain sensing techniques that can be used to monitor deformation within the steel reinforcing shells (Figure 2) of a pulse magnet during operation. Due to the quick nature of these pulses, strain sensors identifying mechanical anomalies are needed to detect the onset of, and potentially avoid, catastrophic failure. High-field pulse magnets operate at the yield point of these structural materials, making full comprehension of deformation paramount to understanding the system’s structural health. During use, the shells of the large magnet deform both axially and radially along the centerline of the structure. Currently, a resistive measurement method is being used to identify mechanical failure within the structure. However, this test gives little information as to the location of the failure as it is integrated over the length of each coil, making it difficult to monitor the life and reliability of individual components. This project works to identify appropriate adhesives that will bond to the stainless steel shells despite the large thermal gradient between cryogenic and room temperatures. It will also test the effect of the extreme temperature environment on RFSG, FBG, PZT and PVDF based sensors. By doing so, this project hopes to uncover a reasonable method of measuring strain within the magnet that can be easily implemented. 2. STRAIN SENSORS In this study four strain sensing technologies are investigated: traditional resistive foil strain gauges, fiber Bragg gratings, piezoelectric ceramics and piezoelectric polymers. While each of these materials has the ability to measure dynamic strain, the optical hardware used in this study was limited to dynamic ranges below 1Hz. Therefore, tests were divided into two categories: static strain measurements using RFSG and FBG sensors, and dynamic strain measurements using RFSG, PZT, and PVDF sensors. The following section provides an overview of each type of sensor and considerations regarding their use in this application. 2.1 Resistive Foil Strain Gauges (RFSGs) Foil resistive strain gauges are the most commonly used strain gauge types. RFSGs quantify strain by the measuring the mechanical deformation of a foil strip under strain. Under a tensile load, the crosssectional geometry of the foil strip is deformed, resulting in a change in the resistance of the foil. This change in resistance directly affects the measured voltage through the foil, and so the strain can be derived from this change [4]. As RFSGs use electrical properties of the foil to measure strain, they are highly affected by electromagnetic fields, which induce currents that are not related to strain. It is possible that once the induced current has been measured, it can be subtracted from future readings. As RFSGs rely on the mechanical properties of the foil, they are also highly susceptible to changes in temperature. Bridge circuits that normalize the output, taking temperature changes into account with a dummy gauge, can be used to compensate for this affect. 2.2 Fiber Bragg Grating (FBG) Fiber Bragg grating is an optical strain sensor that quantifies strain by measuring the wavelength of reflected light. The FBG is a small region of the optical fiber where the fiber core has been treated with a laser to produce a periodic modulation of its refractive index. The separation of each period of this modulation is referred to as the Bragg wavelength [5]. An infrared light is sent down the fiber optic strand and the strain is measured at the FBG sections of the fiber [6]. As the light from the LED hits the grating, only the wavelengths of light that correspond to the Bragg wavelength, that is the separation distances in the FBG, are allowed to be reflected back to the sensor. However, when the fiber strand is deformed as result of external
Figure 3: Schematic of how a Bragg grating functions.
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strain, the Bragg wavelength will be increased, allowing different wavelengths of light to be reflected back. The measured strain can be directly derived from the change in wavelengths of the light reflected from the Bragg grating. Different levels of strain applied to the fiber optic strand will allow a different reflected wavelength, which will correlate to the strain in the material to which the fiber is bonded [4]. When using these gauges, it is vital to ensure that the mounting is effective, in order to measure true strain on the surface of the material. Because of the importance of this aspect, different mounting techniques will be investigated, concentrating on those that can withstand the emersion in liquid nitrogen and the thermal cycling expected during operation. 2.3 Piezoelectric Strain Gauge An alternative method for strain measurement is the use of piezoelectric transducers. The molecular structure of piezoelectric materials provides a coupling of mechanical and electrical domains, meaning that the material produces a mechanical response to an applied electrical field, and vice-versa. [5] By measuring the charge generated in the piezoelectric material, the stress in the material can be calculated [7]. Piezoelectric materials can only be used for dynamic measurements, as in a static strained state they will have no electrical output. For this work, piezoelectric sheet materials will be used to measure strain. These gauges will measure the average strain across the region covered by the sensor’s surface area. Piezoelectric strain sensors have been shown to have advantages of compactness, sensitivity over a large strain bandwidth, lightweight, low power consumption, and ease of integration into structural components.[5,7] It has been demonstrated [5] that it is feasible to use piezoelectric strain sensing to detect both degradation of the mechanical and electrical properties of a piezoelectric gauge, and bonding defects between a piezoelectric gauge and the structure. 3. SENSOR DIAGNOSTICS One component of this study is the reliability of sensor self-diagnostics at cryogenic temperatures. Due to the active nature of piezoelectric sensors, they can be used to interrogate the bonding and condition of the sensor itself. Methods for determining the health of PZT transducers has been examined in previous research, however it has not yet been demonstrated at extreme temperature environments. Saint-Pierre et al. used the shape of the first real impedance resonance and its change to determine the state of the bonding condition [8]. Guirgiutiu and Zagrai proposed a similar technique using the attenuation of the first imaginary impedance resonance for damage detection [9]. Pacou et al. discussed the use of the shift of the first natural frequency of the piezoelectric patch before and after bonding as a possible method for determining bonding condition [10]. There are several disadvantages to the above methods. First, they are not sensitive to small debonding in the PZT. Secondly they all require relatively high frequency measurements to determine the bonded first natural frequency. A standard impedance analyzer can be used to make a measurement into the 600 kHz range and up, which is needed in these methods. This frequency requirement makes these techniques generally unsuitable for field deployment using currently available sensor nodes. Finally the absolute number of data points that need to be collected for sensor diagnosis is quite high compared to the method purposed in this paper. The higher number of data points demands more from the SHM node, which generally have very limited storage capacity and RAM. Bhalla and Soh [11] suggest that the imaginary part of the admittance measurement is more sensitive to bonding conditions, and therefore could be useful in determining the bond health. This relationship was further developed by Park et al. [7, 12], who showed the initial relationship between bonding condition and the slope of the imaginary admittance measurement (susceptance). This technique is based on the electrical admittance (Y) measurement of a free PZT
Y free (Z ) iZ
wl T (H 33 (1 iG )) , tc
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where Z is angular frequency, i is the complex number [-1]1/2, w is sample width, l is sample length, tc is thickness, H33T is strain at constaint stress and G is dielectric loss tangent of the PZT material. When bonded to a substrate the bonding layer’s effect on the electrical admittance takes the form,
Ybond (Z ) iZ
wl T (H 33 (1 iG ) d 312 YPE ) , tc
(2)
where d312 is the piezoelectric coupling constant and YPE is the complex Young’s modulus of the PZT material at zero electric field. Equations 1 and 2 show that the same PZT will have a different capacitive value from a free-free condition to a surface-bonded condition. The bonding of the sensor will cause a downward shift in the electrical admittance of the free PZT by a factor of
d 312 YPE .
(3)
This change in slope would allow for the health of the bonding condition and the physical health of the transducer to be assessed with a measurement of the susceptance [7]. 4. EXPERIMENTAL PROCEDURE Several experiments were conducted as part of this investigation. The overall objective was to identify how each strain sensor reacts to operating in a cryogenic environment, how sensitivities scale, and whether room temperature sensor diagnostic techniques remain valid when submerged in liquid nitrogen. The first series of experiments were conducted to expose beam specimen to an environment similar to that found in the 60T and 100T multi-shot magnets. Due to limited access to user magnets in the NHMFL facility some conditions could not be replicated identically, such as the large currents and magnetic fields supplied by the large scale magnets. Therefore the principal focus of this experiment was on sensor response in extreme thermal conditions. Initial consideration was given to understanding thermal shock effects on sensor response, however further examination indicated that the least intrusive point of instrumentation would be on the outer circumference of the reinforcement shells. And while the coils of the magnet can undergo large thermal cycles from 77K to ~ 200K during a 15ms pulse, the system is refilled with liquid nitrogen as soon as the magnet is de-energized. Initial calculations indicate that the outer surfaces of the reinforcement shells may only see a thermal change of 20°C between pulses, which in most applications is negligible with respect to sensitivity changes. Three separate experiments were conducted in this study. The first of these focused on the response of an instrumented fixed-fixed beam that was excited in the center of each sample (Figure 4). Each sample was 25.4mm x 304.8mm x 3.2mm in size with 254.0 mm between clamping fixtures. Three materials were considered for the test samples: 304 ShakerStinger stainless steel, 6061 aluminum, and carbon fiber. A center hole was drilled in each sample to attach Test a stinger that provided both static and dynamic Sample excitations. In the case of static deformations a 10-32 threaded rod was used as an adjustment screw to impose fixed deformations related to the PVDF PZT Sensor number of turns. The deformations that were Sensor imposed ranged from 0.8mm to 4mm for the samples instrumented with FBG and RFSG sensors. For dynamic measurements, the stinger was attached to a Labworks, Inc. ET-132 electroFixedͲFixedClamp magnetic shaker that applied chirp and harmonic signals to the test samples. Chirp excitations Figure 4: Test fixture used in fixed-fixed experiments. ranged from 10 to 1000 Hz, and were used to Samples were excited by an electromagnetic shaker. characterize RFSG, PVDF, and PZT sensors. In
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each series of experimental samples were first tested at room temperature, then were submerged in a bath of liquid nitrogen (LiN) and allowed to cool to 77K. Once the samples were cooled, the nitrogen bath was lowered and samples were tested as they were suspended approximately 30mm above the LiN. The support stand and full setup are illustrated in Figure 5. In addition to the fixed-fixed condition, samples were tested in a free-free boundary condition by removing the clamping fixture. These tests focused on the dynamic response of the RFSG, PVDF and PZT samples as static deformations could not be imposed using the current test stand. As in the previous series of experiments, samples were first tested at room temperature, then lowered into the LiN bath and allowed to cool, then tested as they were suspended just above the LiN bath.
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Figure 5: Support stand used to suspend the clamping fixture andsample.Testswereconductedatroomtemperatureandat cryogenic temperatures after the samples had been lowered intoaliquidnitrogenbath.
Throughout these experiments data was acquired using three separate analyzers. The static tests using FBG relied on a Micron Optics sm125 Optical Sensing Interrogator to monitor the fiber Bragg grating, while data from RFSGs was collected using an Agilent 34970A BenchLink Data Logger with a Vishay P-3500 Strain Indicator to provide the appropriate signal conditioning. For dynamic tests a Dactron Photon signal analyzer was used to excite the structure and collect time and frequency domain data. Samples were driven in the fixed-fixed and free-free experiments with a Labworks ET-132 electromagnetic shaker and PA-135 amplifier. The reference signal for each test was obtained from a PCB model 208C03 force transducer mounted between the shaker and stinger. ICP conditioning for the force transducer was supplied by the Dactron analyzer. Dynamic tests were conducted using a chirp excitation signal from 10-1000 Hz that was supplied over 80% of the acquisition window. Data was collected up to 500 Hz using a hanning window. Each measurement consisted of 20 linear averages, and the beam’s first natural frequency was identified and used as a harmonic excitation for the sample. This data was collected both at room temperature and LiN temperatures for each sample. During the liquid nitrogen experiments the beam fixture was submerged in the LiN for approximately 10 minutes to ensure that the entire sample had equilibrated to 77K. Initial tests were conducted at this point while the sample remained under the liquid nitrogen solution. It was realized after these initial tests that the fluid was serving to mass load tests samples and provide increased damping in the system. To address this issue the test procedure was altered such that the bath of liquid nitrogen was lowered after the 10 minute cool down cycle so that the test sample was located approximately 30 mm above the surface of the LiN. This allowed the sample temperature to remain near 77K throughout the testing, while eliminating mass loading and damping influences attributed to submersion in the liquid nitrogen. The final series of experiments conducted in this study considered the use of sensor diagnostics at cryogenic temperatures to interrogate bond health for piezoelectric ceramic sensors. In this test ten PZT sensors were bonded to an aluminum substrate using Stycast epoxy (Figure 6). One PZT sensor remained unbonded and served as the control sensor in this experiment. Three PZTs were adhered to the aluminum substrate using even pressure and even coverage of epoxy to serve as a good bonding condition. Three sensors were bonded with wax paper cover small regions of the sensor (~ 3/8, 1/2 and 5/8 of the overall area) to provide a partial bonding condition. The final three sensors were bonded completely to the structure, then cut with an abrasive wheel to emulate varying degrees of damage (1/8, 1/4, and 1/2 removal of material). The test plate was suspended from the same test stand used in the
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fixed-fixed and free-free experiments, and measurements were made at room temperature as well as in the LiN bath. The sensors were cycled through five cooling cycles, in which the plate was allowed to return to room temperature before being resubmerged in the LiN. Due to the high-frequency nature of the impedance measurements used in sensor diagnostics, it was deemed that any mass loading effects of being submerged in the liquid nitrogen would be negligible, as compared to the dynamic tests of the previous study.
Partial Bonding Good Bonding
Broken Sensors
Control Sensor (unbonded)
Figure 6: Test setup used to investigate sensor diagnostics at Data for this series of experiments was cryogenic temperatures. Ten sensors were investigated: one collected using an Agilent 4294A Precision Impedance Analyzer. control sensor, three with good bonding, three with partial Samples were interrogated with a 0.5V bonding,andthreethatwereintentionallybroken. chirp signal from 1,000 – 20,000 Hz, and the electrical impedance response was monitored. Two sets of data were taken for cycle to ensure that any anomalies can be identified during the testing procedure. Following the measurement the imaginary component of the admittance signal was extracted in MATLAB to study how the PZT bonding condition evolves with each thermal cycle . 5. EXPERIMENTAL RESULTS The experimental studies conducted as part of this investigation were carried out over a period of three weeks at the NHMFL facility at LANL. The first experiments were designed to identify the most reliable adhesive for use in subsequent experiments. Once the most suitable adhesive had been identified, the next series of experiments were focused on the performance of FBG and RFSG sensors given low frequency, pseudo-static loads. Following the static study, dynamic tests were performed to evaluate the RFSG, PVDF, and PZT sensors. The final series of experiments consider the application of sensor selfdiagnostics to PZT sensors at cryogenic temperatures.
5.1 Adhesion at Cryogenic Temperatures The criteria for a suitable epoxy proved to be quite specific, as an epoxy withstanding cryogenic temperatures is crucial. Additionally, bond integrity when submerged in a liquid environment was also necessary. Figure 7 illustrates the dynamic response of RFSGs bonded to a 304 stainless steel substrate using two of the principal adhesives considered in this study: MBond 200 and Stycast ES-2-20 epoxy. MBond adhesive was selected due to its wide use in strain gauge mounting, whereas the Stycast epoxy was selected due to recommendations from staff at the NHMFL facility. The benefit of the Stycast epoxy at cryogenic temperatures is that it is doped with fine aluminum particulates that provide thermal expansion properties which are consistent with metallic substrates. These thermal expansion properties help to limit some of the debonding issues observed with other adhesives. One issue that was observed with the Stycast epoxy was a relatively high viscosity relative to the baseline MBond adhesive. From the results shown in Figure 7, it is seen that the samples prepared with Stycast were not adversely affected by the higher viscosity, as the sensitivities are nearly identical with those of the MBond samples. One issue that was evident throughout these experiments was the need for a systematic bonding procedure, particularly for samples with polymeric coatings, such as the PVDF sensor shown in Figure 8. The bonding procedure that was developed as part of this study was to roughen the sensor and substrate surfaces to be bonded, cleaning the surfaces with alcohol prior to bonding, allowing the surfaces to dry, and then applying the adhesive and uniform pressure over the sensors’ entire area for several hours.
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Figure7:DynamicresponseofRFSGsmountedusingMbondandStycastESͲ2Ͳ20epoxyatroom(left)and cryogenic(right)temperatures.Bothadhesivesperformedwellwhensubstrateswereproperlyprepared. Additionally, the sensor wires were taped to the substrate to provide a strain relief to further protect the sensors when submerged in the liquid nitrogen bath. This bonding procedure was utilized for each of the flat sensors used throughout the remainder of these experiments. 5.2 FBG and RFSG Response to Static Deformations Due to bandwidth limitation with the Micron Optics sensing equipment, FBG experiments were limited to pseudo-static deflections below 1 Hz. For comparison purposes a resistive foil strain gauge was mounted on the opposing face of the 304 stainless steel sample beam as shown in Figure 9. Room temperature tests were first performed to compare the sensitivity of the FBG with that of the RFSG, using an adjustment screw to impose pseudo-static deformations from 0-4mm in 0.8mm intervals. The results of this room temperature test are shown in Figure 10. There is good general agreement in the shape of the FBG and RFSG responses; however there are some discrepancies when the sample is relaxed and as it nears a peak deformation of 4mm at the center of the sample. Several possible sources for the deviation between the RFSG and FBG are bond integrity/relaxation and the additional strain imposed by solder joints and wiring on the RFSG. Further testing will be needed to identify the source of this variation between the two sensors at room temperature. The more interesting result; however, is the RFSG response at cryogenic temperatures. As these tests were not to be conducted at an appreciable strain
Figure8:DebondingobservedinPVDFsensors duringtesting.
Figure 9: Resistive foil strain gauge and fiber Bragg gratingmountedtothestainlesssteelsubstrate.
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As part of this test, samples were evaluated in three conditions, 1.) room temperature (in open air), 2) submerged below the liquid nitrogen bath, and 3.) suspended 30mm above the LiN bath after it had been submerged for 10 minutes. The results of this comparison are presented in Figure 12. It is clear that the LiN served to mass load the test sample by st lowering the 1 natural frequency 15%. It is also evident that the fluid served to increase damping in the system as evident in the frequency response results of Figure 12. When removed from the LiN bath the amplitude of the 1st natural frequency returned to a level near what was measured at room temperature for the PZT and RFSG samples, however this was not the case for the PVDF sensor. This deviation in the PVDF sensor is attributed to severe debonding in the sensor, as illustrated in Figure 8. In addition to the loss in performance of the PVDF sensor, it is also noted that the natural frequency at cryogenic temperatures is actually lower than that for the room temperature results. This is counterintuitive to the fact that the substrate contracts at colder temperatures, and thus should exhibit an increase in the first natural frequency. Further investigation indicates that this lowering in frequency can be attributed to the clamping mechanism used to apply the fixed-fixed boundary conditions. It was observed during testing that the clamping condition actually loosened upon cooling, thus relaxing the boundary condition and thus causing the natural frequency to decrease.
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illustrated at both room and cryogenic temperatures. It is evident from the results in Figure 11 that the PZT sensor provides the largest sensitivity to strain, followed by the PVDF and RFSG sensor. Limitations in the RFSG at higher frequencies may, however, be attributed to the Vishay signal conditioning hardware that was being used. It can also be seen from the figure that cryogenic temperatures influence the sensitivity of each transducer, particularly near the first natural frequency of the fixed-fixed beam sample. In addition, initial tests demonstrated the influence of the liquid nitrogen on test results.
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To investigate this issue further a series of free-free experiments were conducted to examine how the boundary condition influences the system’s first natural frequency. For the free-free experiment the sample was mounted to the stinger attachment through the center hole as shown in Figure 13. The results of this study for the 304 stainless steel sample are shown in Figure 14. As stated previously, a chirp signal from 10-1000 Hz was used to excite the beam at room temperature. In Figure 14, the graph depicts a very slight difference in the frequency of the first bending mode, though the second bending mode is less apparent near 600 Hz in the fixed-fixed structure. From st this response it can be seen that the free-free condition has a lower 1 natural frequency as expected, however it is only a reduction of 3%, whereas the cryogenic temperature produced a reduction of approximately 5%. Further tests will need to be conducted to determine why the cryogenic temperature
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produces such a noticeable shift in the natural frequency, particularly since the shift is larger than that observed between the fixed-fixed and free-free conditions. 5.4 Sensor diagnostics Another advantage of the PZT sensor, in addition to its high sensitivity, is that it can be used to apply high frequency structural health monitoring (SHM) approaches to a structure, as well as its ability to interrogate the bond integrity between sensor and substrate, a process referred to as sensor self-diagnostics. In SHM applications it can be difficult to ascertain whether an anomalous result is related to structural or sensor related damage. The objective of performing this series of initial sensor diagnostics experiments is to characterize the response of likely modes of damage in sensors at room and cryogenic temperatures. The results of these tests can later be used to distinguish between sensor and structural damage should piezoelectric sensors be selected as possibilities for monitoring the health of pulsed magnets. This application is especially relevant to the measurement of strain in high field pulse magnets as any potential sensor network embedded within a high pulse magnet for SHM will likely be inaccessible, and the cycling of a cryogenic harsh environment will likely make sensor damage and debonding a real issue. As described in the experimental procedure, an array of nine PZT sensors was mounted on a 5mm thick aluminum plate. The patches are divided into three groups of three, each bonded using the Stycast ES-2-20 epoxy. The first series of three gauges were all properly bonded to the substrate. The second series of three gauges were partially debonded, using wax paper under half the gauge so that the epoxy only adhered to a portion of the sensor. The third series of gauges were intentionally damaged; as an abrasive cutting wheel was used to cut the gauges at 1/4, 3/8, and 1/2 diameters across. A tenth gauge was used as a baseline, and was not bonded to the plate. Figure 15
Figure15:ElectricaladmittancemeasurementsofafreePZT sensorbefore,during,andaftersubmersioninliquidnitrogen.
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Figure16:Electricaladmittancemeasurementsofsensorsthatwerecompletelybondedtothealuminumplate. presents the results for the baseline sensor as the PZT was cycled from room temperature to 77K and back to room temperature. It is evident that the cryogenic environment greatly influences the sensitivity of the PZT as indicated by the significant change in slope; however, it is most important to note that there is a residual change in the room temperature response of the piezoelectric. Bond integrity is denoted by the slope of this admittance curve, and would need to be monitored relative to a baseline measurement to properly account for the temperature induced shift that is apparent in Figure 15. In general terms, the unbonded piezoelectric sensor will have the greatest slope as the slope is related closely to the capacitance of the PZT. As the sensor is bonded to a substrate, the electrical impedance becomes coupled to the mechanical impedance of the structure through the bonding layer. The mechanical boundary condition imposed by the structure through the bonding layer serves to reduce the electrical capacitance of the PZT, resulting in a decrease in the slope of the admittance. If a sensor is properly bonded to a structure, then a positive change in slope will be indicative of debonding, as the sensor begins to gain capacitance due to a relaxation in the mechanical boundary condition. Conversely, damage to the sensor will reduce the overall physical size of the remaining ‘operable’ region, resulting in a loss of capacitance. Thus, damage to the sensor will generally be reflected as a decrease in slope as the overall electrical capacitance drops. The three sample sets used in this study were configured to help illustrate this behavior, and to prove the technique’s effectiveness at cryogenic temperatures. The first series of piezoelectrics (sensors 1-3) were applied to the aluminum substrate using consistent bonding techniques, and are considered to constitute a ‘good’ bonding condition. Figure 16 presents the admittance curves for each of the sensors throughout the five cycles between room and cryogenic temperatures. The initial room temperature measurements throughout this discussion are presented as the blue set of curves. It is apparent from these results that the residual shift in slope continues to accumulate throughout the subsequent cooling cycles. While this is a feature that appears to be related to the sensor itself, it is a behavior that will need to be investigated further before the long term deployment of these sensors could be undertaken in applications where large thermal cycles are expected. One of the more interesting results of Figure 16 relates to sensor 3. Based upon the response it is seen that the sensor underwent some debonding during the first warming cycle as the plate was brought back to room temperature following the first submersion in LiN. This debonding is evident in the positive change in slope; however, following this first instance of debonding, the sensor behaved consistently throughout subsequent cooling cycles. Additionally, the admittance response to debonding is apparent in both room
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Figure17:Electricaladmittancemeasurementssensorsthatwerepartiallybondedtothealuminumplate. temperature and cryogenic data sets, as the positive change in slope is present in both temperature regimes. The second series of PZTs (sensor 4-6) were partially bonded to the aluminum substrate using wax paper to prevent adhesion of 1/4, 1/2, and 3/4 of the sensor diameter. This change in bonding area can be seen in the solid blue admittance curves of Figure 17. One interesting feature of this series of curves is that the residual shift in admittance values is in the positive direction, rather than the negative direction as seen in the baseline sample. This behavior is attributed to the fact that the partial bonding condition serves to constrain the PZT in a pseudo-cantilevered configuration. In such a configuration it is assumed that the impedance measurement itself causes the sensor to deform as a voltage signal is applied to the PZT, progressively deteriorating the bond, giving rise to subsequent debonding with each measurement at the interface with the wax paper. X-ray or imaging analysis should be conducted to provide support of this assumption. In addition to the residual changes in the admittance curve, it is also interesting to observe the change in response exhibited by sensor 4 in Figure 17. During the first submersion cycle the sensor behaves as the other two sensors do; however, on the second submersion it is seen that the sensor suffered some damage. Upon completion of the tests, visual inspection found that there was a significant crack that developed in the sensor along the solder joints of the PZT as seen in Figure 18. And, as in the case with sensors 1-3, this characteristic was observed in the data collected at room and cryogenic temperatures. The final series of PZTs (sensors 7-9) were bonded to the substrate in a consistent manner, and then damage was induced using an abrasive cutting tool. Varying degrees of
Evidenceof Crackingin Sensor4 Figure18:ImageofSensor4followingthe damageinducedduringthecoolingcycle.A crack is evident just above the solder joints.
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Figure19:Electricaladmittancemeasurementsforsensorswithvaryinglevelsofimposeddamage. damage were inflicted on the sensors, detaching 1/4, 3/8, and 1/2 diameter sections. The results obtained from these damaged sensors are shown in Figure 19. It is evident from this figure that the more significant damage produces a larger reduction in the slope of the electrical admittance as is expected from Equations 1-2, and this reduction in slope can be seen in both room temperature and cryogenic datasets. Another feature of this data that should be commented on is the residual increase in slope throughout the cooling cycles. As in the previous case this is counter to what was observed with the baseline PZT, however it is attributed to changes in the bond layer introduced when damage was inflicted on the samples. While this is currently only a hypothesis of what happened, future tests in which damage is inflicted before bonding should help to clarify what is being seen. 5. SUMMARY AND CONCLUSIONS
In this investigation a series of strain sensing technologies have been investigated under cryogenic conditions. The goal of this study is to aid in the identification of a suitable sensing approach that can be integrated within future designs of the high field pulsed magnets at the National High Magnetic Field Laboratory at Los Alamos National Laboratory. Tests indicated that fiber optic sensors provide clear strain signals with little noise when submerged in liquid nitrogen. The fiber optic interrogation hardware used in this experiment was limited to dynamic ranges below 1Hz, and future tests should incorporate more capable hardware that can achieve the higher bandwidths (> 1000Hz) needed to identify anomalies during the operation of pulsed magnets. Piezoelectric ceramics were seen to provide the highest sensitivity to strain of the sensors used in this study. They also provide a much larger dynamic range, and the ability to implement structural health monitoring algorithms and sensor diagnostics techniques to provide high resolution inspection of reinforcement shells as well as the bond integrity between the sensor and substrate. PVDF sensors also provided high sensitivity measurements; however they were generally plagued with bonding issues, as the mylar protective coating was difficult to adhere to, and submersions in liquid nitrogen tended to cause sensors to debond. While this study has provided a first step in identifying suitable sensors for the NHMFL application, additional work must be undertaken to understand how each sensor responds to the large magnetic fields and eddy currents induced during the pulse cycle of these large scale magnets. Electromagnetic interference may prove to be the deciding factor in selecting a suitable sensing technology. Future studies will be designed to focus on these issues, and to select the most appropriate sensing technology.
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ACKNOWLEDGMENTS The authors would like to thank the Los Alamos Dynamic Summer School and Dr. Charles Farrar for the opportunity to participate in this research program. We would also like to thank Vibrant Technologies, SIMULIA, and Enthought for providing software used in modeling and data analysis. The authors would like to thank James Sims, Curtt Ammerman, Gretchen Ellis, and Josef Schillig of Los Alamos National Laboratory for their input and guidance as part of this research. Finally, we would like to thank Charles Mielke and Jonathan Betts of the National High Magnetic Field Laboratory for providing assistance and access to their laboratory facilities. The experiments carried out at the NHMFL were done under the auspices of the National Science Foundation, the State of Florida and the US Department of Energy. References
1. Sims, J.R., Rickel, D.G., Swenson, C.A., Schillig, J.B., Ellis, G.W., and Ammerman, C.N., “Assembly, Commissioning and Operation of the NHMFL 100 Tesla Multi-Pulse Magnet System” in IEEE Trans. Appl. Supercond., vol. 18, NO. 2, June 2008 2. Sims, J.R., Schillig, J.B., Boebinger, G.S., Coe, H., Paris, A.W., Gordon, M.J., Pacheco, M.D., Abeln, T.G., Hoagland, R.G., Mataya, M.C., Han, K., and Ishmaku, A., “The U.S. NHMFL 60 T Long Pulse Magnet Failure” in IEEE Trans. Appl. Supercond., vol. 12, no. 1, March 2002. 3. Swenson, C.A., Gavrilin, A.V., Han, K., Walsh, R.P., Schneider-Muntau, H.J., Rickel, D.G., Schillig, J.B., Ammerman, C.N., and Sims, J.R., “Performance of 75 T Prototype Pulsed Magnet” in IEEE Trans. Appl. Supercond., vol. 16, no. 2, June 2006 4. Evans, J. E., Dulieu-Barton, J.M., Burguete, R.L., “Electrical Resistance Strain Gauges,” Modern Stress and Strain Analysis, pp. 4,5. 2009. 5. Evans, J. E., Dulieu-Barton, J.M., Burguete, R.L., Optical Fibre Bragg Grating Strain Sensors,” Modern Stress and Strain Analysis, pp. 2,3. 2009. 6. Kashyap, R., “Fiber Bragg Grating Band-Pass Filters” in Fiber Bragg Grating, no. 2 pp. 237-245, 2004. 7. Park, G., Farrar, C.R., Rutherford, A.C., Robertson, A.N., “Piezoelectric Active Sensor SelfDiagnostics Using Electrical Admittance Measurements” Journal of Vibration and Acoustics AUGUST 2006, Vol. 128 / 469 8. Saint-Pierre, N., Jaye, Y., Perrissin-Fabert, I., and Baboux, J. C., “The influence of bonding defects on the electric impedance of a piezoelectric embedded element,” Journal of Physics D: Applied Physics 29, pp. 2976–2982, December 1996. 9. Giurgiutiu, V., Zagrai, A. N., “Embedded self-sensing piezoelectric active sensors for on-line structural identification,” Transactions of the ASME 124, pp. 116–125, January 2002. 10.Pacou, D., Pernice, M., Dupont, M., and Osmont, D., “Study of the interaction between bonded piezoelectric devices and plates,” in 1st European Workshop on Structural Health Monitoring, (155), 2002. 11.Bhalla S., and Soh, C.K., “Electromechanical impedance modeling for adhesively bonded piezotransducers,” Journal of Intelligent Material Systems and Structures 15, pp. 955–972, December 2004. 12.Park, G., Farrar, C. R., di Scalea, F. L., and Corria, S., “Performance assessment and validation of piezoelectric active-sensors in structural health monitoring,” Smart Materials and Structures 15, pp. 1673–1683, December 2006.
BookID 214574_ChapID FM_Proof# 1 - 23/04/2011
BookID 214574_ChapID 139_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Wireless Noise and Vibration Management System for Construction Sites Kurt Veggeberg, Business, Development Manager ([email protected]) National Instruments,11500 N. Mopac C, Austin, TX 78759 ABSTRACT - Construction often generates community noise and vibration complaints particularly in established urban areas. This is an overview of a portable noise and vibration management system with multiple front ends connected wirelessly throughout a construction site to be compliant with local regulations and contractual obligations for noise and vibration levels. The acoustic requirements and analysis of a system can be computationally intensive and are becoming more stringent. Construction sites can generate vibration levels which can cause human annoyance in the audible and tactile range and possible building damage. The architecture implemented in this study consists of multiple noise and vibration monitoring stations or clients with acquisition of noise and vibration from a microphone, accelerometer and a geophone, a central server for supervision, and a database for post processing of the data. It is designed to distribute the various tasks of monitoring for maximum flexibility and performance. The monitoring stations are autonomous with respect to the central server allowing critical tasks such as alarms to be carried out locally in the event of a disruption of the network. Introduction
Construction often generates community noise and vibration issues. Construction sites can generate levels of vibration that can cause human annoyance and discomfort and possible building damage. Research on human response to vibrations suggests that people have an annoyance threshold far lower than any building’s susceptibility to damage, even under the worst of circumstances, which is why it is important to have a system in place to monitor the effect of construction on the surrounding environment. In France, as many other countries, legislation like the Public Health Code (Décret n° 2006-1099) or the German DIN 4150-3 Vibration in Buildings, regulates noise and vibration from construction sites. The construction company may also be subject to contractual obligations for monitoring. This makes it important to monitor noise and vibration to avoid misunderstandings and maintain the tranquility of the neighborhood by identifying problems in a timely fashion. A key element is that the environmental monitoring has to establish a level for the ambient for the site before the construction is introduced. Legislative Requirements for Monitoring The acoustic requirements and analysis of a system for construction site management can be computationally intensive and are becoming more detailed. This can require more advanced instrumentation, data logging, analysis and reporting. When the Public Health Code on Neighborhood Noise in France was updated in 2006, limits were set on overall sound levels of no more than 5 dB(A) over the ambient during the daytime from 7 am to 10 pm and 3 dB(A) at night from 10 pm to 7 am. [1] Measures of ambient noise have to be made for at least 30 minutes and base figures have to be established for daytime and nighttime operation. There are also specifications and adjustments for the duration of noise. [Table 1]
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_139, © The Society for Experimental Mechanics, Inc. 2011
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T <= 1 minute 1 minute < T <= 5 minutes 5 minutes < T <= 20 minutes 20 minutes < T <= 2 hours 2 hours < T <= 4 hours 4 hours < T <= 8 hours T > 8 hours
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There are also new specifications on levels in 1/3 octave bands. For example, in the octave bands centered around 125 Hz and 250 Hz, the level over ambient noise in those frequency bands is 7 dB(A) and 5 dB (A) centered on the octave bands around 500 Hz, 1000 Hz, 2000 Hz and 4000 Hz. Construction sites can generate vibration levels which can cause human annoyance in the tactile range and possible building damage. A variety of construction machinery used for soil excavation, modification and improvement are a source of noise and vibration. In France, the vibration from construction equipment use is governed by legislation from 1986 relating to mechanical vibrations emitted into the environment. This regulation sets the threshold based on the environment before the construction begins. Depending on the situation and the criticality of the site, other constraints can be incorporated into construction permits, such as the inclusion of human vibration perception thresholds by man, or verification of the absence of prominent tones. These vibration measurements are typically made in accordance with ISO 2631 for whole body vibration. ISO 2631-2:2003 concerns human exposure to whole-body vibration and shock in buildings with respect to the comfort and annoyance of the occupants. It specifies a method for measurement and evaluation, comprising the determination of the measurement direction and measurement location. It defines the frequency weighting Wm which is applicable in the frequency range 1 Hz to 80 Hz where the posture of an occupant does not need to be defined. [2] Vibration standards come in two varieties; those dealing with human comfort and those dealing with cosmetic or structural damage to buildings. In both instances, the magnitude of vibration is expressed in terms of Peak Particle Velocity (PPV) and millimeters per second (mm/s). These guidelines for vibration relate to relatively modern buildings and are normally reduced to 50% or less for more critical buildings. Critical buildings include premises with machinery that is highly sensitive to vibration or historic buildings that may be in poor repair, including residential properties. Typical guidelines state that that there should typically be no cosmetic damage if transient vibration does not exceed 15mm/s at low frequencies rising to 20mm/s at 15Hz and 50mm/s at 40Hz and above. These guidelines relate to relatively modern buildings and are normally reduced by 50% or less for more critical buildings. Critical buildings include premises with machinery that is highly sensitive to vibration or historic buildings that may be in poor repair, including residential properties. The German standard DIN4150-3 Vibration in Buildings: Effects on Structures, provides limits below which it is very unlikely that there will be any cosmetic damage to buildings. For structures that are of great intrinsic value and are particularly sensitive to vibration, transient vibration should not exceed 3mm/s at low frequencies. Allowable levels increase to 8mm/s at 50Hz and 10mm/s at 100Hz and above. [3]
The Solution SAVE (Surveillance of Acoustics and Vibration in the Environment) was conceived by dB Vib Consulting and implemented by SAPHIR of France as a noise and vibration management system for construction site managers. It is based on National Instruments LabVIEW and the Sound and Vibration Measurement Suite, with multiple
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1555 laptop based front ends using a Compact DAQ dynamic signal acquisition system to interface to an outdoor Type 1 level microphone, accelerometer and geophone. These are placed throughout a construction site or a suitable representative of the nearest structures not connected to the site. They are then connected wirelessly via 801.11g incorporated in the laptop to a central server to insure compliance with local regulation and contractual obligations. The Architecture of SAVE The SAVE architecture consists of multiple noise and vibration monitoring stations or clients, a central server for supervision and a database for post processing of the data. It is designed to distribute the various tasks of monitoring for flexibility and performance. The monitoring stations are autonomous with respect to the central server allowing critical tasks such as alarms to be carried out locally in the event of a disruption of the network. The noise and vibration monitoring stations contain a laptop computer in a weatherproof case connected wirelessly to a central server. This is connected to a NI-9234 24 bit Dynamic Signal Analyzer with the proper signal conditioning for providing power for IEPE devices via a USB carrier for the acquisition of noise (sound level and 1/3 octave analysis) and vibration from an outdoor microphone, accelerometer and a geophone. A NI-9481 High Voltage Relay output module is used for the activation of audio and visual alarms for immediate attention. [Figure 1]
Figure 1.Components of SAVE system The central server is connected to the monitoring stations throughout the construction site in locations sensitive to noise and vibration. The system automatically detects and configures the monitoring stations, allows display of the data and alarms of several stations, downloads the files from the monitoring stations and allows remote
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1556 access and the sending of alarms or notices to remote locations. The database allows post-processing of the data files for different applications and analysis. [Figure 2]
Figure 2.SAVE distributed throughout construction site. The SAVE software was developed in LabVIEW with the Sound and Vibration toolkit for the front ends, central server and database due to its compliance with international standards for sound level measurements, weighting filters, octave analysis and human vibration measurements according to IEC 61260 (Electroacoustics - Octaveband and fractional-octave-band filters), IEC 61672 (Electroacoustics - Sound Level Meters), and ISO 2631(Mechanical vibration and shock -- Evaluation of human exposure to whole-body vibration). Wireless Technology Significant time and money has been invested into researching the use of wireless technology for remote monitoring. Yet, significant wireless deployments are just beginning to materialize in industry such as construction site management and building acoustics where running wires can be difficult. There are many advantages to eliminating cables in remote monitoring applications, but there are also many challenges. As standards such as Wi-Fi (IEEE 802.11) continue to mature, those challenges are being addressed. IEEE 802.11 has a variety of advantages for remote data acquisition and data streaming for dynamic signal acquisition as compared to other standards such as IEEE 802.15 (Zigbee) including range and security. IEEE 802.11 typically operates on 2.4 Ghz and 5 Ghz. It is typically specified with a range from 30 to 100 meters with data rates from 54 to 600 Mbps. The range depends on a variety of factors and can be extended significantly through a variety of network topologies and high gain antennas. In this particular application, a star topology was used with routers, switches, wireless access points, and wireless distribution systems (access points repurposed as repeaters) providing adequate coverage for a typical site. Off the shelf mesh access points can extend this further. Security is the number one concern for many engineers and scientists considering wireless. The reasoning behind this is due in large part to the failings of early wireless standards such as wired equivalent privacy (WEP), which did not prevent unauthorized access well. There are two main components of network security that must be addressed before wireless is widely adopted: authentication and encryption.
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1557 A wireless network is inherently more accessible than a wired network (such as Ethernet) because it is not a closed system: data travels through the air. IEEE 802.11X has evolved to provide authentication on wireless networks based on the Extensible Authentication Protocol (EAP). Clients on the network must identify themselves before being granted access to the network. There are other less sophisticated strategies for preventing unauthorized network access as well. Good security practice for wireless networks includes MAC and/or Internet Protocol (IP) address filtering and service set identifier (SSID) suppression. Even if data is accessible to an unauthorized user, it is not necessarily intelligible. Data encryption on wireless networks has evolved significantly over the last decade from clear-text broadcasts to 128-bit cryptography. The Advanced Encryption Standard (AES) is now an NIST standard and a requirement for all U.S. government installations. The advantages of using a standards-based wireless network include lower costs, interchangeable products from different suppliers, and established best practices. Also, standards such as IEEE 802.11 are readily incorporated with existing Ethernet-based systems. The IEEE ratified IEEE-802.11n-2009 on September 11, 2009 which offers significantly improved data rates (4 – 5 faster than 802.11g) and ranges for wireless local area networks with potential up to 100 Mbps comparable to wired networks. [4]
Conclusion The SAVE system was designed to distribute the various tasks of monitoring for maximum flexibility and performance using wireless networks. The monitoring stations are autonomous with respect to the central server allowing critical tasks such as alarms to be carried out locally in the event of a disruption of the network. The capabilities of the architecture of SAVE for monitoring and archiving make it adaptable to other applications where continual monitoring of noise and vibration needs to be implemented in a distributed environment.
Acknowledgements Thanks to Jean-Michel Chalons of Saphir and dB Vib for sharing information about their application.
References [1] Décret n° 2006-1099 du 31 août 2006 relatif à la lutte contre les bruits de voisinage et modifiant le code de la santé publique (dispositions réglementaires), JORF n°202 du 1 septembre 2006. [2] Deutsches Institute Fur Normug E.V. (German National Standard), DIN 4150-3 Vibration in Buildings – Part 3: Effects on Structures, 01 February 1999. [3] International Organization for Standardization, ISO 2631-2:2003 Mechanical Vibration and Shock – Evaluation of Human Response to Whole Body Vibration – Part 2: Vibration in Buildings (1 hz to 80 hz)., ISO 2003. [4] IEEE Ratifies 802.11n, Wireless LAN Specification to Provide Significantly Improved Data Throughput and Range, Piscataway, NJ, IEEE, 11 September 2009.
BookID 214574_ChapID FM_Proof# 1 - 23/04/2011
BookID 214574_ChapID 140_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Estimation of Rigid Body Properties from the Results of Operational Modal Analysis
A. Malekjafarian, M.R. Ashory, M.M. Khatibi Department of Mechanical Engineering, Semnan University, P.O.Box:35195-363, Semnan, Iran
Abstract The accurate knowledge of the rigid body properties of a structure including the mass, the center of mass location and the moments of inertia is crucial in machine design, vibration analysis, optimization and modeling of mechanical systems. Estimation of these properties through the theoretical methods is difficult when the structure has a complicated shape. Experimentally, the inertia properties of structures can be estimated using the conventional modal testing methods by extracting the rigid body modes. However, all the rigid body modes are not always available, due to the fact that the structure is not excited at the proper degrees of freedom. In Operational Modal Analysis (OMA) the structure can be excited at any point and in different directions. This suggests that the data from OMA is adequate to extract all the rigid body modes of structure. In this paper, a new approach for estimating the rigid body properties of structure using OMA is proposed. The method is applied to the numerical model of a steel beam and the accuracy of results is evaluated. Also a real beam was tested in order to evaluate the performance of method in practice.
1. Introduction Estimation of ten inertia properties (mass, center of gravity location and inertia tensor) is necessary and important in the design of engineering structures. The inertia properties can be estimated from the theoretical methods such as Finite Element (FE) Method. However, in most practical cases an accurate model is not available or establishment of such a model is time consuming, especially for complicated structures. However, it is possible to estimate the inertia properties of structure using measured experimental data of modal testing. Different methods have been proposed by researchers to estimate the inertia properties of structures. There are two main categories; time domain methods and frequency domain methods [1]. Pendulum method [2, 3] is one of the first proposed time domain methods in which the structure is hanged and forced to oscillate in order to estimate the period of oscillation which is used to evaluate the inertia properties in turn. Although it is still used extensively, the Pendulum method is not accurate due to the friction, the air around the structure and the extra masses. The Pendulum method has been improved by Hou Zhi-Chao et al [4] which is less expensive, with more simplicity and reliability. Pandit et al [5] devised a time domain method to obtain the inertia properties of rigid bodies with damped boundary conditions. Pandit [5] also used transformation matrix to transform the translational motion to the rotational and translational motion in order to evaluate the inertia properties. In another work [6], time domain data was used in six axes for estimation of rigid body properties from time domain impact data. There are some problems in excitation of structure in this method [17]. Frequency domain methods are separated in to three general categories; Inertia Restrain Method (IRM), Method of Direct Physical Parameter Identification (MDPPI), Modal Methods (MM). IRM or Mass Line (ML) method is based on the principle that the dynamic response of a free-free structure in low frequency region can be
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1560 characterized by a constant term [7-10]. MDPPI directly uses the measured FRFs to determine the mass, stiffness and damping ratios [11, 12]. The Modal Method uses the mass-normalized rigid body mode shapes and the orthogonality properties to estimate the inertia parameters [13, 14]. In this method a modal identification procedure is required to extract all the six rigid body modes. Debility in exciting all the six rigid body modes during an experimental test is the most important difficulty associated with frequency domain methods [1]. These methods require as many exciter locations and directions to significantly excite all the six rigid body modes [15]. Due to the limitations in excitation it is not possible to easily extract all the rigid body modes of structures using current methods. Besides in some structures, the excitations are too complicated to be measured. The objective of Operational Modal Analysis (OMA) is to identify the modal parameters (natural frequencies, damping ratio and mode shapes), using only measured responses without the knowledge of inputs. Thus in OMA there is no limitation in exciting the structure in different directions. On the other hand, OMA encounters some limitations in the determination of mode shapes. Due to this fact that the excitation is not measured in OMA, the extracted mode shapes are required to be scaled. In this paper OMA is used to extract the rigid body properties of structure. As there are no limitations in exciting the structure, all the rigid body modes may be extracted. Mass-change method is applied for scaling the mode shapes. Then the Modal Method is used to obtain the inertia properties of structure from the measured data.
2. Theory 2-1- Operational modal analysis Due to the difficulties related to the excitation of structure in conventional modal analysis, the new methods called Operational Modal Analysis were proposed. In OMA, the structure is excited by the ambient forces and there is no need to measure the force. There are two categories of OMA methods; time domain methods and frequency domain methods. The most important frequency domain method is Frequency Domain Decomposition (FDD) method and the most known time domain method is Stochastic Subspace Identification (SSI) method [18, 19, 20].
2-2- Scaling of mode shapes in OMA One of the important problems in OMA is that the mode shapes are not scaled. This is due the fact that the excitation forces are not measured. This problem leads to some limitation in OMA. Thus, the mode shapes are required to be scaled before using in other applications. One of the methods of scaling the mode shapes obtained form OMA is the mass change method [21, 22] According to this method, the structure is modified by adding a number of masses at some chosen points of the structure and the new data is used to compute the scaling factors of mode shapes. The relation between the unscaled mode shape and the scaled mode shape can be given by:
φ α ψ
1
For a system of mass and stiffness, the equations of motion governing the system can be written as:
Μ Z&& C Ζ& Κ Ζ F (t )
2
The eigenvalue equation of the system is:
Μ φ1 ω12 Κ φ1
3
If the mass of system is changed, the eigenvalue equation is changed to:
Μ ΔΜ φ 2 ω 22 Κ φ 2 If the added masses have a small effect on the modal properties of system, we have:
4
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φ 2 ≅ φ1 φ
5
Consequently, the following equation can be extracted [23]:
Μ Ψω12 − ω 22 ΔΜ Ψω 22 φ T
Premultiplying eq. (6) by
6
and considering the orthogonality of modes, the scaling factor of modes can be
obtained as:
α
ω
2 1 T
− ω 22
7
ω ψ ΔΜ ψ 2
Equation (7) is an approximate derivation for the scaling factor based on the assumption [22]. There are other approximate derivations for scaling factors as [23-24]:
α 01
ω
2 0 T
− ω12
8
ω ψ 0 ⋅ Δm ⋅ ψ 1 2 1
2-3- Modal method The Modal Method is one of the methods which is used for estimation of rigid body properties. This method is used in this work for extraction of the inertia properties. The Modal Method is based on the orthogonality property of mode shapes. Establishing the origin of physical coordinate system as a reference, for mass-normalized mode shapes:
φ0T M 0φ0 I
9
where φ0 is the 6×6 mass-normalized mode shape matrix which contains six rigid body modes of structure with respect to a selected origin.
M 0 is a 6×6 mass matrix of structure and is defined as:
⎡ m ⎢ 0 ⎢ ⎢ 0 M0 ⎢ ⎢ 0 ⎢ mz cm ⎢ ⎣⎢− mycm
0 m
0 0
0 − mz cm
mz cm 0
0
m
mycm
− mxcm
− mz cm
mycm
J xx 0
− J xy 0
0
− mxcm 0
− J yx 0 − J zx 0
J yy 0
mxcm
− J zy 0
− mycm ⎤ mxcm ⎥⎥ 0 ⎥ ⎥ − J xz 0 ⎥ − J yz 0 ⎥ ⎥ J zz 0 ⎦⎥
10
As the rigid body mode shapes are linearly independent, mode shape matrix is invertible and the inertia matrix can be derived as:
M 0 φ0−T φ0−1
11
The mode shape matrix extracted from experimental results is related to physical coordinates which is shown by
φ . The relation between these two mode shape matrices φ
and
φ0
is given by:
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φ R0φ0 where
12
R0 is the transformation matrix of the rigid body modes corresponding to the N triaxial accelerometers
[16]: , can be estimated as:
φ0 ( R0T R0 ) −1 R0T φ where
13
( R0T R0 ) −1 is the Psedu inverse of R0 . From theoretical point of view, only two triaxial accelerometers are
needed which measure six responses. Lee et al [18] showed that at least three accelerometers are needed for identification of rigid body inertia properties. Using the upper right quadrant and the lower left quadrant of the mass matrix in Eq. (10), the mass of structure and the center of gravity of structure can be calculated from the members of mass matrix. Using the lower right quadrant of mass matrix, the elements of inertia tensor are calculated.
3- Numerical case study The Finite Element model of a steel beam was built using the 2DOFs beam elements (Fig. 1)
Fig. 1: The FEM model of beam The specifications of beam are given in Table 1.
Length of beam (mm) 653
Width (mm) 49
Table1: Specifications of beam. Thickness Density Youngs modulus 3 (mm) (Kg/m ) (GPA) 6 7850 200
No. of elements 5
The beam was first tested in a simulated measurement in the free-free boundary conditions. The first elastic natural frequency of free-free beam was calculated to be 79.95 Hz. The beam was suspended by two springs K1 and K2 with the stiffness of 7500 N/m. The beam was excited at all DOFs by random excitation in the simulated test and the responses of beam were calculated. SSI and FDD methods were applied on the measured responses to estimate the natural frequencies (Table 2) and mode shapes (Fig. 2). Table 2: Natural frequencies of the beam. No. of mode 1 2 FEM natural frequency (Hz) 14.46 27.11 SSI natural Frequency (Hz) 14.38 27.09 FDD natural Frequency (Hz) 14.46 27.09
3 79.86 79.90 79.86
As can be seen in table 2 the third natural frequency of the suspended beam is close to the first natural frequency of beam in the free-free boundary condition. It can be concluded that the first two natural frequencies of beam in the suspended boundary conditions are related to the rigid body modes of beam
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1563 Mass-change method was used for scaling the first two mode shapes of beam. Table 3 shows the amount of added masses at each node based on 5% of the mass of beam.
No. of points Mass (gr)
1 5.6
Table 3: Specification of added masses 2 3 4 11.2 11.2 11.2
5 11.2
6 5.6
Fig. 2: Comparison of scaled and unscaled mode shapes The inertia properties of beam were calculated using Modal Method presented in section 2-3 (Eq. 10) and compared with the exact values as shown in table 4. The results show that the method proposed in this paper can estimate the inertia properties of system. Table 4: Comparison of the extracted rigid body properties and exact values. 2 Method Mass (Kg) Center of mass (m) Inertia momentum (Kg.m ) Exact 1.529 0.326 0.2173 Modal Method from SSI 1.590 0.325 0.2254 Modal Method from FDD 1.537 0.326 0.2206
Fig. 3: Inertia properties error in some added masses
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1564 Fig. 3 compares the error percentage of inertia properties for different amount of mass change with respect to the mass of beam. Also it can be concluded from Fig. 3 that 5% mass change leads to the best estimation for the inertia properties.
4- Experimental case study In order to illustrate the practical capability of method, a steel beam was considered, with the same geometrical and material specifications of numerical case study as shown in table 1. The beam was suspended by two springs, which are limited to move in one direction by a guide rod, as shown in Fig.6. The stiffness of two springs were chosen to be 7500 N/m in order to separate the first two rigid body modes from the first elastic mode. Six accelerometers type A123E and an analyser type BK3560D were used for the test setup (Fig. 4). Pulse software [25] was used to extract modal parameters from measured outputs. The beam was excited using the random excitation and its responses were measured. Natural frequencies and mode shapes of rigid body modes were estimated using SSI and FDD methods.
Fig. 4: Suspention of steel beam by two springs. Table 5: First two natural frequencies of beam No. of mode 1 2 SSI Frequency (Hz) 15.31 26.94 FDD Frequency (Hz) 15.86 26.53
3 78.44 78.32
Fig. 5 shows the stabilization diagram from SSI method. The first two modes are due to the rigid body modes at 15.31 and 26.94 Hz.
Fig. 5: Stabilization diagram from SSI method.
Fig. 6: Normalized Singular Values of Spectral Densities from FDD method.
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1565 Fig. 6 shows the normalized singular values of spectral densities using FDD method. The first two modes are due to the rigid body modes at 15.86 and 26.53 Hz. The mode shapes were scaled using mass change method. Masses were added at each point according to table 3. The mode shapes were used in Modal Method presented in section 2-3 was applied to extract the inertia properties of beam. Table 6 shows the obtained inertia properties compared to exact values. The bar diagram of error of inertia properties is shown in Fig. 7.
Fig. 7: Error of the inertia properties. Table 6: Comparison of the experimental rigid body properties and exact values. Method Mass (Kg) Center of mass (m) Inertia momentum (Kg*m^2) Exact 1.5291 0.3265 0.2173 MM from FDD 1.4466 0.3169 0.1831 MM from SSI 1.4653 0.3178 0.1859
5- Conclusion In this paper a new approach is suggested for estimating of the inertia properties of a structure using the test data from ambient excitation. The advantage of the proposed approach is that the structure can be excited at all points at different directions. Therefore, the data are enough to reveal all the rigid body modes of structure. The mass change method is used to scale the derived mode shapes from FDD and SSI methods. The scaled mode shapes are used in Modal Method to extract the inertia properties of structure. It is shown that for this case the 5% of added masses leads to the most accurate results. More study is required to be undertaken in the future for more complicated case studies. The comparison of the experimental results and the exact values show that the method can be used for estimation of inertia properties. The main problem is the precision of the scaled mode shapes from operational modal analysis. The accuracy of the derived mode shapes is related to the amount of added masses to the structure. Therefore, the added masses are required to be chosen properly for the test structure.
6- References [1] R.A.B Almeida, A.P.V. Urgueira, N.M.M. Maia, Identification of rigid body properties from vibration measurements, Journal of sound and vibration 299 (4-5), 2007, pp. 884-899. [2] G.W. Hughes, Trifilar pendulum and its application to experimental determination of moments of inertia, ASME Papers 57-SA-51, 1957.
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1566 [3] F. Holzweissig, H. Dresig, Lehrbuch der Maschinedynamic: Grundlagen und Praxisorientierte Beispiele, mit 40 Aufgaben mit\Losungen und 53 Talellen, Fachbuchverlag neubearb, Aufl Leipzig, Koln, 1994. [4] H. Zhi-Chao, L. Yi-ning, L. Yao-xin, L. Dan, A new trifilar pendulum approach to identify all inertia parameters of a rigid body or assembly, Mechanism and Machine Theory 44, 2009, 1270-1280. [5] S. Pandit, Z. Hu, Determination of rigid body characteristics from time domain modal test data, Journal of sound and vibration, V5, 1994, (52-61). [6] S.M. Pandit, Z. Hu and Y. Yao, Experimental technique for accurate determination of rigid body characteristics, th Proc. of the 10 Int. Modal Analysis Conference, IMAC, 1992, 307-311. th
[7] Y.S. Wei, J. Reis, Experimental Determination of Rigid Body Inertia Properties, Proc. of the 7 Int. Modal Analysis Conference (IMAC VII), 1989, pp. 603-606. [8] U. Fullekrug, C. Schedlinski, Inertia Parameter Identification from Base Excitation Test data, Proc. of the 5th International Symposium on Environmental Testing for Space Programmes, Noordwijk, Netherlands, 2004. [9] A. Fregolent, A. Sestieri, Identification of Rigid Body Inertia Properties from Experimental data, Mechanical System and Signal Processing, 10(6), 1996, pp. 697-709. th
[10] A.P.V. Urgueira, On the Rigid Body Properties Estimation from Modal Testing, Proc. of the 13 Int. Modal Analysis Conference (IMAC XIII), 1995, pp.1479-1483. [11] J.A. Mangus, C. Passarello, C. Vankarsen, Estimation Rigid Body Properties from Frequency Reaction th Measurements, Proc. of the 11 Int. Modal Analysis Conference (IMAC XI), 1993, pp. 469-472. [12] S.J. Huang, G. Lallement, Direct Estimation of Rigid Body Properties from Harmonic Forced Responses, th Proc. of the 15 Int. Modal Analysis Conference (IMAC XV), 1997, pp. 175-180. [13] J. Bretl, P. Conti, Rigid body mass properties from test data, Proc. of Fifth Int. Modal Analysis Conference, 1987, pp. 655-659. [14] P. Conti, J. Bretl, Mount stiffness and inertia properties from modal test data, Journal of Vibration, Acoustic, Stress and Reability in Design 111, 1089, 134-138. [15] R.A.B. Almeida, A.P.V. Urgueira, N.M.M. Maia, Evaluation of the Performance of Three Different Methods used in the Identification of Rigid Body Properties, Shock and Vibration, V 15, 2008. [16] R. Almeida, Evaluation of the Dynamic Characteristic of Rigid Bodies Based on Experimental Results, PhD Thesis, Department of Mechanical Engineering and Industrial, Faculty of Science and technology, New University of Lisbon, Portugal, 2006. [17] H. Lee, Y. Park and Y. Lee, Response and Excitation Points Selection for Accurate Rigid Body Inertia Properties Identification, Mechanical Systems and Signal Processing, 1(4), 1999, pp. 571-592. [18] R. Brincker, L. Zhang, P. Andersen, Modal identification from ambient responses using frequency domain th decomposition, Proc. of 18 Int. Modal analysis Conference (IMAC XXI), p. 625-630. [19] P. Van Overschee and B. De Moor, Subspace identification for linear systems: Theory, implementation, application, Kluwer Academic Publisher, 1996. [20] R. Brincker, P. Andersen, Understanding Stochastic Subspace Identification, Proc. of Int. Modal Analysis Conference, (IMAC), 2006.
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1567 [21] Parloo, E., Verboven, P., Guillaume, P., Van Overmeire, M., Sensitivity- Based Operational Mode Shape Normalization, In Mech. Systems and Signal Proc., 16(5), pp. 757-767, 2002. [22] Brincker, R., Andersen, P., A Way of Getting Scaled Mode Shapes in Output Only Modal Analysis. In Proc. of the International Modal Analysis Conference (IMAC) XXI, paper 141, February 2003. [23] López Aenlle, M., Brincker, R., Fernández Canteli, A., Some Methods to Determine Scaled Mode Shapes in Natural Input Modal Analysis, In Proc. of the International Modal Analysis Conference (IMAC) XXIII, paper, 2005. [24] López Aenlle, M., Brincker, R., Fernández Canteli, A., Villa, L. M., Scaling Factor Estimation by the Mass Change Method, Proc.Of the International Operational Modal Analysis Conference (IOMAC), Copenhague, 2005. [25] PULSE, Version 8.0, Brüel & Kjær, Sound & Vibration Measurement A/S. 1996-2003.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Mass-Stiffness Change Method for Scaling of Operational Mode Shapes: Experimental Results
M.M. Khatibi, M.R. Ashory, A. Malekjafarian Department of Mechanical Engineering, Semnan University, P.O.Box: 35195-363, Semnan, Iran
ABSTRACT In Operational Modal Analysis (OMA) the test structure is excited by the unknown forces and only the response signals are measured. Therefore, the mode shapes can not be scaled straightforward from the test. The mass change method is a well known technique to estimate the scaling factors of mode shapes by adding masses to the selected points of structure. A new method have already been proposed for scaling of the operational mode shapes by changing the mass and stiffness of the test structure by the authors. In this paper the experimental results of the application of the proposed method on a beam is presented. It is shown that the first mode of structure is more affected compared to mass change method and consequently the corresponding scaled mode shape is more accurate. Moreover, the experimental results show that the accuracy of the scaled mode shapes improves by using the proposed mass-stiffness change method.
1 INTRODUCTION Modal testing is a well-known experimental tool to evaluate the dynamic properties of structures. In conventional modal testing, both the input and output signals are measured and the modal properties based on natural frequencies, damping factors and mode shapes, are extracted using various methods [1]. In practice some structures can not be excited by a controlled force due to their accessibility, dimensions or boundary conditions. Also, applying a large force for exciting a large structure may damage it or cause nonlinear behavior of the structure [2]. Moreover, the force signals can easily be contaminated in the noisy environment. Recently, the identification of modal parameters using Operational Modal Analysis (OMA) has received a considerable attention. In OMA only the response signals are measured and the test structure is excited by the ambient forces. A disadvantage of this method is that the obtained mode shapes are unscaled. However, the scaled mode shapes are required in some applications such as damage detection, model updating and structural modification. In some methods, the Finite Element Method (FEM) is used for scaling of the operational mode shapes [3]. In the other methods of scaling some restrictions are considered for the excitation [4] or excitation at specific points [5]. The mass change method is an experimental method for scaling the mode shapes [6, 7]. The results of mass change method were improved by the other researches, as reported in [8, 9, and 10]. In the mass change method, the structure is changed by attaching the masses to determine the scaling factors of mode shapes. Previously a new mass-stiffness change method by the authors was proposed for scaling the operational mode shapes in which the structure is modified by adding both mass and spring to it [11]. It was shown that by using the proposed method the accuracy of the scaling especially for the first mode improves. The method was validated by the numerical case study of a beam. In this paper, the experimental results of a clamped-clamped beam are used for validation of the mass-stiffness change method.
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2 THEORY 2.1 Frequency Domain Decomposition (FDD) method FDD is an Operational Modal Analysis method in the frequency domain [2]. This method is based on one of the modal testing techniques known as CMIF [1] and was first proposed in [12]. For r input x and m output y , the equation of motion is given by:
G yy ( jω ) H ( jω ).G xx ( jω ).H T ( jω )
(1)
G xx is the Power Spectral Density (PSD) matrix of the input and G yy is the PSD matrix of response, H ( jω ) is the transpose of Frequency Response Function (FRF) matrix, where the over bar on H denotes the
Where
conjugate of the FRF matrix. The fractional form of the PSD matrix of output can be written as [12]: n ⎡ Qk Qk G yy ( jω ) ∑ ⎢ k 1 ⎢ jω − λ k jω − λ k ⎣
⎤ ⎡ n Qs Qs ⎤ ⎥ .Gxx( jω ).⎢ ∑ ⎥ jω − λ s ⎥⎦ ⎥⎦ ⎢⎣ s 1 jω − λ s
(2)
Where n is the number of modes, Qk is the residue term of the k mode, λk is the k natural frequency of the k th mode and n is the number of system modes. If the input force is assumed to be a white signal and for low damping, Eq. (2) can be written as: th
th
t
d φ φt d kφkφk G yy ( jω ) ∑ k k k K 1 jω − λ k jω − λ k n
(3)
Where d k is a scalar and φk is the k th mode shape vector. The dynamic behavior of a structure is dominated by one of its mode close to the corresponding natural frequency. Therefore, the response of structure is similar to its mode shape close to each natural frequency [12]. The power spectral density matrix of the re sponse in this frequency is decomposed by taking Singular Value Decomposition (SVD) of the matrix as:
G yy ( jωi ) U i S iU iH
(4)
Where the matrix U i [ui1 , u i 2 ,..., u im ] is the unitary matrix including the singular vectors uij and matrix including the singular values sij .
S i is a diagonal
2.2 Mass-stiffness change method The relation between
φ α ψ
ψ the unscaled mode shape and φ the scaled mode shape can be given by: (5)
Where α is the scaling factor. Different relations have been proposed for scaling of the mode shapes using mass change method. The most accurate one is [8, 9]:
α 12
ω
2 1 T
− ω 22
ω ψ 1 ⋅ Δm⋅ ψ 2 2 2
(6)
where α 12 is the scaling factor, Δm is the mass change matrix. ω1 and ψ 1 are the natural frequency and mode shape before scaling and ω2 ,ψ 2 are the natural frequency and mode shape after scaling. For an undamped structure or a structure with proportional damping, the following relation holds:
Μ φ1 ω12 Κ φ1 (7) Where Μ and Κ are the mass and stiffness matrices respectively and φ1 is the scaled mode shapes before modification. If in addition to the mass, the stiffness of structure is changed, Eq. (7) can be written as:
Μ ΔΜφ2 ω22 Κ ΔΚ φ2
(8)
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Where ΔΚ is the stiffness change matrix and φ2 is the scaled mode shapes after modification. By Subtracting of Eq. (8) from Eq. (7), the following relation is obtained:
Μ φ1 ω12 − φ2 ω 22 − ΔΜ φ 2 ω 22 Κ φ1 − φ 2 − ΔΚ φ2
(9)
Considering small mass and stiffness changes, the mode shapes before and after modification change negligibly, i.e.:
φ2 ≅ φ1 φ
(10)
Considering Eq. (10), Eq. (9) can be simplified as:
Μ φ ω12 − ω 22 ΔΜ ω 22 − ΔΚ φ T Pre-multiplying by φ and considering orthogonality of mode shapes, Eq. (11) becomes: ω12 − ω22 φT ΔΜω 22 − ΔΚ φ
(11)
(12)
By substituting Eq. (5) in to Eq. (12) the following equation is obtained:
α
ω
2 1
− ω 22
ψ ΔΜ ω T
2 2
− ΔΚ ψ
(13)
By substituting Eq. (10) in to Eq. (13) three following equations are derived:
α11 α 12 α 22
− ω 22 ΔΜ ω22 − ΔΚ ψ 1
ψ 1 T
ω
2 1
ω
2 1
− ω 22
ψ 1 ΔΜ ω T
ω
2 1
2 2
2 2
(14)
(15)
(16)
− ΔΚ ψ 2
− ω 22
ψ 2 ΔΜ ω T
− ΔΚ ψ 2
In equations (14) to (16) two types of normalization can be considered: the normalization of one coordinate (the largest component is assumed to be one) and the normalization to length (the length of mode shape is assumed to be one) [8].
3 EXPERIMENTAL CASE STUDY In order to validate the mass-stiffness change technique experimentally, the method was applied to a cantilever beam. The beam was made of steel and had dimensions 6×4×700 mm. The beam was discretized to eight elements as shown in Fig. 1. The modal parameters of beam were extracted using two methods: first the conventional hammer modal testing and second the operational modal testing. The unscaled mode shapes from OMA were scaled using the mass change method as well as the proposed mass-stiffness change method.
Figure 1: Discritization of the cantilever beam.
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3.1 Hammer modal test The beam was subjected to the classical hammer test allowing the estimation of the scaled mode shapes. Eight accelerometers type A/120V were mounted on the eight points of beam (Fig. 2). Point 3 was chosen for the excitation of beam by analysing the theoretical data using Modplan software. The theoretical relations for choosing the best point for excitation are given in [13].
Figure 2: Hammer modal testing on the cantilever beam The beam was excited by a hammer type BK8202 and the force signal was amplified using an amplifier type BK2647A. The response and force signals were measured and the frequency response functions were extracted by using Pulse 8 software (Fig. 3). The Modent module of Icats software was used to obtain the first eight natural frequencies and scaled mode shapes of the beam. The results of test are given in Fig. 5 and Table 1.
No. of Mode Natural Frequency (Hz)
Table 1: Natural frequencies from Hammer test 1 2 3 4 5 8.1 50.7 144.31 281.56 463.09
Figure 3: FRFs of hammer test from Pulse software
6 691.95
7 963.74
8 1278.37
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3.2 Operational modal test The ambient modal testing of beam consisted of roving hammer excitation in some arbitrary points of beam. Eight accelerometers were attached at eight points of beam (Fig. 2). The acceleration signals were measured and FDD method was applied to extract the first eight natural frequencies and unscaled mode shapes of beam (Fig. 4).
Figure 4: Singular Value curves from Operational Modal Analysis Pro. Software The comparison of the natural frequencies from the hammer test and those of the operational modal testing are given in Table 4. Fig.5 shows the comparison of the first and fourth mode shapes from hammer test and OMA showing the differences between the two mode shapes. The comparison of the mode shapes using the MAC criterion shows that the two sets of extracted mode shapes are completely correlated (Fig. 6). Therefore, the unscaled mode shapes can be corrected by introducing a scale factor.
Figure 5: Comparison of the first and fourth mode shapes from hammer test and OMA
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Figure 6: 3D MAC criterion between mode shapes from OMA and hammer test
3.3 Mass change method The mass change method was applied to scale the unscaled mode shapes from OMA. The added masses at different nodes of the beam are given in Table 2. The experimental setup is shown in Fig 7.
No. of Node ΔM (gr)
Table 2: The amount of added masses to the beam. 1 2 3 4 5 8.73 8.97 8.85 8.95 8.92
6 9.05
7 8.77
8 4.5
Figure 7: OMA test on the cantilever beam using mass change method The operational modal test was repeated and the new natural frequencies and mode shapes were obtained The mode shapes were scaled using Eq.(6).
3.4 Mass - stiffness change method The proposed mass-stiffness change method was applied to the cantilever beam in order to validate the technique (Fig 8). The amount of mass and stiffness changes are given in Table 3. Only two springs were added to the beam at points 6 and 7. The beam was excited by a hammer at arbitrary points and the natural frequencies and mode shapes of beam were calculated using the FDD method. The scaling factor of unscaled mode shapes were calculated using Eq. (15) and the extracted mode shapes were scaled (Table 4).
BookID 214574_ChapID 141_Proof# 1 - 23/04/2011
1575 Table 3: The amount of mass and stiffness change of the beam. 1 2 3 4 5 6 11.03 11.27 11.15 11.25 11.22 11.35 ----------300
No. of Node ΔM (gr) ΔK (N.m)
7 11.07 300
8 5.2 ---
Figure 8: OMA test on the cantilever beam using mass-stiffness change method
3.5 Results and discussion The natural frequencies from four different tests in the pervious sections are compared in Table 4. Fig. 9 shows the comparison of natural frequency shifts due to the mass change and mass-stiffness change methods. Table 4: natural frequencies of the tests No. of Mode Hammer Test (Hz) OMA Test (Hz) OMA Test(Mass Change) (Hz) OMA Test(Mass-Stiffness Change) (Hz)
1 8.1 8 7.75 9.25
2 50.7 51 49.5 49.25
3 144.31 144 141.25 140.5
4 281.56 282 274.5 273.75
5 463.09 463 451.25 449.25
6 691.95 692 676.75 671.75
7 963.74 964 939.25 934
8 1278.37 1277 1241.25 1238
The mode shapes were compared based on the MSF factor defined as:
MSFi
ϕ
t ϕ Scaled −i ∗ ϕ Scaled −i t ϕ Hammer −i ∗ ϕ Hammer −i
(17)
ϕ
Where Scaled −i is the ith scaled mode shape and Hammer−i is the ith hammer test mode shape. MSF factor shows how much the scaled mode shapes are correlated to the exact mode shapes. Table 5 shows the comparison of the MSF factor of unscaled and scaled mode shapes from OMA test using the mass change and mass-stiffness change methods. The total error based on Eq. (17) were calculated for both set of mode shapes and compared as shown in Fig.10. Table 5: MSF factors of unscaled and scaled mode shapes No. of Mode MSF (Mass Change Method) MSF (Mass-Stiffness Change Method)
1
2
3
4
5
6
7
8
1.0839
1.1198
0.8359
1.1657
1.2433
1.2529
1.71
1.3956
0.9846
1.1006
0.8628
1.0063
1.1713
1.3608
1.669
1.2069
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Figure 9: Natural frequency shift of beam
Figure 10: Error of scaling of mode shapes
Fig.9 shows that the first natural frequency is considerably affected by the mass-stiffness changes compared to the results of the mass change method. This causes that the total error is much less than the first mode shape compared to the other mode shapes as shown in Fig. 10. Also the total error of mass-stiffness change method is less than the mass-change method except for the sixth mode shape. Figures 11 and 12 show that the proposed mass- stiffness change method can accurately scale the first and fourth mode shapes compared to the mass change method.
Figure 11: first Mode shape of cantilever beam from test
Figure 12: Fourth Mode shape of cantilever beam from test
4 CONCLUSIONS In this paper, a new method is presented for scaling of the unscaled mode shapes obtained from OMA by changing both the mass and stiffness of structure. The natural frequencies and mode shapes of the FE model of a beam are calculated. In the first step, mode shapes are scaled using classic mass change method. Next, in addition to the mass, the stiffness of beam is changed and the mode shapes are scaled. It is shown that the mass-stiffness change method can produce more accurate results compared to the mass change method especially for the first mode shape. The efficiency of the method has been demonstrated experimentally on the output-only data from a cantilever beam. The comparison of the scaled mode shapes with the exact mode shapes from the classical modal testing shows a good agreement between the results.
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5 REFFERENCES nd
[1] Ewins, D.J., Modal Testing: Theory, Practice and Application (2 Ed), Research Studies Press Ltd., 2000. [2] Hanson, D., Operational Modal Analysis and Model Updating with a Cyclostationary Input. PhD. Thesis, 2006. [3] Pandey, A.K., Biswas, M., Damage detection in structures using changes in flexibility. J. Sound and Vibration 169(1), 317, 1994. [4] GAO, Y., Randall, R.B., Determination of frequency response functions from response meas urem ments. Part i: extraction of poles and zeros from response measurements, J. Mechanical Systems and Signal Processing 10 (3), p. 293317, 1996a. [5] GAO, Y., Randall, R.B., Determination of frequency response functions from response meas urem ments. Part ii: regeneration of frequency response functions from poles and zeros, J. Mechanical Sys tems and Signal Processing 10 (3), p. 319340, 1996b. [6] Parloo, E., Verboven, P., Guillaume, P., Van Overmeire, M., Sensitivity- Based Operational Mode Shape Normalization, J. Mechanical Systems and Signal Processing, 16(5), p. 757-767, 2002. st [7] Brincker, R., Andersen, P., A way of Getting Scaled Mode Shapes in Output Only Modal Analysis. Proc. 21 Int. Modal Analysis Conference (IMAC-XXI), paper 141, 2003. [8] López Aenlle, M., Brincker, R., Fernández Canteli, A., Some Methods to Determine Scaled Mode Shapes in Natural Input Modal Analysis, Proc. 21st Int. Modal Analysis Conference (IMAC-XXIII), 2005a. [9] López Aenlle, M., Brincker, R., Fernández Canteli, A., Villa, L. M., Scaling Factor Estimation by the Mass st Change Method, Proc. 1 Int. Operational Modal Analysis Conference (IOMAC-I), 2005b. [10] López Aenlle, M., Fernández, P., Brincker, R., Fernández Canteli, A., Scaling Factor Estimation Using an nd Optimized Mass Change Strategy. Part 1: Theory, Proc. 2 Int. Operational Modal Analysis Conference (IOMAC-II), 2007. [11] Khatibi, M.M., Ashory, M.R., Malekjafarian, A., Scaling of mode shapes using mass-stiffness change method, rd Proc. 3 Int. Operational Modal Analysis Conference (IOMAC-III), Italy, 2009. [12] Brincker, R., Zhang, L., and Andersen, P., Modal identification from ambient responses using fre quency domain decomposition, Proc. 18th Int. Modal Analysis Conference (IMAC-XXI), p. 625-630, 2000. [13] Imamovic, N., Validation of large structural dynamics models using modal test data, Ph.D. Thesis, Imperial College of Science, Technology & Medicine London, 1998.
BookID 214574_ChapID FM_Proof# 1 - 23/04/2011
BookID 214574_ChapID 142_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Sensitivity Analysis of Rigid Body Property Estimation from Modal Method
M. Masoumi, S. Shahbazmohamadi, M.R. Ashory, Department of Mechanical Engineering, Semnan University, P.O.Box: 35195-363, Semnan, Iran
ABSTRACT Determination of the ten inertia parameters of mass, center of gravity coordinates, moments of inertia and products of inertia is important in the structural dynamic applications such as optimization, vibration control or structural modification. For the structures with complicated shapes, the inertia properties cannot be easily identified by the theoretical tools. However, there are appropriate experimental methods to evaluate these properties. The experimental route is via to obtain the rigid body modes from the measured Frequency Response Functions (FRFs). The objective of the present paper is the evaluation of the performance of one of the identification methods of rigid body properties based on the modal data, namely Modal Model Method (MMM). The accuracy of the MMM is evaluated by estimating the sensitivity of calculated rigid body properties. The error analysis of the corresponding calculations in MMM is performed mathematically. Based on this study, guidelines are suggested for improving the accuracy of results from MMM. Nomenclature: The mass normalized mode shapes with respect to the center of coordinates Unit matrix The mass matrix with respect to the center of mass Kronika delta The i th mass normalized mode shape The condition number of matrix P The moment of inertia with respect to X axis The moment of inertia with respect to Y axis The moment of inertia with respect to Z axis ,
,
The product of moments of inertia
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_142, © The Society for Experimental Mechanics, Inc. 2011
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1580 1. Introduction In order to analyze and investigate the dynamic behavior of a structure the mass, the center of gravity and the inertial tensor are required to be obtained. Knowing these parameters have a considerable range of usages in designing, balancing, optimizing and car manufacturing. There have been diverse methods introduced to determine the inertial parameters of a body, all of which were based on the equations of motion of a rigid body [1].A number of these methods are the ones in the Modal area. Generally, the methods introduced in this area can be classified into three main sets: the Modal Model Method, Direct Identification and the Inertial Method [2].The fundamentals of the Modal Model Method is based on the orthogonality of Mass and Mode shape Matrices. Moreover, one of the main advantages of this method is their ability to identify the body inertial parameters when its flexible modes and rigid modes are not clearly separate. Two main sets of methods that identify the body inertial modes have been introduced so far. One is Conti and Bretls Method [3] and requires the extraction of each six modes of a rigid body and the latter is first introduced by Nuutila and Toivola [2] and according to that, theoretically, having at least four modes of the rigid body, we can determine the inertial parameters. In this paper ,a sensitivity function has been introduced in order to evaluate the error in the method of Nuutila and Toivola. Then through the use of this function a set of accelerometers have been placed in different positions on the structure and the optimum result which has the least sensitivity to the error has been determined. The application of the function is evaluated by a numerical case study. 2. Theory In the Modal Model Method, the inertial parameters of a rigid body are obtained utilizing the orthogonality principle between the mass and mode shape matrices. In this methods, initially the body will be excited through the forces in various directions and points. Then the obtained modes are made orthogonal to the mass matrix and at the next stage, the modes that have been determined in the place of accelerometers are then translated to the center of coordinates through the transformation matrix. Eventually through applying the principle of orthogonality between the mass and mode shape matrices, all the elements of mass matrix that are also related to the inertial parameters of the body can be determined. We take the center of the system physical coordinates as the center of coordinates, the orthogonal properties of mode shapes can be shown as below [4]; I
(1)
Where M is:
M
m 0 0 0 mz my
0 0 m 0 0 m mz my 0 mx mx 0
0 mz my J J J
mz 0 mx J J J
my mx 0 J J J
(2)
The first three rows of are related to the translational motions and the second rows are related to the rotations of the rigid body, we can obtain the Mass matrix from Eq.(1): (3) Using Eq.(3) the inertia parameters of structures can be obtained [3]. Six modes are required for extracting the mass matrix, resulting in finding the inertial properties of the body, while during experiment, it may be at times difficult to excite all 6 modes of a rigid body and the excitation of some of the mode shapes can be difficult. In the other method presented by Nuutila and Toivolo, in order to determine the inertial parameters of a rigid body, we are in need for obtaining only four modes of the rigid body comparing with all six. In this method, the two-by-two orthogonality between the mode shapes and mass matrices have been utilized, resulting that:
BookID 214574_ChapID 142_Proof# 1 - 23/04/2011
1581 ���� �� ��� � ��� � � ��� � �
������ � �� ������ � �
(4)
According to Eq.(4) and by analyzing the mass matrix, it can be concluded that if n modes have been estimated independent equations can be obtained in the way that 10 required unknowns (mass, moments of inertia and the coordinates of center of mass) can be extracted only through the four mode shapes. In this method, the obtained moments of inertia are according to the center of coordinates and must be translated to the center of mass. For a more clarification on Eq.(4), two mode shapes of i and j are expanded as:
(5) Eq.(5) can be written as :
(6) Where
is the th coefficient of Eq.(5) between modes shapes i and j. In this equation
and
are i th and j
th mode shapes respectively. If five or six modes are estimated, the Pseudu inverse method can be used to determine the 10 unknowns. Eq.(6) can be rewritten for n mode shapes, as: ∆ Where P,
(7) and ∆ are defined as follows;
,
1, … , ;
(8)
(9) ∆
1
…
,
1, 0,
(10)
Where subscript T indicates the transform of a matrix. For determining matrix containing the unknown parameters of the rigid body as it has already been stated, through Pseudu Inverse method, we have ∆ Where
is the inertial parameters of the rigid body
(11)
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1582 3. Sensitivity Analysis Here the sensitivity analysis of Nuutila and Toivola method in determining the inertial properties of a body is analyzed. In this method, the vector ∆ that is made up of 0 and 1 and is free of any error. So in order to analyze the sensitivity of this method we suppose that Matrix P which is made of mode shapes has some deviations from its exact value and consequently there is some error in the calculated vector. Hence, the solution of the eq.7 is as follows [5]: ∆
(12)
Following the deduction of the ordinary solution from Eq.(12) and some simplifications, it can be concluded: ∆
(13)
Or (14) Using the procedure explained in [5] Eq.(13) can be simplified as: ||
||
2
||
||
(15)
In the above relation is the angle between ∆ and y that is the smallest angle between ∆ and an element of R(A) in which R(A) is obtained from the decomposition of Matrix P. In addition || || is the norm of matrix, which is defined as
∑
|x |
and
presents the condition
number defined as: || || ||
||
(16)
If we define the residual as r as follows: ∆
(17)
As a result we will ultimately have: ||
||
||∆||
and by replacing this number in Eq.(15) and by simplifying this equation
3
In Eq.(17),
(18) can be ignored and
can be obtained from the following equation : (19)
According to the Eq.(18), it can be seen that the sensitivity of determining the inertial properties of a body can be regarded as proportional to since this amount is in fact an upper limit for the sensitivity in calculation and therefore the less is this number the less is the sensitivity and as a result the error in the calculated mode shapes can create less error in the obtained results. Therefore the sensitivity function can be defined as: (20)
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1583 4. Determination of the most appropriate position for the accelerometer In order to obtain the most proper position for placing the accelerometers on a body due to reaching the most exact results, the sensitivity function defined in Eq.(20) can be used. According to the guidelines published by Lee [6] the most appropriate conditions for the installation of accelerometers in the case of three accelerometers is applying the triangular form. However, in order to reach the most appropriate formation, we can use the sensitivity function. As it has earlier been mentioned this function is an upper limit of the sensitivity of the calculated inertial parameters. Therefore by applying the least amount of this function for the position of accelerometers we can obtain the formation with the minimum sensitivity to error. 5. Numerical case study In order to validate the method for determining the position of the accelerometers, this technique has been applied on a steel frame that has been modeled using FEM and the results are being compared with the exact values. The case study as it can be seen in Fig.1 is a steel frame with the shape of rectangular cube with the length of 0.5 meter and the width and height of o.435 and 0.335 meters respectively. The inertial properties of the body have been fully presented in Fig.1.
Fig.1 Numerical case study
In order to obtain the most appropriate place for the accelerometers, Five different configurations for the accelerometers were considered: 3 6 8
,
4 5 8
,
1 3 , 6
3 5 6
,
1 2 6
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1584 The overall error in obtaining the inertial properties can be calculated as:
The results obtained from the test simulation on the case study in the form that the mode shapes are calculated in the absence of error using the five mentioned collections of accelerometers in Fig.2. As it can be observed in Fig.2, sensitivity function is minimum for the second set of accelerometers.
Fig.2 The error and amount of sensitivity function for different set of accelometers in the presense of 1 % error in estimated mode shapes In presence of 1% error the minimum error in calculations of inertia parameters are due to the second set (Fig.2) and it can be regarded as the best formation for obtaining the inertia parameters. In the application of this method however it must be born in mind that obtaining reliable results in the first series of accelerometers requires careful selection and must be the best choice according to results presented by Lee in order to be able to find the best combination for the elimination of error in the estimation of inertia parameters.
6. Conclusions In this paper, a new sensitivity function is derived for obtaining the inertia properties of a structure, based on the method introduced by Nuutila and Toivola. This function can give the optimum configuration of accelerometers among a set of configurations in order to minimize the error in the calculations of inertia properties. A numerical case study on a steel frame shows the effectiveness of method. However, a careful selection of the first series of accelerometers is required and must be selected by the methods proposed by the other authors.
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1585 References: [1] Carsten Schedlinksli, Michel Link, A survey of current inertia parameter identification methods, Mechanical systems and signal processing; Vol. 15(1), pp189-211, 2001 [2] Toivola, J., Nuutila,O., Comparison of three methods for determining rigid body inertia properties frequency response functions, IMAC 1993
from
[3] Bretl, J., Conti, P., Rigid body mass properties from test data; IMAC 1987 [4] Almeida, R.A.B., Urgueira, A.P.V., Maia, N.M.M., Identification of rigid body properties from vibration measurements, Journal of sound and vibration; Vol. (299), pp 884-899, 2007 [5] Watkins, D.S., Fundamentals of Matrix Computations, JOHN WILEY & SONS, New York, 2002 [6] Lee, H., Lee, Y.B., Park, Y.S., Response and excitation points selection for accurate rigid-body inertia properties identification; Mechanical systems and signal processing; Vol. 13(4), pp 571-592, 1999 [7] Rao, S.S., The Finite Element Method in Engineering, Elsevier Science & Technology, 2004 [8] Ewins, D.J. Modal Testing: Theory and practice, JOHN WILEY & SONS, 1984
BookID 214574_ChapID FM_Proof# 1 - 23/04/2011
BookID 214574_ChapID 143_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Vibration Absorber Design via Frequency Response Function Measurements
N. Nematipoor, M.R. Ashory, E. Jamshidi Department of Mechanical Engineering, Semnan University, P.O. Box: 35195-363, Semnan, Iran
ABSTRACT Vibration absorbers are usually designed using the Finite Element (FE) model of structures. However, the FE models of vibrating systems are not always available due to the complexity of structure. Moreover, the FE models of structures are not accurate due to the joint problem or numerical errors. Modal testing is an experimental approach to build the mathematical model of structures. It is generally believed that the modal models are more accurate than FE models, because the test structure is modeled by direct measurement from the structure. In this paper, a method is proposed to design the vibration absorbers using the measured Frequency Response Functions (FRFs) of the primary structure. A translational absorber is designed using the method proposed in this paper to absorb the vibration amplitude of a cantilever beam. The experimental results show that the designed absorber is effective and suppresses the amplitude of vibration considerably.
1. Introduction In recent years, control of the vibration amplitude of flexible structures using tuned vibration absorbers has attracted much attention andhas been studied by many authors. Jacquot[1] proposed a method eliminates vibration of harmonically excited Euler-Bernoulli beam. The method gives the vibration absorber parameters based on a single mode of beam. So it was limited in application.Özgüven and Candir[2]extended the previous method to suppress any two resonances. They performed a min-max optimization for the response in any desired mode using assumed-modes approach.ManiKanahally and Crocker [3] give the optimized stiffness and damping parameters for a certain chosen mass to suppress significant modes in which the absorbers are tuned to operate.Keltie and Cheng [4]used point masses to absorb the vibration amplitude of any location of structure. Their method find the optimized location of certain point mass to reduce the vibration level at the desired location of the structure. Ozer and Royston[5]proposed a method gives the vibration absorber parameters mounted to a damped multi-degrees-of-freedom structure based on Sherman-Morrison matrix inversion formula. The method is capable to minimize the overall vibration amplitude of a multi-degrees-of-freedom system.Cha and Pierre[6] proposed a method to impose a single node using normal modes of a supported linear structure by mounting a chain of absorbers.Cha[7]extended his work by using a set of sprung masses and rotational absorbers to enforce one or more fixed nodes for any supported linear structure subjected to harmonic excitations. In this paper, a method is proposed to suppress the vibration amplitude of an arbitrary location on a linear structure subjected to harmonic excitation. The method is beneficial because there is no need to have theoretical or FE model and it is not related to a certain geometry.In order to validate the proposed method a numerical and experimental case studies were performed. In both cases a tuned sprung mass absorber was considered as dynamic vibration absorber and for chosen absorber stiffness, its mass value was calculated.
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_143, © The Society for Experimental Mechanics, Inc. 2011
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2. Theory Changes in a Frequency Response Function (FRF) of a system due to mounting an absorber, using direct substructuring technique SMURF, can be estimated as follows: Fig. 1(a) indicates a structure which has been rigidly connected to the second system of an absorber. The connection is at point j and in x direction. The force excites the structure at point i and the response is measured at point l. Fig. 1(b) indicates the free body diagram of the system.
Figure1.Modification of the structure by attaching an absorber The system is presumed to be linear, andthe equations governing the coupled system are: = + = + = Where !"# isthe receptance for DOFs i and j,and$%& is the receptance of the attached absorber. and the constraining equations are: '( = )* +, + -. = 0 Elimination of the reaction forces and displacements at the connection point results in: 567 89: (G) /0 = (123 4 ; >? )BC = DEF . HI <=
(M)
@A
(1) (2) (3) (4) (5) (6)
in which JKL is the modified receptance function between points l and i when the second system absorber is (Q) connected to the original system at point j. From equation (6) one can immediately relate NOP and the original receptances, as: (U)
RST = RVW 4
XYZ [\]^ _`a
(7)
Whereαli, αji, αlj andαjj are the original receptances, namely for the system without the absorber. Multiplying both sides by –ω2, and the numerator and denominator of the fraction on the right hand side of equation (9) by –ω2,the relation between accelerances is: ( )
= 4
[
(8)
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1589 ( )
As the aim of mounting, of the absorber is vanishing the vibration at node l, is assumed to be zero consequently: 0= !4
"#$ [&'(
(9)
)*+ ,-./
Therefore, Amm can be given as: 345 [789 012 = : 4 =>?
(10)
;<
So the vibration absorber impedance is calculated from equation (10). Equation (10) is a general equation for modifying a system by the connection of an absorber. For a simple sprung mass absorber (Figure 2), the dynamic stiffness matrix is: [@] = ACDB
EF L GHIJ K
(11)
Figure2.Sprung mass absorber The receptance matrix is obtained by inversing the dynamic stiffness matrix, as: R
[M] = [N]OP = QS
U ] VWX ^_` Y a Z[\
T
Therefore, αmm can be given as:
= 4 And Amm can be given by:
= 4 = 4 SubstitutingAmm from equation (14) into equation (10) we have: ) [" $ ! 4 '( = *+ ./ 4 345 # %& 0 12
(12) (13) (14) (15)
For a certain j andk, m can be obtained from equation (15). 3. Numerical case study A theoretical cantilever steel beam was considered for the numerical case study. Thebeam had dimensions 3 1000×48×6 mm and the density and modulus of elasticity were assumed to be respectively 7870 kg /m and 206 2 GN/m . The beam was discretized to 30 elements of 2 nodded 4 Degrees Of Freedom (DOFs) beam elements. The beam was excited by a harmonic forcePoint i(Figure 3) with the frequency ofω=1180 rad/s (187.8 Hz). The absorber mounted at point j.In order to suppress the vibration amplitude at point l.
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Figure3.Numerical case study of the beam Substituting αij,αjj, αli and αljin equation (15) and considering the absorber stiffness to be 10000 N/m, the absorber mass(m) was obtained. Figure (4) shows the FRF (αli) before and after mounting the absorber. It can be seen that the absorber is capable tovanish the vibration amplitude in the exaction frequency ω=1180 rad/s (187.8 Hz). 0
-50
-100
X/F (dB)
-150
-200
-250
-300
-350
-400 0
50
100
150
200
250
300
350
Frequency (Hz)
Figure4.Computed αli with (dotted line) and without (solid line) absorber.
4. Experimental Case Study A steel beam withthe dimensions of 1000×48×6 mm was tested in the free-free boundary conditions as shown in Figure (5).The beam was excited by a shaker type 4808at point I, j and the responses were measured at two points l and j by accelerometer type DJBA120V.
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Figure5. Schematic of the test configuration The beam was excited at the frequency of 79 Hz at point i. The aim was to suppress the vibration amplitude at point l by attaching a sprung mass absorber at point j. Two testswere carried out on the beam. At first the beam was excited at point l to obtainαli and αij. Then the beam was excited at point j to obtain αljand αjj. Substituting αij, αjj, αli and αlj in equation (15) andconsidering the absorber stiffness kj to be 10000 N/m, the absorber mass (m) was calculated to be 42gr.To assess the method,αli is plotted before and after mounting the vibration absorber in Figure(6).It can be seen from Figure (6) that the amplitude vibration decreases considerably around the excitation frequency 79 Hz. Particularly the vibration amplitude at the excitation frequency 79 Hz decreases approximately 85%.
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30 Without absorber With absorber 20
X/F (dB)
10
0
-10
-20
-30 10
20
30
40
50
60
70
80
90
100
Frequency (Hz)
Figure6.αliobtained from experiment with (dotted line) and without (solid line) absorber. 5. Conclusions In this paper, a new method was proposedto design a vibration absorber using the measured FRFs of the primary structure.The advantage of this method is that the vibration absorber can be designed by using the experimental results of modal testing and there is no need to have any theoretical or finite element model. Also, the method is not restricted to any certain geometry or boundary conditions. It can be used for any linear structure with any complexity. In addition, the only required data of the structure to design the vibration absorber is four measured FRFs from the modal testing. One numerical and one experimental case studywere carried out showing the effectiveness of method in suppressing the vibrationamplitude at the desired point. The vibration amplitude decreased by 100% in the numerical case study. However, vibration amplitude decrement in the experimental case study wasabout 85%.
6. References [1]JacquotR.G., Optimal dynamic vibration absorbers for general beam systems, Journal of Sound and Vibration 60 (4),535–542, 1978 [2] zg venH.N., CandirB., Suppressing the first and second resonances of beams by dynamic vibration absorbers, Journal of Soundand Vibration 111 (3), 377–390,.1986 [3] ManikanahallyD.N., CrockerM.J., Vibration absorbers for hysteretically damped mass-loaded beams, Journal of Vibration andAcoustics 113, 116–122,.1991 [4]KeltieR.F., ChengC.C., Vibration reduction of a mass-loaded beam, Journal of Sound and Vibration 187 (2), 213–228,.1995
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[5] OzerM.B., RoystonT.J., Application of Sherman–Morrison matrix inversion formula to damped vibration absorbers attached tomulti-degree of freedom systems, Journal of Sound and Vibration 283 (3–5), 1235– 1249,.2004 [6] ChaP.D., PierreC., Imposing nodes to the normal modes of a linear elastic structure, Journal of Sound and Vibration 219 (4), 669–687,.1998 [7] ChaP.D., ZhouX., Imposing points of zero displacements and zero slopes along any linear structure during harmonic excitations, Journal of Sound and Vibration 297(1-2),55-71,2006
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BookID 214574_ChapID 144_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Non contact eddy current dampers for control systems
I.Shahini1 , M.R.Ashory2, S.Shahbazmohamadi2, A.A.Maddah3 , M.M.Khatibi2 Department of Mechanical Engineering, Islamic Azad University of Semnan, IRAN 2 Department of Mechanical Engineering, Semnan University, Semnan, IRAN 3 Department of Mechanical Engineering,Tarbiat Modares University, Tehran, IRAN
1
Abstract: A time-varying magnetic field induces eddy current in a conductive structure. For a vibrating system, the conductor is moving in the magnetic field, generating eddy currents that will decay the vibration of the dynamic system. This process causes the system to function as a damper. In this paper an eddy current damper is designed for controlling the vibrating system. The concept and theoretical model of the eddy current damper is developed to predict the amount of damping induced on the structure. As the eddy current damper is a noncontacting system, it can be easily applied to any conductive vibrating system. The designed eddy current damper is applied to a vibrating beam. Modal testing is conducted in order to estimate the damping factors of a beam. The accuracy of theoretical model of eddy current damper is evaluated using the experimental data. Nomenclature A Magnetic potential As Cross-section area B Magnetic flux density b Radius of the circular magnet C Damping matrix Cb Damping of beam Ce Eddy current damping coefficient D Non-conservative force δ Thickness of the conductor and Dirac delta function E Electric field E Modulus of elasticity F Damping force F Concentrated forces FT Transformer eddy current damping force FM Motional eddy current damping force f Distributed forces I Moment of inertia I(t) Electric current J Eddy current density K Stiffness matrix ℓ Continuous line ℓs Length of the conductor
M µo Ø(x) Q ρ ρs r(t) rc S σ T t U u V Vs v vb ω XL
Mass matrix Permeability of the free space Mode shapes External force Area density Electrical resistivity Temporal coordinate Equivalent radius of the conductor Continuous surface Conductivity Kinetic energy Time Potential energy Displacement Volume Voltage Velocity of the conductor Velocity of the beam in the z direction Frequency Inductive reactance
1. Introduction: When a non-magnetic conductive metal is placed in a magnetic field, eddy currents are generated. These eddy currents circulate in such a way that they induce their own magnetic field with opposite polarity of the applied field causing a resistive force. However, due to the electrical resistance of the metal, the induced currents will be 2 dissipated into heat at the rate of RI and the force will disappear. In the case of a dynamic system, the conductive metal is continuously moving in the magnetic field and experiences a continuous change in the flux that induces an electromotive force (emf) allowing the induced currents to regenerate and in turn produce a
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_144, © The Society for Experimental Mechanics, Inc. 2011
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1596 repulsive force that is proportional to the velocity of the conductive metal. This process causes the eddy currents to function as a viscous damper and dissipate energy causing the vibrations to die out faster. There are many applications of eddy current dampers. Karnopp[1] introduced the idea that a linear electrodynamic motor consisting of coils of copper wire and permanent magnets could be used as an electromechanical damper for vehicle suspension systems. Schmid and Varga [2] studied a vibration-reducing system with Eddy Current Dampers (ECDs) for high resolution and nanotechnology devices such as a Scanning Tunneling Microscope (STM). Takagi et al. [3] studied the deflection of a thin copper plate subjected to magnetic fields both analytically and experimentally. Matsuzaki et al.[6] proposed the concept of a new vibration control system in which the vibration of a beam periodically magnetized along the span, is suppressed by using electromagnetic forces generated by a current passing between the magnetized sections. To confirm the vibration suppression capabilities of their proposed system, they performed a theoretical analysis of a thin beam with two magnetized segments subjected to an impulsive force and showed the concept to suppress the beams first three modes of vibration. Graves et al. [7] derived the mathematical model of electromagnetic dampers based on a motional emf and transformer emf devices and presented a theoretical comparison between these two devices. Kwak et al. [10] investigated the effects of an eddy current damper on the vibration of a cantilever beam and their experimental results showed that the eddy current damper can be an effective device for vibration suppression. Bae et al. [11] modified and developed the theoretical model of the eddy current damper constructed by Kwak et al. Using this new model, the authors investigated the damping characteristics of the ECD and simulated the vibration suppression capabilities of a cantilever beam with an attached ECD numerically. Sodano and Bae [12] has reviewed the papers on the eddy currents on a structure. Sodano and et al [14] (2006) introduced a better concept of ECD that has utilized two stable magnetic in two sides of an aluminium conductor that two copper planes are placed at its tips and creates a magnetic force. Sodano and Inman [15] (2007) developed a variety of passive and active eddy current dampers that utilized the radial magnetic flux of the permanent magnet rather than the flux in the poling direction. The research presented theoretical models of each damper and performed experiments to validate their accuracy. Using this novel configuration Sodano et al. [15] were able to generate significant damping and in the case of the passive system upwards of critical damping in a cantilever beam. The non-contact nature of this system also made it well suited for ultra flexible membranes and through experimental testing the system was shown to generate damping levels as high as 34% of critical damping. 2. Mathematical modeling : 2-1. Modeling of the eddy currents The fundamental of the performance in the eddy current is that a magnetic field around a coil will induce eddy currents in a conductor near the coil due to an alternative current. The characteristic of eddy currents is that they absorb the system energy through the magnetic field and when the conductor has a greater density or it has a greater magnetic field, the energy will be more powerful. The relation between the voltage and the alternative current is given by: (1) (2) We also have : (3) 2-2. Force determination of the eddy current: The general form of the Faraday's law for the movement in a closed circuit and a time-varying electromagnetic field is:
.
ℓ
·
· ℓ
(4)
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1597 The calculations can be then simplified as mentioned by Sodano,et al [15] Magnetic potential: ⁄ ⁄
(5)
Damping force:
J , 2
,
,
2
(6)
,
,
(7)
2-3. Beam model under the eddy current force The dynamic response of the beam can be formulated as [15].
∑
,
(8)
We then organize the following equations as follows:
∑
∑
(9)
∑
∑
(10)
∑
∑
(11)
∑
∑
,
(12)
The equation of motion can be obtained using the Lagrangian method as:
Q
(13)
As a result:
,
"
2
"
∑
(14)
(15)
(16) (17)
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2 0,
,
(18)
0
(19)
3. Finite Element Model(FEM) of system In order to validate the obtained results a FEM model of system utilizing Matlab software is used. The number of elements is 6 and at the third node the force transducer and the accelerometer and at the last node, the copper conductor plate has been applied in order to obtain the optimum results. In the FEM software viscous damper has been considered in the last node that we have modeled the current in the test through adjusting this damper. Figure 1 shows the designed finite element model.
Figure 1: Experimental setup 4.
Experimental Analysis
Following the FEM modeling in order to validate the results of the model we have utilized the experimental analysis of a cantilever beam made of steel. The dimensions of beam can be found in Fig.2.
Figure 2: A schematic of the experimented beam The electromagnetism that has been applied has been made of a copper with the diameter of 0.6 mm and a soft iron core with the diameter of 20 mm. The external diameter of coil is 67mm and the length is 25mm. The properties of the materials of the beam, conductor and the electromagnetism have been summarized in Table.1:
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1599 Table.1. The properties of the used Electromagnetism Property Value Youngs Modulus of beam 210Gpa 3 Density of beam 7900 kg/m 5 Conductivity of the beam 11.6*10 Ω/m The thickness of the copper conductor 1.4 mm 7 Conductivity of the copper conductor 5.80*10 Ω/m Number of turns of the coil 1100 The resistance of the coil 12Ω Relative permeability of core material 1500 The eddy current in the conductor beam has been produced using the electromagnetic coil attached to the current source. We then adjust the amount of the current manually and the difference in the field, and consequently the ECDs force can be observed in the experiments while it has no contact with the structure. The excitation used in the conductor beam is a prolonged harmonic one in a specific frequency range applied by the shaker. Figure 3 shows The experimental set-up.
Figure 3: Experimental setup 5. Results and Discussion Through experiments the natural frequencies of the beam has been measured and the amplitudes in these frequencies and in the absence of any current has been saved. The amplitudes of the FRF can now be obtained, Then the results of the cantilever beam modeled by the FEM software have been compared to the results obtained by experiments in the absence of any damper. The amplitudes of the FRFS in the FEM software must resemble the amplitudes of the experimented FRFS in natural frequencies. After the reassurance of the validity of the model, the model has been developed through adding a viscous damper at the end of the beam and the results are compared with the experimental results of different currents. And for every current, different damper coefficients have been obtained. The more we increase the current, the field will be more powerful and consequently the more powerful dissipating force can be obtained. This will lead to a decrease in the amplitudes of the FRFs. This means the damping coefficient has been increased. We then adjust the damping coefficient in the way that the peaks for the FRFS in the FEM model will be the same as the peak of FRFs obtained from the experiment. This trend will be applied for all measured currents and the damping coefficients relating to each current has been saved. In this paper the first two vibration modes have been experimented in the currents: 0.2, 0.4, 0.6, 0.8, 1.0 and 1.24 and the conforming FEM model have also been selected. The results can be viewed in Figures 1-4 and Tables 1 and 2. In Table.1 the results obtained from FEM and experiments have been verified and the damping coefficient related to first peak. In Table.2 the results obtained from FEM and experiments have been verified and the damping coefficient related to second peak.
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1600 Table.2. Comparison of the results from FEM and test at the first mode of beam Test FEM Model Damper(C)(Ns/m) Current(I)(Amp) Frequency(Hz) 6.813 6.249 No damp _ Amplitude(dB) 41.6 41.63 Frequency(Hz) 6.813 6.249 0.206 0.2 Amplitude(dB) 37.4 37.4 Frequency(Hz) 6.813 6.249 0.229 0.4 Amplitude(dB) 36.8 36.81 Frequency(Hz) 6.813 6.249 0.275 0.6 Amplitude(dB) 35.6 35.68 Frequency(Hz) 6.813 6.249 0.335 0.8 Amplitude(dB) 34.3 34.34 Frequency(Hz) 6.813 6.249 0.403 1.00 Amplitude(dB) 33.0 33.0 Frequency(Hz) 6.813 6.249 0.471 1.24 Amplitude(dB) 31.8 31.82
Figure 4: The measured FRFs of beam around the first mode for different amount of damper
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Figure 5: The FRFs of beam from FEM for different amount of damping
Table.3. Comparison of the results from FEM and test at the second mode of beam Test FEM model Damper©(Ns/m) Current(I)(Amp) Frequency(Hz) 38.88 39.05 No damp _ Amplitude(dB) 66.7 66.76 Frequency(Hz) 38.88 39.05 0.0132 0.2 Amplitude(dB) 66.4 66.4 Frequency(Hz) 38.88 39.05 0.0536 0.4 Amplitude(dB) 63.0 63.0 Frequency(Hz) 38.88 39.05 0.113 0.6 Amplitude(dB) 58.2 58.2 Frequency(Hz) 39 39.05 0.14 0.8 Amplitude(dB) Frequency(Hz) Amplitude(dB) Frequency(Hz) Amplitude(dB)
56.6 39 54.4 38.88 53.9
56.6 39.05 54.4 39.05 53.9
0.183
1.00
0.195
1.24
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Figure 6: The measured FRFs of beam around the second mode for different amount of damper
Figure 7: The FRFs of beam from FEM for different amount of damping 6. Conclusions An eddy current damper designed as a non-contact system for decaying the vibration amplitude of a vibrating structure. The designed damper was calibrated by comparison of the effects of damper on a cantilever beam and its FEM model. The results show that the designed damper can be used for controlling a vibrating system. As a future work the damper is to be used in a closed or open circuit controlling system of a vibrating structure.
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1603 7. References [1] Karnopp, M., 1989, Permanent Magnet Linear Motors Used as Variable Mechanical Damper for Vehicle Suspensions, Vehicle System Dynamics, Vol. 18, 187200. [2] Schmid, M. and Varga, P., 1992, Analysis of Vibration-Isolating Systems for Scanning Tunneling Microscopes, Ultramicroscopy, Vol. 4244, Part B, 16101615. [3] Takagi, T., Tani, J., Matsuda, S., and Kawamura, S., 1992, Analysis and Experiment of Dynamic Deflection of a Thin Plate with a Coupling Effect, IEEE Transactions on Magnetics, Vol. 28, No. 2, 12591262. [4] Larose, G. L., Larsen, A., and Svensson, E., 1995, Modeling of Tuned Mass Dampers for Wind Tunnel Tests on a Full-bridge Aeroelastic Model, Journal of Wind Engineering and Industrial Aerodynamics,Vol. 54/55, 427 437. [5] Teshima H., Tanaka, M., Miyamoto,K., Nohguchi, K., and Hinata, K.,1997, Effect of Eddy Current Dampers on the Vibrational Properties in Superconducting Levitation Using Melt-Processed YBaCuO Bulk Superconductors, Physica C, Vol. 274, 1723. [6] Matsuzaki, T., Ikeda, T., Nae, A., and Sasaki, T., 2000, Electromagnetic Forces for a New Vibration Control System: Experimental Verification, Smart Materials and Structures, Vol. 9, No. 2, 127131. [7] Graves, K. E., Toncich, D., and Ionvenitti, P. G., 2000, Theoretical Comparison of the Motional and Transformer EMF Device Damping Efficiency, Journal of Sound and Vibration, Vol. 233, No. 3, 441453. [8] Zheng, X. J., Zhou, Y-H., and Miya, K., 2001, An Analysis of Variable Magnetic Damping of a Cantilever Beam-Plate with End Coils in Transverse Magnetic Fields, Fusion Engineering and Design, Vol.55, 457465. [9] Zheng, H., Li, M., and He, Z., 2003, Active and Passive Magnetic Constrained Damping Treatment, International Journal of Solids and Structures, Vol. 40, 67676779. [10] Kwak, M. K., Lee, M. I., and Heo, S., 2003, Vibration Suppression Using Eddy Current Damper, Korean Society for Noise and Vibration Engineering, vol. 13, 10, 760766. [11] Bae, J. S., Kwak, M. K., and Inman, D. J., 2004, Vibration Suppression of Cantilever Beam Using Eddy Current Damper, Journal of Sound and Vibration, in press. [12] H.A. Sodano, J.S. Bae, Eddy current damping in structures, Shock and Vibration Digest 36 (6) (2004) 469 478. [13] Sodano, H. A., Bae, J.-S., Inman, D. J., & Keith Belvin, W. (2005). Concept and model of eddy current damper for vibration suppression of a beam. Journal of Sound and Vibration, 288(4-5), 1177-1196. [14] Sodano, H. A., Bae, J. S., Researcher, S., Inman, D. J., & Belvin, W. K. (2006). Improved concept and model of eddy current damper. Journal of Vibration and Acoustics, 128, 294. [15] Sodano, H. A., & Inman, D. J. (2007). Non-contact vibration control system employing an active eddy current damper. Journal of Sound and Vibration, 305(4-5), 596-613.
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Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Damping Augmentation of Nanocomposites Using Carbon Nanotube/Epoxy
Naser Kordani*, Abdolhosein Fereidoon, Mohammadreza Ashoori Department of Mechanical Engineering, Semnan University, P.O.Box: 35195-363, Semnan, Iran, e-mail:[email protected], Telephone/fax: +98-0231-3354122
ABSTRACT In a nanotube-based polymeric composite structure, it is anticipated that high damping can be achieved by taking advantage of the interfacial friction between the nanotubes and the polymer. The purpose of this paper is to investigate the structural damping characteristics of polymeric composites containing carbon nanotubes with various kinds and amounts. The damping characteristics of the specimens with 0, 0.5 wt% nanotube contents were computed experimentally. Through comparing with neat resin specimens, the study shows that one can enhance damping by adding CNT fillers into polymeric resins. Similarly experiment showed that the maximum value of damping ratio was obtained at 0.5 wt%. Keywords: epoxy; damping; nano composites; nanotubes; polymer
Table: Nomenclature F0
excitation amplitude damping ratio
1
2
frequencies corresponding to the half-power point frequencies corresponding to the half-power point excitation frequency
n
a X RES
natural frequency constant Amplitude at resonance
K
stiffness of the spring
X
amplitude
m
mass
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_145, © The Society for Experimental Mechanics, Inc. 2011
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1606 1. INTRODUCTION Recently, nanoparticles have been attracting increasing attention in the composite community because they are capable of improving the mechanical and physical properties of traditional fiber-reinforced composites [1- 4]. 2
Their nanometer size, which leads to high speci c surface areas of up to more than 1000 m /g, and extra ordinary mechanical, electrical and thermal properties make them unique nano- llers for structural and multifunctional composites. Commonly used nanoparticles in nanocomposites include multiwalled carbon nanotubes (MWCNTs), singlewalled carbon nanotubes (SWCNTs), carbon nanofibers (CNFs), montmorillonite (MMT) nanoclays, and polyhedral oligomeric silsesquioxanes (POSS). Other nanoparticles, such as SiO2, Al2 O3, TiO2, and nanosilica are also used in the nanocomposites. Compared to other particulate additives, carbon nanotubes and carbon nanofibers offer more advantages. The addition of small size and low loading of carbon nanotubes and carbon nanofibers can enhance the matrixdominated properties of composites, such as stiffness, fracture toughness, and interlaminar shear strength [5- 9]. These exceptional properties have been substantiated by a variety of experimental procedures [10- 12]. As produced, SWCNTs are found either in parallel bundles referred to as ropes or in concentric bundles known as multi-walled nanotubes (MWCNTs) [13, 14]. In each bundle arrangement, the SWCNTs are held together with relatively weak van der Waals forces. It has been found that interlayer sliding of MWCNTs is comparable to that of graphene layers in crystalline graphite [14, 15]. New fabrication and purification techniques have enhanced the production of CNTs [16, 17], leading to the possibility that lightweight structural polymers with excellent mechanical properties can be produced using small weight/volume fractions of CNTs as a reinforcing phase. For example, with the addition of only 1% nanotubes by weight, a 36 42% increase in elastic modulus has been observed [18]. Experimental results also demonstrated that the improvement of material properties relies on nanotube dispersion and resin/nanotube interfacial bonding [18- 22]. To analyze nano-structures, molecular dynamics (MD) methods are often used [23- 25]. However, for large and complex systems, MD simulations require expensive computational facilities as well as extensive computation time. Most of the research on CNT-based composites has focused on their elastic properties. Relatively little attention has been given to their damping mechanisms and ability. While Koratkar et al. [26, 27] recently observed promising damping ability of a densely packed MWCNT thin film (no matrix); however, damping characteristics of CNT filled composites have not been investigated in any detail. Previous research has explored the effects of nanoscale particle fillers on the damping properties of polymer composites. For elastomeric materials, it has been found that rod-like aggregates of roughly spherical carbon black particles increase the material damping in the strain range in which the breakdown and reformation of carbon black aggregates occurs [28, 29]. This strain dependent damping enhancement in particle-filled elastomers is known as the Payne Effect. Analogous effects can be expected for composites containing CNT fillers. Recently, Buldum and Lu [23] investigated the interfacial sliding and rolling of carbon nanotubes using MD methods. It was found that a nanotube first sticks and then slips suddenly when the force exerted on it is sufficiently large. In this paper, in order to have direct impact to the field of vibration damping, a structure, or system, level approach was used to examine the damping mechanism and characteristics of CNT-based composites.
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1607 1. THEORY The damping of a structure can be estimated using the well known half-power point method [30]. This theory measures the frequency response function (FRF) of a structure. The sharpness of resonance related to one of the modes can be used to estimate the damping ratio of that mode. Figure 1 shows the half- power points related to one resonance peak of the structure.
Figure1. The half-power pointes technique related to one resonance peak of the structure. Damping can be calculated from equation (1) =
2
2
2
1
4 n
2
2
2
1
(1)
n
The half-power point method is mainly applicable for lightly damped structures. For heavily damped structures, equation (1) can be modified as follows.
Figure2. Indicates a heavily damped resonance peak of a vibrating system. Around a mode, the dynamic behavior of system can be considered as a single degree of freedom system (SDOF) [30]. For a SDOF system we have: F0 X
2 2
(1 (
) ) n
Where
K
(2) (2
)
2
n
is the excitation frequency;
1,
2
are the frequencies corresponding to the half-power point;
the natural frequency; m and k are the mass and stiffness of the spring; excitation amplitude.
n
is
is the damping ratio; and F0 is the
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1608 If we consider the level of X RES -1.5db instead of X RES -3 db in the half-power point method, we have: 20Log a = -1.5 db
(3)
a=0.84 (4)
X= a* X RES F0 X RES
K
(5)
2
Replacing equations (5) and (4) into equation (2), for <<1, we have: /
(
n
) 2 = 1± 1.3
For the first time, 2
2.6
=
(6) =
1
and then
=
2
, and with subtraction we have:
2
2
1 n
(7)
2
2. EXPERIMENT 3-1. Specimen Preparation An epoxy resin (LY 564 from Huntsman) with a polyamide hardener (HY 560 from huntsman) employed as matrix was used in this study, in addition to unidirectional carbon fiber (T300) as filament. In order to evaluate the CNT effect on fiber reinforced polymer composite, different types of carbon nanotube (CNT) were dispersed in the matrix. CVD- technique-produced CNT from the Research Institute of Petroleum Industry (RIPI) were used. The MWCNT outer diameter is 23 nm, and the inner diameter is 11 nm; moreover, these CNT were functionalized with Oxidation and Ultraviolet ray technique. Figure 3(a) and Figure 3(b) show a carbon nanotube TEM image before and after functionalization. b
a
Figure3. (a) Carbon nanotube micrograph for 0.5 wt%; after functionalization; (b) Epoxy micrograph for 0.5 wt% CNT The dark points represent nanotube additions because of strong van der Waals forces. The diameters of the dark points, which are estimated to vary from 5 to 25 nanometers, are shown in Figure 3(a).
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1609 For simplicity, one weight fraction each of SWCNT and MWCNT_COOH (0.5wt %) was dispersed directly into hardener (HY560) with ultrasonic for 10 min and with 60% power. After sonication, a high shear mixer used at 700 rpm for 30 min was used to achieve better dispersion. The epoxy resin (LY 564) was degassed first and then mixed with the HY560/CNT mixture. A matrix was chosen to satisfy the fiber volume fraction (50%). A composite plate was left for 15 hours at 50 C under a vacuum bag after it was prepared with hand lay up. Specimens prepared by cutting the composite plate into small pieces with a saw (220 mm × 13-20 mm × 5 mm) are shown in Figure 4.
Figure4. Specimens prepared by cutting a composite plate. The processes were the same for MWCNT-COOH and SWCNT. The formulations of the nanocomposites are presented in Table 1. Scanning electron microscope (SEM) photos were taken to evaluate the nanotube dispersion in the resin, as shown in Figure 5, which is for a 0.5% nanotube weight ratio.
Figure5. SEM photo of epoxy/CNT composite with 0.5 wt% CNT material.
3-2. Modal Testing and Analysis The regular composite beam without nanotube and the nanocomposite beam with carbon nanotube were used as the specimens for damping test. To investigate the damping characteristics, an experimental study was performed. A force transducer (BK8200) was attached on one side of the specimen to measure the input force. An accelerometer (A/123E) was attached on the other side of the specimen to detect the acceleration, as shown in Figure 6. The specimen was in a free-free boundary condition and was excited by a shaker (BK4808). Sweep sinusoidal signals were used as the excitation source for the shaker, and the frequency response function (FRFs) was derived using an Analyzer in a conventional modal testing procedure.
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Figure6. Experimental setup for the damping test. Figure 7-9 shows the measured FRFs for specimens with different nanotube material contents (0, 0.5 wt%). The damping factors for these specimens were computed according to the theoretical procedure described in section 2.
Figure7. 0.5 wt % SWCNT nanocomposite.
Figure8. 0.5 wt % MWCNT-COOH nanocomposite.
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Figure9. 0 wt % CNT.
3. RESULTS and DISCUSSION Experimental investigations were performed to obtain the damping ratio of the neat specimen, as shown in Figure 9, and the damping characteristics of the CNT-based composites. The damping characteristics of the specimens with 0.5 wt% nanotube content were computed experimentally, as shown in Figure 7-8. It as can be observed in these figures, that the damping characteristics of the beam vary significantly different between specimens with and without nanotubes. It is also apparent that damping can be enhanced by adding CNTs. The maximum damping ratio of the 0.5 wt% MWCNT-COOH-specimen is much higher than those of the 0.5 wt% SWCNT-specimens and the neat specimen. Figure 10 shows that the damping ratio of the 0.5 wt% MWCNT-COOH_ specimen is greater than that of the 0.5 wt% SWCNT because of surface modification. The damping ratio of the regular composite beam and the nanocomposite beam are compared in Figure 11. 0.35 0.3 Damping ratio(
)
0.25 0.2
0.5wt% SWCNT
0.15
0.5wt% MWCNT-COOH
0.1 0.05 0
1st mod
2nd mod
3rd mod
Figure10. Damping ratio of the 0.5 wt% SWCNT, MWCNT-COOH specimens are compare for the first, second, and third natural frequencies.
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1612 0.35 Damping ratio(
)
0.3 0.25 0.2
0w t%
0.15
0.5w t%MWCNT-COOH
0.1 0.05 0 1st m od
2nd m od
3rd m od
Figure11. The regular composite beam and the nanocomposite beam for the first, second, and third natural frequencies The peak value in the FRF represents resonance at a certain frequency. From the FRF, it can be clearly seen that the sharp peak of the first mode, second mode, and third mode are significantly reduced for the nanocomposite beam, which indicates that the nanocomposite beam has improved the damping property. To further demonstrate the improved damping of the nanocomposite beam, the frequency responses of the regular Composite beam and the nanocomposite beam are compared in Table 1. Table1. Damping ratio calculated by modified half-power method 1st mode Frequency (Hz)
1st mode damping ratio
2st mode Frequency (Hz)
2st mode damping ratio
3st mode Frequency (Hz)
3st mode damping ratio
Regular composite beam
118
0.036
272
0.028
580
0.023
Nanocomposite beam Epoxy /SW/0.5 wt% Nanocomposite beam Epoxy /MWCOOH/0.5 wt%
129
0.180
318
0.21
679
0.102
122
0.075
314
0.322
685
0.142
This comparison demonstrates that the damping ratio values of the nanocomposite beam at these three natural frequencies are much greater than those of the regular composite beam. The damping ratio is calculated by using equation (7). Table 1 show the first three modal frequencies and associated damping ratios of the five beams. From the damping ratio comparison, it is clear that the damping ratio of the nanocomposite beam increased up to 100 1200% at the 2nd mode and 3rd mode frequencies. However, there is little change in the mode frequencies, which means that there is only a slight change in the stiffness of the composites. This demonstrates an advantage of nanocomposite over regular composite. Therefore, it is concluded that the incorporation of carbon nanotubes could result in a significant increase in structural damping of conventional fiber-reinforced composites.
4. CONCLUSION Given that more energy will be dissipated with a greater frictional force, the interfacial area between the nanotubes and the resin is extremely large because of the small size of carbon nanotubes, which will cause greater frictional force and structural damping.
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1613 It can be observed that the composites with MWCNT-COOHs have much higher damping ratios than that of the neat resin specimen. This higher damping ratio is due to the fact that nanotube-based composites have the largest interfacial contact area with the resin, and the highest stiffness among all the different fillers considered. Note that with greater stiffness, fillers have an increased capability to resist applied loading, which could lead to more slippage and energy dissipation through friction. In other words, the increase in contact area dramatically increases the potential for energy dissipation due to interfacial friction. Nevertheless, it can be concluded that, by taking advantage of the large interfacial contact area between CNTs and resins, as well as the high stiffness and low density properties of CNTs, high performance in energy dissipation and structural damping can be achieved by the proposed treatment. Because of the higher aspect ratio and high surface area of the nanotubes, it seems to be impossible to disperse the MWCNT properly when the greater amounts of filler are incorporated into a matrix, especially by the meltmixing method. Similarly experiments showed that the maximum damping ratio was obtained at 0.5 wt%-COOH. From Figure 10, it can be observed that the composites with MWCNT-COOH have a higher damping ratio than the composites with SWCNTs.
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1614 REFERENCES 1. Breuer, O. Sundararaj, U. " Big returns from small fibers: a review of polymer/carbon nanotube composites," Polymer Composites, vol. 25, no. 6, pp. 630 645, 2004. 2. Thostenson, E.T. Ren, Z. Chou, T.W. "Advances in the science and technology of carbon nanotubes and their composites: a review," Composites Science and Technology, vol. 61, no. 13, pp.1899 1912, 2001. 3. Lau, K.T. Hui, D. "The revolutionary creation of new advanced materials-carbon nanotube composites," Composites Part B: Engineering, vol. 33, no. 4, pp. 263 277, 2002. 4. Gojny, F.H. Wichmann, M.H.G. Fiedler, B. Bauhofer, W. Schulte, K. "Influence of nano-modification on the mechanical and electrical properties of conventional fibre-reinforced composites," Composites Part A: Applied Science and Manufacturing, vol 36, no. 11, pp.1525 1535, 2005. 5. Gojny, F.H. Wichmann, M.H.G. K¨opke, U. Fiedler, B. Schulte, K. "Carbon nanotube-reinforced epoxycomposites: enhanced stiffness and fracture toughness at low nanotube content," Composites Science and Technology, vol. 64, no. 15, pp. 2363 2371, 2004. 6. Qian, D. Dickey, E.C. Andrews, R. Rantell, T. "Load transfer and deformation mechanisms in carbon nanotube polystyrene composites," Applied Physics Letters, vol. 76, no.20, pp. 2868 2870, 2000. 7. Schadler, L.S. Giannaris, S.C. Ajayan, P.M. "Load transfer in carbon nanotube epoxy composites," Applied Physics Letters, vol. 73, no. 26, pp. 3842 3844, 1998. 8. Ma, H. Zeng, J. Realff, M.L. Kumar, S. Schiraldi, D.A. "Processing, structure, and properties of fibers from polyester/carbon nanofiber composites," Composites Science and Technology, vol. 63, no. 11, pp.1617 1628, 2003. 9. Bower, C. Rosen, R. Jin, L. Han, J. Zhou, O. "Deformation of carbon nanotubes in nanotube-polymer composites," Applied Physics Letters, vol. 74, no. 22, pp. 3317 3319, 1999. 10. Salvetat, J.P. Bonard, J.M. Thomson, N.H. Kulik, A.J. Forro´, L. Benoit, W. et al, "Mechanical properties of carbon nanotubes," Apply Physique A, vol. 69, no. 3, pp. 255 260, 1999. 11. Treacy, M.M.J. Ebbesen, T.W. Gibson, J.M. "Exceptionally high Young s modulus observed for individual carbon nanotubes," Nature, vol. 381, pp. 678 680, 1996. 12. Wong, E.W. Sheehan, P.E. Lieber, C.M. "Nanobeam mechanics: elasticity, strength, and toughness of nanorods and nanotubes," Science, Vol. 277, no. 5334, pp. 1971 1975, 1997. 13. Wagner, H.D. Lourie, O. Feldman, Y. Tenne, R. "Stress-induced fragmentation of multiwall carbon nanotubes in a polymer Matrix," Applied Physics Letters, vol. 72, no. 2, pp. 188 190, 1998. 14. Yu, M.F. Lourie, O. Moloni, K. Dyer, M.J. Kelly, T.F. Ruoff, R.S. "Strength and breaking mechanism of multiwalled carbon nanotubes under tensile load," Science, vol. 287, no. 5453, pp. 637 640, 2000. 15. Ru, C.Q. "Effect of van der Waals forces on axial buckling of a double-walled carbon nanotube," Journal of Apply Physique, vol. 87, no. 10, pp. 7227 7231, 2000. 16. Chiang, I.W. Brinson, B.E. Smalley, R.E. Margrave, J.L. Hauge, R.H. "Purification and characterization of single-wall carbon nanotubes," Journal of Physical Chemistry B, vol. 105, no. 6 pp.1157 1161, 2001. 17. Colomer, J.F. Stephan, C. Lefant, S. Van Tendeloo, G. Willem, I. Ko´nya, Z. et al, "Large-scale synthesis of single-wall Carbon Nanotubes by Catalytic Vapor Depostion (CVD) method," Chemical Physics Letters, vol. 317, no. 1-2, pp. 83 89, 2000. 18. Qian, D. Dickey, C. Andrews, R. Rantell, T. "Load transfer and deformation mechanism in carbon nanotube polystyrene composites," Applied Physics Letters, vol. 76, no.20, pp.2868 2870, 2000.
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1615 19. Ajayan, P.M. Schadler, L.S. Giannaris, C. Rubio, A. "Single-walled carbon nanotube-polymer composites: strength and weakness," Advanced Materials, vol. 12, no. 10, pp. 750 753, 2000. 20. Sandler, J. Shaffer, M.S.P. Prasse, T. Bauhofer, W. Schulte, K. Windle, A.H. "Development of a dispersion process for carbon nanotubes in an epoxy matrix and the resulting electrical properties," Polymer, vol. 40, no. 21, pp.5967 5971, 1999. 21. Schadler, L.S. Giannaris, S.C. Ajayan, P.M. "Load transfer in carbon nanotube epoxy composites," Applied Physics Letters, vol. 73, no. 26, pp. 3842 3844, 1998. 22. Thostenson, E.T. Ren, Z.F. Chou, T.W. "Advances in the science and technology of carbon nanotubes and their composites: a review," Composites Science and Technology, vol. 61, no.13, pp.1899 1912, 2001. 23. Buldum, A. Lu, J.P. "Atomic scale sliding and rolling of carbon nanotubes," Physical Review Online Archive, vol. 83, no.24, pp. 5050-5053, 1999. 24. Odegard, G.M. Gates, T.S. Nicholson, L.M. Wise, K.E. "Equivalent continuum modeling of nanostructured materials," Composites Science and Technology, vol. 62, no. 14, pp. 1869-1880, 2002. 25. Vincenzo, L. Yao, N. "Molecular mechanics of binding in carbon-nanotube-polymer composites," Journal of Materials Research Society, vol. 15, no. 12, pp. 2770 2779, 2000. 26. Koratkar, N.A. Wei, B. Ajayan, P.M. "Multifunctional structural reinforcement featuring carbon nanotube films," Composites Science and Technology, vol. 63, no. 11, pp.1525 1531, 2003. 27. Koratkar, N.A. Wei, B. Ajayan, P.M. "Carbon nanotube films for damping applications," Advanced Materials, vol.14, no. 13-14, pp. 997 1000, 2002. 28. Payne, A. R. Whittaker, R.E. "Low strain dynamic properties of filler rubbers," Rubber Chemistry and Technology, vol. 44, pp. 440-478, 1971. 29. Slo¨sberg, M. Kari, L. "Testing of nonlinear interaction effects of sinusoidal and noise excitation on rubber isolator stiffness," Polymer Testing, vol. 22, pp. 343-351, 2003. 30. Thomson, W.T. Teylor, W. "Theory of Vibration with Applications," George Allen & Unwin, London.
BookID 214574_ChapID FM_Proof# 1 - 23/04/2011
Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc.
Identification of bolted joints under repeatable loads
1
R.khodadadberomy , 2M.R.Ashory 2E. Jamshidi
1
Islamic Azad University, 2Department of Mechanical Engineering, Semnan University, P.O. Box: 35195-363, Semnan, Iran
Abstract: With the advent of delicate and high speed structures like aircraft and rotating machines, researchers have been discovering the importance of joint effects on the structural dynamic response. Joints and fasteners often have a significant effect on the dynamic behavior of assembled mechanical structures and the analytical prediction of structural responses. Previous researchers on bolted joints did not count on the repeatability of the measured data under the same load. In this paper the bolted joint is modeled using Iwan model. On the other hand the bolted joint is tested experimentally under repeatable loading unloading conditions with and without gasket. The differences in the results are examined in the numerical model of joint. The variations in the parameters of model are investigated to obtain a consistent model of joint under repeatable loads. The modified Iwan model shows more accuracy due to considering the effects of the loading conditions of joint. 1. introduction: Joints are effective on the dynamic response of the assembled structures. Mothershead et. al. [1] developed an algorithm to identify the joint parameters based on the Frequency Response Functions (FRFs) The algorithm is only limited to theory. Yang and Fan extended the theory to determine the joint stiffness using FRFs. Damjan and Boltazar [2] identified dynamic behavior of joints using FRFs. Song[3] simulated the dynamic behavior of jointed structures using adjusted Iwan beam element. Gangadharan et. Al. [5] proposed two coupled and uncoupled model constructed from torsional springs. Yang and Park [6] proposed a model consists of a spring and a damper. The method calculates the joint parameters comparing FRFs obtained from the experiment and those of mathematical model which is modeled with out considering the joint. Salehzade Nobari et.al. [7] modified the method of measuring natural frequencies and mode shapes of the structure. Ahmadian et. al. [8-9] proposed a general method to identify the joint parameters. Li [13] developed a model for jointed structure and updated the the joint stiffness parameters. Iwan [10-11] developed a model to predict the non linear behavior of joint. The model consists of the series and paralleled springs and sliders and gives the hysteresis loops . The Iwan model consists of Jenkis elements , in which there are a spring and a damper. In the literature, the repeatability of jointed structures has not been considered yet. In this paper a jointed structure tested under different loads . The structure simulated using Iwan model. The joint stiffness identified in each experiment and the repeatability of joint stiffness is investigated.
T. Proulx (ed.), Structural Dynamics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 12, DOI 10.1007/978-1-4419-9834-7_146, © The Society for Experimental Mechanics, Inc. 2011
1617
1618 2. Iwan Modeling This model consists of a series of spring-slider units arranged in a parallel or series systems [16]. This model is used for modeling the dynamic behavior of joints. Figure 1. Shows the Iwan modeling of a joint. Actually Iwan model composed of a number of jenkins elements which is shown in the Figure 1.
Figure 1. (a) a parallel-series Iwan system, (b)Jenkins Elements. Mathematically, the equation governing the Iwan model can be given by [12]: ∞
F t
ρ
̎ ku t
t,
d
Where U is the imposed displacement F(t) is the applied force is the population of Jenkins elements of strength K is the stiffness common to all of the Jenkins elements
1
1619 The displacements of the sliders y t, u ıf u 0 if u
y t, Here y t,
from the imposed system displacement can be written as:
⁄k ⁄k
y t, y t,
2
is considered to be zero initially for all
. Considering the changes of variable as:
⁄k ρ y t,
3
k ρ k
4
y t, k
5
The parameter k can be removed and equations (1) and (2) are changed to: ∞
F t
ρ
u t
y t,
u if u 0 if u
y t,
d
6
y t, y t,
(7)
The new variables have different dimensions than the original values. Fs is the force required for slipping of the whole interface (macro-slip) and Kt indicates the stiffness of the joint under small applied load. For the parallelseries Iwan model , macro slip is given by every element sliding as: u t For all
y t,
8
, equation (7) can be written as: ∞
F t
d
ρ
9
No sliding at the inception of loading happens for the parallel-series Iwan system Therefore : y t,
In t=0
0
Consequently: ∞
K t
ρ
d
10
Therefore the slope of the hysteresis curve just after the reversal can be give by kt. Iwan used his model to predict the response of a lap type joint under the small or large excitation force.
1620 In Iwan model the stiffness of joint tends to be zero after the beginning of the large sliding. 3. Liner elastic beam elements The liner elastic beam element can be reduced to the rigid bars and two springs with constant stiffness k1 and k2 as shown in Figure.2, where h and L are the fixed height and length of the element, respectively. There are one translational and one rotational degree of freedom at each end of the element; i.e. ωj and θ (i=1,2) The corresponding shear forces and bending moments are x1; x2 and M1; M2; respectively. The extensional deformations of the two springs are given by d
θ
θ
ω
ω
,d
θ
θ
11
which causes internal linear elastic forces in the springs F
k d
,F
k d
12
The shear forces and bending moments at the two ends of the element are due to the internal spring forces F1 and F2. They are related by Q
F
M
,Q F
F
F ,
M
F
F
13
Choosing the stiffnesses of the two springs to be K
12
,K
4
14
Figure 2. Linear elastic beam element
1621 4. Experimental case study Two steel beams were jointed as shown in figure3. The dimensions of beam were 220×50×6 mm and 190×50×6 (mm3). A shaker type 4808 excited the beam at 130mm far from the free end of it. (see figure 4.) and accelerometer placed at 110mm far from the clamped end of the beam. The jointed location is 180mm far from the free end.
Figure 3. Test configuration The structure excited harmonically by a sine wave. The excitation force and the response measured by the accelerometer plotted in Figures 5. And 6.
1622
Figure 4. Dimensions of beam and shaker and accelerometer locations
8 6 4
force(N)
2 0 -2 -4 -6 -8
0
0.5
1 time(s)
Figure 5. Measured force signal
1.5
2
1623
25 20 15
Acceleration(m/s2)
10 5 0 -5 -10 -15 -20 -25
0
0.5
1 time(s)
1.5
2
Figure 6. Measured accelerator signal The experiment carried out for the torques of 20,30 and 40 (N.m) applied to the joint , using two excitation frequencies 70 and 300HZ. The test repeated four times for each torque value to investigate the repeatability of the identified torsional stiffness of the joint. Figure 7. and 8. Show the torsional stiffness identified using different torques for non-gasket and gasket joints when the excitation force has the amplitude of 50N and frequency of 70HZ. 7
3
The retability graph with out gasket at 50(N) and 70HZ
x 10
2.5
kt (N/m)
2
1.5
1
0.5
0 20
22
24
26
28 30 32 Torque (N.m)
34
36
38
40
Figure 7. Repeatability diagram for non-gasket jointed under different torques
1624 7
1.8
The retability graph with gasket at 50(N) and 70HZ
x 10
1.6
kt (N/m)
1.4
1.2
1
0.8
0.6
0.4 20
22
24
26
28 30 32 Torque (N.m)
34
36
38
40
Figure 8. Repeatability diagram for gasket jointed under different torques It was observed from Figures 7. and 8. that the identified torsional stiffness for the gasket joint has small variations in identical conditions compared to the non-gasket joint. Table 1. Percentage of tortional stiffness variation in 70HZ Kt variation 58.8% 10.5%
Max. value 1.020E+07 6.26E+06
Min. value 4.2E+06 5.60E+06
70HZ Non- gasket joint Gasket joint
torque 40(N.m)
Table 1. shows that the percentage of variation of torsional stiffness for non- gasket joint is 58.8%compared to 10.5% for gasket joint. So the repeatability of identified joint stiffness reduces by 48.3%. The experiment carried out for torques of 20,30 and 40 (N.m) applied to the joint , using two excitation frequencies 300HZ and the joint stiffness identified. Figure 9. and 10 show the torsional stiffness identified using different torques for non-gasket and gasket joints when the excitation force has the amplitude of 50N and frequency of 300HZ.
1625 7
2.5
The retability graph with out gasket at 50(N) and 300HZ
x 10
2
kt (N/m)
1.5
1
0.5
0 20
22
24
26
28 30 32 Torque (N.m)
34
36
38
40
Figure 9. Repeatability diagram for non-gasket joint under different torques 7
1.8
The retability graph with gasket at 50(N) and 300HZ
x 10
1.6
kt (N/m)
1.4
1.2
1
0.8
0.6
0.4 20
22
24
26
28 30 32 Torque (N.m)
34
36
38
40
Figure 10. Repeatability diagram for gasket joint under different torques
It can be seen from Figures 9. and 10. That the identified torsional stiffness for gasket joint has small variations in identical conditions compared to the non-gasket joint.
1626 Table 2. Percentage of tortional stiffness variation in 300HZ Kt variation 65.35% 10.76%
Max. value 1.010E+07 6.6E+06
Min. value 3.5E+06 5.89E+06
300HZ Non- gasket joint Gasket joint
torque 40(N.m)
Table 2. shows that the percentage of variation of torsional stiffness for non- gasket joint is 65.35% compared to 10.76% for gasket joint. So the repeatability of identified joint stiffness reduces by 54.59%. 5. Conclusions In this paper the repeatability of the identification of bolted joint parameters was investigated using Iwan model. For this purpose the repeatability checked four times and for different torques applied to the bolted joint for gasket and non-gasket joints. It was observed that the percentage of variation of torsional stiffness identified for non-gasket joint is much more than gasket joint and the repeatability of the bolted joint stiffness reduces by 50% for the non-gasket joint compared to the gasket joint. 6. References [1] H. Ouyang, M.J. Oldfield, J.E. Mottershead, Experimental and theoretical studies of a bolted joint excited by a torsional dynamic load, International Journal of Mechanical Sciences 48 (2006) 14471455. [2] Damjan Cˇelic ˇ, Miha Boltez ˇar, Identification of the dynamic properties of joints using frequencyresponse functions, Journal of Sound and Vibration 317 (2008) 158174. [3] Y. Songa, C.J. Hartwigsenb,1, D.M. McFarlanda, A.F. Vakakisb,c,L.A. Bergmana, Simulation of dynamics of beam structures with bolted joints using adjusted Iwan beam elements, Journal of Sound and Vibration 273 (2004) 249276. [4] R.A. Ibrahima, C.L. Pettit, Uncertainties and dynamic problems of bolted joints and other fastenersb, Journal of Sound and Vibration 279 (2005) 857936. [5] Tachung Yang , Shuo-Hao Fan, Chorng-Shyan Lin, Joint sti ness identification using FRF measurements, Computers and Structures 81 (2003) 25492556. [6] Matthew S. Young, Mayank Tiwari, Rajendra Singh, Identification of Joint Stiffness Matrix Using a Decomposition Technique, The Ohio State University Department of Mechanical Engineering Acoustics and Dynamics Laboratory Center for Automotive Research, Columbus, Ohio 43210-1107, USA. [7] A. Salehzadeh Nobari, Identification of the dynamic characteristics of structural joints, Department of Mechanical Engineering, Imperial College of Science, Technology and Medicine, London SW7, U.K. December 1991. [8] Hamid Ahmadian, Hassan Jalali, Generic element formulation for modelling bolted lap joints, Mechanical Systems and Signal Processing 21 (2007) 23182334. [9] Keivan Ahmadi, Hamid Ahmadian, Modelling machine tool dynamics using a distributed parameter toolholder joint interface, International Journal of Machine Tools & Manufacture 47 (2007) 19161928. [10] Daniel J. Segalman, A Four-Parameter Iwan Model for Lap-Type Joints, Structural Dynamics Research Department Sandia National Laboratories, November 2002.
1627 [11] Daniel J. Segalman, An Initial Overview of Iwan Modeling for Mechanical Joints, Structural Mechanics and Vibration Control Division Sandia National Laboratories, SAND2001-0811 Unlimited Release March 2001. [12] Yaxin Song , C. J. Hartwigsen, Lawrence A. Bergman and Alexander F. Vakakis, A threedimensional nonlinear reduced-order predictive joint model, Earthquake engineering and Engineering vibration, Article ID: 1671-3664 ( 2003 ) 01 - 0059- 15. [13] W. L. Li, A new method for structural model updating and joint stiffness identification, Mechanical Systems and Signal Processing (2002) 16(1), 155 167. [14] D. J. Segalman , Modelling joint friction in structural dynamics, Struct. Control Health Monit. 2006, 13:430 453. [15] David O. Smallwood , Danny L. Gregory, Ronald G. Coleman, Damping Investigations of a Simplified Frictional Shear Joint, Engineering Sciences Center Sandia National Laboratories PO Box 5800 , Albuquerque, NM 87185 (505) 844-9743. [16] Matthew Oldfield, Huajiang Ouyang, John E Mottershead and Andreas Kyprianou, Modelling and Simulation of Bolted Joints under Harmonic Excitation, Materials Science Forum Vols. 440-441 (2003) pp 421-428
