Advanced Aerospace Applications, Volume 1

Conference Proceedings of the Society for Experimental Mechanics Series

For other titles published in this series, go to www.springer.com/series/8922

Tom Proulx Editor

Advanced Aerospace Applications, Volume 1 Proceedings of the 29th IMAC, A Conference on Structural Dynamics, 2011

Editor Tom Proulx Society for Experimental Mechanics, Inc. 7 School Street Bethel, CT 06801-1405 USA [email protected]

ISBN 978-1-4419-9301-4 e-ISBN 978-1-4419-9302-1 DOI 10.1007/978-1-4419-9302-1 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011922492 © 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)

Preface

Advanced Aerospace Applications represents one of six clusters of technical papers presented at the 29th IMAC, A Conference and Exposition on Structural Dynamics, 2011, organized by the Society for Experimental Mechanics, and held in Jacksonville, Florida, January 31 - February 3, 2011. The full proceedings also include volumes on Linking Models and Experiments, Modal Analysis; Civil Engineering; Rotating Machinery, Structural Health Monitoring, and Shock and Vibration; and Sensors, Instrumentation, and Special Topics. Each collection presents early findings from experimental and computational investigations on an important area within structural dynamics. The current volume on Advanced Aerospace Applications includes studies on Aeroelasticity, Ground Testing, and Dynamic Testing of Aerospace Structures. It could be said that many early developments in the field of structural dynamics were motivated by the needs of aviation, and later aerospace. By their very nature aerospace products are susceptible to vibration and they operate in high vibration environments. Structural dynamics plays a key role in aerospace design and testing, impacting flight safety, product durability, performance, and comfort. As in other industries, today’s aerospace products are pushing the limits of performance, pose increased demands for structural dynamic analysis and testing, and will benefit greatly from the recent developments in this field that are the topics of technical sessions at IMAC. 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

Contents

1

High Frequency Optimisation of an Aerospace Structure Through Sensitivity to SEA Parameters A. Culla, Università di Roma ’La Sapienza’; W. D’Ambrogio, Università dell’Aquila; A. Fregolent, Università di Roma ’La Sapienza’

1

2

Benefit of Acoustic Particle Velocity Based Reverberant Room Testing of Spacecraft E.H.G. Tijs, Microflown Technologies; J.J. Wijker, Dutch Space BV; A. Grillenbeck, IABG mbH

15

3

Ultrasonic Vibration Modal Analysis Technique (UMAT) for Defect Detection J.L. Rose, Pennsylvania State University/FBS, Inc.; F. Yan, FBS, Inc.; C. Borigo, Y. Liang, Pennsylvania State University

25

4

Acoustic Testing and Response Prediction of the CASSIOPE Spacecraft V. Wickramasinghe, A. Grewal, D. Zimcik, National Research Council Canada, Institute for Aerospace Research; A. Woronko, P. Le Rossignol, V-O. Philie, MDA Space Missions; M. O'Grady, R. Singhal, Canadian Space Agency

33

5

Force Limited Vibration Testing Applied to the JWST FGS OA Y. Soucy, Canadian Space Agency; P. Klimas, COM DEV Canada

45

6

On Force Limited Vibration for Testing Space Hardware Y. Soucy, Canadian Space Agency

63

7

Calculation of Rigid Body Mass Properties of Flexible Structures K. Napolitano, M. Schlosser, ATA Engineering, Inc.

73

8

Simulating Base-shake Environmental Testing J. Steedman, NAVCON Engineering Network; B. Schwarz, M. Richardson, Vibrant Technology, Inc.

95

9

Geometry-based Updating of 3D Solid Finite Element Models T. Lauwagie, E. Dascotte, Dynamic Design Solutions

103

10

A PZT-based Technique for SHM Using the Coherence Function J. Vieira Filho, F.G. Baptista, Sao Paulo State University; D.J. Inman, Virginia Polytechnic Institute and State University

111

11

The Best Force Design of Pure Modal Test Based Upon a Singular Value Decomposition Approach J.M. Liu, Tsinghua University/China Orient Institute of Noise & Vibration; Q.H. Lu, Tsinghua University; H.Q. Ying, China Orient Institute of Noise & Vibration

119

viii 12

Modal Identification and Model Updating of Pleiades F. Buffe, CNES; N. Roy, TOP MODAL; S. Cogan, FEMTO-ST Institute

131

13

Aircraft GVT Advances and Application – Gulfstream G650 R. Brillhart, K. Napolitano, ATA Engineering, Inc.; L. Morgan, R. LeBlanc, Gulfstream Aerospace Corporation

145

14

Aircraft Dynamics and Payload Interaction – SOFIA Telescope R. Brillhart, K. Napolitano, ATA Engineering, Inc., T. Duvall, L-3 Communications, Integrated Systems

159

15

Application of Modal Analysis for Evaluation of the Impact Resistance of Aerospace Sandwich Materials A. Shahdin, J. Morlier, G. Michon, L. Mezeix, C. Bouvet, Y. Gourinat, Université de Toulouse

171

16

An Integrated Procedure for Estimating Modal Parameters During Flight Testing W. Fladung, G. Hoople, ATA Engineering, Inc.

179

17

Multiple-site Damage Location Using Single-site Training Data R.J. Barthorpe, K. Worden, The University of Sheffield

195

18

Assessment of Nonlinear System Identification Methods Using the SmallSat Spacecraft Structure G. Kerschen, University of Lige; L. Soula, J.B. Vergniaud, Astrium Satellites; A. Newerla, European Space Agency (ESTEC)

19

Ground Vibration Testing Master Class: Modern Testing and Analysis Concepts Applied to an F-16 Aircraft J. Lau, B. Peeters, J. Debille, Q. Guzek, W. Flynn, LMS International; D.S. Lange, Air Force Flight Test Center; T. Kahlmann, AICON 3D Systems GmbH

20

Advanced Shaker Excitation Signals for Aerospace Testing B. Peeters, J. Lau, LMS International; A. Carrella, University of Bristol; M. Gatto, G. Coppotelli, Università di Roma “La Sapienza”

21

System and Method for Compensating Structural Vibrations of an Aircraft Caused by Outside Disturbances W. Luber, J. Becker, CASSIDIAN - Air Systems

203

221

229

243

22

Operational Modal Analysis on a Modified Helicopter E. Camargo, D. Strafacci, Centro Técnico Aeroespacial; N-J. Jacobsen, Brüel & Kjær Sound & Vibration Measurement A/S

265

23

Development of New Discrete Wavelet Families for Structural Dynamic Analysis J.R. Foley, J.C. Dodson, U.S. Air Force Research Laboratory; A.J. Dick, Q.M. Phan, P.D. Spanos, Rice University; J.C. Van Karsen, Michigan Technological University; G.L. Falbo, LMS Americas, Inc.

275

24

Model Updating With Neural Networks and Genetic Optimization M.E. Yumer, E. Cigeroglu, H.N. Özgüven, Middle East Technical University

285

25

A Piezoelectric Actuated Stabilization Mount for Payloads Onboard Small UAS K.J. Stuckel, W.H. Semke, University of North Dakota

295

ix 26

Extraction of Modal Parameters From Spacecraft Flight Data G.H. James, T.T. Cao, V.A. Fogt, R.L. Wilson, NASA Johnson Space Center; T.J. Bartkowicz, The Boeing Company

307

27

Dynamic Characterization of Satellite Components Through Non-invasive Methods D. Macknelly, Imperial College London; J. Mullins, Vanderbilt University; H. Wiest, Rose-Hulman Institute of Technology; D. Mascarenas, G. Park, Los Alamos National Laboratory

321

28

An Inertially Referenced Non-contact Sensor for Ground Vibration Tests B. Allen, Moog CSA Engineering; C. Harris, D. Lange, Edwards Flight Test Center

339

29

Reliability of Experimental Modal Data Determined on Large Spaceflight Structures A. Grillenbeck, S. Dillinger, IABG mbH

351

30

Operational Modal Analysis of a Spacecraft Vibration Test M. O’Grady, R. Singhal, Canadian Space Agency

363

High frequency optimisation of an aerospace structure through sensitivity to SEA parameters Antonio Culla, Walter D'Ambrogio and Annalisa Fregolent

Abstract Classical (FEM, BEM) structural optimisation techniques fail to solve medium high frequency dynamic problems because too many DoFs are involved and eigenvalues and eigenvectors loose the significance due to high modal density. Using a SEA model, the subsystem energies are controlled by (coupling) loss factors, under the same loading conditions. In turn, coupling loss factors (CLF) depend on physical parameters of the subsystems. The idea is to determine an approximate relation between CLF and physical parameters that can be modified in the structural optimisation process, for instance, by using Design of Experiment (DoE). Starting from this relation, an optimisation problem can be formulated in order to bring the subsystem energies under prescribed levels. A preliminary analysis of subsystem energy sensitivity to CLF can be performed to save time in looking for the approximate relationship between CLF’'s and physical parameters. The approach is applied on a typical aerospace structure.

1 Introduction In Statistical Energy Analysis, the studied systems belong to a random population of similar systems [9]. Systems are considered similar if their physical parameters are slightly different. SEA considers a structure as the union of several subsystems. Antonio Culla Dipartimento di Meccanica e Aeronautica, Università di Roma 'La ’ Sapienza', ’ Via Eudossiana 18, 00184 Roma, Italy, e-mail: [email protected] Walter D’'Ambrogio Dipartimento di Ingegneria Meccanica, Energetica e Gestionale, Università dell'Aquila, ’ Piazza E. Pontieri 2, 67040 Roio Poggio (AQ), Italy e-mail: [email protected] Annalisa Fregolent Dipartimento di Meccanica e Aeronautica, Università di Roma 'La ’ Sapienza', ’ Via Eudossiana 18, 00184 Roma, Italy, e-mail: [email protected]

T. Proulx (ed.), Advanced Aerospace Applications, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 4, DOI 10.1007/978-1-4419-9302-1_1, © The Society for Experimental Mechanics, Inc. 2011

1

2 Each of them is a modal group, i.e. a set of similar modes. For instance, considering two plates welded together, six modal groups can be identified, one set of flexural modes and two sets of in plane modes for each plate. SEA estimates the mean value of the energy stored in the modal groups constituting the studied system. The mean value provided by SEA equations is in principle the average response of a set of similar systems. However, SEA equations are represented by a linear system of equations for each frequency, or better for each frequency band. The solution of each linear system gives the energy of each subsystem in a given frequency band. No average operation is explicitly performed, but all the statistics is not visible to the user. In general, this is not a problem because many simple relationships used by physicists and engineers are the result of more complicated mathematical procedures. Unfortunately, in this case this simple model holds only under many strong hypotheses, listed in Section 2. The linear system results from some mathematical manipulations, that include also averages on frequency bands, on the classical equations of motion of multi degrees of freedom systems, and the observance of the strong hypotheses mentioned before. The coefficients of the linear system, named coupling loss factor (CLF) and internal loss factor (ILF), are the result of these average processes and account for the parameters of the native physical system. Therefore, SEA gives the energy of each modal group belonging to the studied system. This energy is the most representative sample of a statistical population of similar systems and on a frequency band. No information is given about the dispersion of the data around the result. In order to provide a true statistic solution it is necessary, at least, to know the variance of the result. All the methods to approach the problem to calculate this variance use either parametric models [11, 2, 5, 1] or nonparametric models [9, 8, 6, 7]. In the first case the physical properties of the system are considered to be uncertain. The uncertainty of the response is calculated by considering the propagation of the physical uncertainties through the model. If the system is very complex and random, its natural frequencies statistics can be assumed, that is the statistics of the response can be considered independent of the statistics of the physical quantities. This is the nonparametric case. In this paper a parametric approach is adopted. However, the study on the effect of SEA parameter variations on SEA results is an opportunity to perform an optimization problem in order to reduce the energy level of some critical subsystem of a complex structure. The effect of uncertainties is modeled by using a sensitivity approach [4, 3]. The sensitivity of the energies stored into the subsystems is calculated by considering uncertainties on ILF’'s and CLF’'s. The goal is to preliminarily understand how much the energies (SEA solution) depend on variations of CLF’'s and ILF’'s. After discarding CLF's ’ and ILF’'s that are less effective on subsystem energies, a model of the remaining SEA coefficients as function of selected physical parameters is developed. The selected physical parameters are that can be modified. Subsequently, a simple mathematical model of how they affect CLF’'s and ILF’'s can be obtained using Design of Experiments [10].

3 A constrained optimisation problem is formulated in order to bring the subsystem energies under prescribed levels. The variables of the problem are the relative deviation of the selected physical parameters from their nominal value. Upper and lower bounds of these design variables are defined. Finally, the energy of the subsystems are constrained to be lower than a prescribed value. In section 4 the procedure described before is performed to optimize the energy of three subsytems of an aerospace structure by modifying three physical parameters. First, a preliminary sensitivity analysis is performed in order to select the more significant CLF’'s to which the considered energies are more sensitive. Then, by using DoE an approximate relation between the more sensitive CLF’'s and the three selected physical parameters is obtained. Finally, the results of the optimisation gives the set of physical parameters values corresponding to the constrain on the energies.

2 SEA equations Under some particular hypotheses, it is possible to assume that the transmitted power between two subsystems is proportional to the difference of the energy stored in each subsystem. A list of these hypotheses is presented below: • all the modes of a subsystem must be similar (i.e. they must have almost the same energy, damping, coupling with the other subsystems and they must be almost excited by the same input power), • the coupling between the subsystems must be conservative, • the eigenfrequencies must be uniformly probable in the frequency range, • the force exciting the subsystems must be random and uncorrelated, • the interactions between the subsystems must be weak. Thus, the SEA equations of Nsub coupled subsystems can be written as follows: Nsub

Pi,in j = ω ηi Ei + ω

∑

(ηi j Ei − η ji E j )

(1)

j=1, j6=i

where i and j are indexes of the subsystems, ηi and ηi j are the internal loss factors (ILF) and the coupling loss factors (CLF), respectively, Pi,in j is the power injected into the subsystem i, E is the energy in a given subsystem and ω is the central frequency of the considered band. Equations (1) represents the energy balance of the subsystems. The power dissipated in the subsystem i is: Pi,d = ω ηi Ei

(2)

The power transmitted from subsystem i to the subsystem j is: Pi j = ω (ηi j Ei − η ji E j )

(3)

4 The solution of the linear system (1) provides the energy stored in each subsystem. The set of equations (1) can be rewritten in a more convenient way as follows: p = ω Ce

(4)

where the coefficients of matrix C are combinations of ILF’'s and CLF’'s as shown in the following equations: ( Ci j = −η ji i, j = 1 . . . Nsub , i 6= j (5) Nsub C j j = η j + ∑i=1, i6= j η ji If ni and n j are the modal densities of subsystems i and j, the following reciprocity relationship holds: ηi j ni = η ji n j (6) By enforcing reciprocity, under the assumption that only the η ji with j > i are known, it is:    −η ji if j > i    nj  Ci j = if j < i −η ji (7) ni      C j j = η j − Nsub Ci j ∑i=1, i6= j

3 Effect of SEA parameter variation on subsystem energies Here a study on the effect of SEA parameter variations on SEA results is performed. Therefore, the goal is to preliminarily understand how much the energies (SEA solution) depend on variations of CLF’'s and ILF’'s. The energy of each subsystem is calculated by solving equation (4) with the obvious implication that energies depend on the CLF’'s and the ILF’'s of the considered system. By defining a range of variability of CLF's ’ and ILF's, ’ a sensitivity approach is used to account for the dependence of the energy on the variations of SEA parameters.

3.1 Sensitivity to SEA parameter variation Sensitivity to loss factors is evaluated in correspondence to nominal values ηˆ of the CLF’'s and ILF’'s. To compare different sensitivity factors, it is assumed that changes ∆ ηkl in the coupling loss factors are a given fraction of nominal value or, with additional effort, they are calculated in agreement with prescribed variations of the physical parameters.

5 ¯ ∂ e ¯¯ ∆ ηkl ∂ ηkl ¯η =ηˆ

∆ ekl =

(8)

and similarly for ILF’'s. To find ∂ e/∂ ηkl , it is necessary to differentiate the solution of Eq. (4): e=

1 −1 C p ω

⇒

1 ∂ C−1 ∂e = p ∂ ηkl ω ∂ ηkl

(9)

and similarly if internal loss factor ηk are considered instead of ηkl . Here it is assumed that the injected power is not affected by changes in CLF’'s and ILF’'s. The derivative of C−1 can be easily obtained from the identity C C−1 = I

∂ C−1 ∂ C −1 = −C−1 C ∂ ηkl ∂ ηkl

(10)

where ∂ C/∂ ηkl can be computed from Eq. (7):  1    nk     ∂ Ci j  nl = −1 ∂ ηkl   − nk      nl 0

if i = j and i = k if i = l and j = l if i = l and j = k

(11)

if i = k and j = l else

and similarly for ∂ C/∂ ηk :

∂ Ci j = ∂ ηk

( 1 0

if i = j and i = k else

(12)

At the end of this stage, CLF’'s and ILF’'s that are less effective in changing the subsystem energies, i.e. for which ∆ ekl is lower, can be discarded.

3.2 CLF’s and ILF’s as functions of physical parameters After discarding CLF’'s and ILF’'s that are less effective on subsystem energies, a model of the remaining SEA coefficients as function of selected physical parameters must be developed. First of all, physical parameters that can be modified are selected; subsequently, a simple mathematical model of how they affect CLF’'s and ILF’'s can be obtained using Design of Experiments (see appendix). At the end of this stage, the most effective CLF’'s and ILF’'s are expressed as:

ηi j = ηi j (x)

(13)

6 where x contains the relative deviations of the selected physical parameters from their nominal values. Therefore, from Eq. (5) or Eq. (7) it is possible to express Ci j as: Ci j = Ci j (x)

⇒

C = C(x)

(14)

Finally, the subsystem energies can be obtained as function of x as: e(x) =

1 −1 C (x) p ω

(15)

3.3 Modification of physical parameters to reduce subsystems energies A constrained optimization problem can be defined in order to reduce the energy level of some critical subsystems by varying the selected physical parameters: Np

arg min f (x) = ∑ wi xi2 x

subject to

(

i=1

Eˆlk (x) ≤ Elk∗

l = 1, . . . , NE

xLi ≤ xi ≤ xUi

i = 1, . . . , N p

k = 1, . . . , N f

(16)

where x contains the relative deviations of the selected physical parameters from their nominal values, wi are weights associated to possible costs of modifications, N p is the total number of physical parameters that can be modified. NE is the number of subsystems for which it is required to reduce or control the energy level, N f is the number of frequency bands. Ekl (x) is the energy of subsystem l at the frequency band k that must be lower than a prescribed value, Elk∗ : it defines N f ×NE constraints. Moreover, upper and lower bounds of the design variable x are defined, xL and xU , so that the solution is always in the range xL ≤ x ≤ xU .

4 Results The studied structure is a launcher fairing. A sketch of the system is shown in figure 1. Subsystems 1, 2 and 3 are made of composite material: the core is aluminum honeycomb and the skins are carbon fiber layers. Subsystem 4 is made of ribbed aluminum shell. The scheme in Figure 2 shows the coupling between the SEA subsystems. The system is excited by an external acoustic diffuse field. The SEA solution is sought at 18 third octave bands between 25 Hz and 1250 Hz.

7

1

5 2

3

4 6

Fig. 1 Launcher fairing: 1 ogive, 2 cylinder, 3 payload adapter, 4 interstage, 5 top cavity, 6 bottom cavity.

!"#"$%&'("

)"#"*+,&-.(/"

0"#"&-1(/213%("

4"#"1$5"*3'&1+"

6"#"53+,$3." 3.351(/"

7"#"8$9$:" *3'&1+"

Fig. 2 SEA subsystems coupling

The considered problem concerns the optimization of the SEA model by imposing an upper bound on the SEA solution when the variability of some CLF’'s is taken into account. In particular, the analysis is focused on the energy of the subsystems 3, 4 and 5, represented by the flexural modes of the structural subsystems payload

8 adapter and interstage skirt and by the acoustic modes of the cavity 5, because they are considered the most critical system of the studied structure. A preliminary sensitivity analysis is performed in order to select the more significant CLF’'s to which the considered energies are more sensitive. A range of variability of ±10% around the nominal values is considered for some selected physical parameters that might be subjected to design modifications: the Young’'s modulus of the aluminum core, the thickness of the carbon fiber skins, the centroid offset of the ribbed shell. The reciprocity relationship (6) is used to get the ηi j and the ∆ ηi j with i > j. The modal densities in the reciprocity relationship are those corresponding to the nominal values of the physical parameters. The ILF’'s are varied of ±10% around their nominal values. Sensitivities of energies in the three critical subsystems, with respect to coupling loss factors and internal loss factors are evaluated according to the procedure outlined in section 3.1. In practice, each value represents the first order approximation of variation of the energy stored in a given subsystem due to a change ∆ ηi j (∆ ηi ) of a given CLF (ILF). Table 1 shows the norm of the sensitivity of each studied energy with respect to CLF’'s and ILF’'s, on all the frequency bands. Since the analysis if performed on the CLF’'s, the ILF’'s contribution is not considered also if the sensitivity values are very high. CLF’'s η21 , η32 , η42 and η43 are identified as the most important, because their correspondent sensitivity values are globally the higher over the three energies. Table 1 Norm of subsystem energies sensitivities with respect to coupling and internal loss factors Sensitivity to η21 η32 η42 η43 η51 η52 η53 η63 η64 η1 η2 η3 η4 η5 η6

kE3 k · 105 1258.6021 73.935694 29.4092 94.7500 12.6385 0.5420 0.0383 0.0002 0.0516 4191.6864 11765.0668 24006.9360 1340.5043 2.6775 0.0710

kE4 k · 105 802.1298 6.149145 899.4531 1080.4808 7.1617 0.3844 0.0072 0.0025 0.65275 2706.9862 7410.2918 13225.5494 985.2324 1.7235 0.0497

kE5 k · 105 161.5913 5.2900 3.1386 1.4237 14.6329 126.2108 1.3447 0.0012 0.0705 552.0421 1501.4571 2670.0953 174.0167 1.7052 0.0139

Figures 3 and 4 show the sensitivity of the studied energies with respect to the most significant CLF’'s, for all third octave bands. Consequently, the optimisation of energies E3 , E4 and E5 is performed by taking into account the variability of these CLF’'s.

9 −3

0

x 10

Sη21

−2 −4 −6 −8 25

50

100

200 [Hz]

400

800

100

200 [Hz]

400

800

−4

6

x 10

Sη32

4 2 0 −2 25

50

Fig. 3 Sensitivities of subsystem energies E3 (o), E4 (∗) and E5 (¦) to coupling loss factors η21 and η 32

By using Design of Experiment an approximate relation between CLF’'s and physical parameters is obtained. A set of numerical experiments is performed, by varying of ±10% around the nominal values, the previuosly selected physical parameters: the Young’'s modulus of the aluminum core, the thickness of the carbon fiber skins, the centroid offset of the ribbed shell. Starting from this relation, an optimisation problem is formulated by following the scheme described in section 3.3. The energies are constrained to assume values lower than the 90% of their nominal values. The physical parameters are bounded between ±10% of their nominal values, the bounds considered for the DoE.

10 −3

2

x 10

Sη42

0 −2 −4 −6 −8 25

50

100

200 [Hz]

400

800

100

200 [Hz]

400

800

−3

2

x 10

Sη43

0 −2 −4 −6 −8 25

50

Fig. 4 Sensitivities of subsystem energies E3 (o), E4 (∗) and E5 (¦) to coupling loss factors η42 and η 43

The result of the optimisation gives the following set of physical parameters values: Young’'s modulus of the aluminum core -10% of its nominal value, thickness of the carbon fiber skins +10% of its nominal value and the centroid offset of the ribbed shell +10% of its nominal value. Therefore, the results correspond to the bounds selected for the optimisation procedure, because not all energies satisfy the prescribed constraints as shown in figure 5.

11

Energy decrease [%]

25 20 15 10 5 0 25

50

100

200 [Hz]

400

800

Fig. 5 Percent energy decrease after design optimisation

5 Conclusions Since classical structural optimisation techniques fail to solve medium high frequency dynamic problems an optimisation problem can be formulated by using a SEA model in order to bring the subsystem energies under prescribed levels. In this paper a constrained optimisation problem is formulated on a typical aerospace structure. The variables of the problem are the relative deviation of the selected physical parameters from their nominal value. Upper and lower bounds of these design variables are defined. Finally, the energy of the subsystems are constrained to be lower than a prescribed value. A preliminary sensitivity analysis is performed in order to select the more significant CLF’'s. By using DoE an approximate relation between the more sensitive CLF’'s and the three selected physical parameters is obtained. Finally, the results of the optimisation correspond to the bounds selected for the optimisation procedure, because not all energies satisfy the prescribed constraints. Next activities will consider the study of more complicated optimisation problem by involving more energies and more physical parameters by considering also the junctions dynamics. Acknowledgements This research is supported by University of Rome La Sapienza grants.

12

Appendix: short background on Design of Experiment In Design of Experiments (DoE), the values of the variables that affect an output response are appropriately modified by a series of tests, to identify the reasons for changes in the response. This does not prevent from performing numerical tests whenever this may be convenient for a better understanding of the numerical problem under investigation. Since many experiments involve the study of the effects of two or more variables or factors, it is necessary to investigate all possible combinations of the levels of the factors. This is performed by factorial designs which are very efficient for this task. To account for possible non linear effects, Central Composite Design (CCD) can be used, that starts from factorial design (2 p observations for p factors) and augments it with the center point i.e. a single observation with all factors at intermediate level (denoted as 0), and axial runs where each factor is considered at two levels (the low level −1 and the high level +1) while the remaining factors are at the intermediate level, for a total of 2p observations (see Figure 6).

(-1,1,1) s

(1,1,1)

z

6

s

s(0,0,1)

(-1,-1,1)

(1,-1,1) s y

s

s

(0,1,0) s

s

(-1,0,0)

x -

s

(0,0,0) (1,0,0)

s s (0,-1,0) (-1,-1,1) s(0,0,-1) s

s

(-1,-1,-1)

(1,-1,-1)

s

(1,-1,1)

Fig. 6 Central Composite Design for p = 3

Overall, a central composite design for p factors requires n = 2 p + 2p + 1 observations. For p control factors, the experimental response can be expressed as a regression model representation of a 2 p full factorial experiment (involving 2 p terms), augmented with p quadratic terms:

13 p

p i−1

i=1 p i−1 m−1

i=1 j=1

f = α0 + ∑ αi xi + ∑ ∑ α ji x j xi + . . . + + ∑ ∑ ··· i=1 j=1

p

∑ αnm··· ji xn xm · · · x j xi + ∑

n=1

(17)

αii xi2 + ε

i=1

The expression contains 2 p + p parameters α , each one providing an estimate of the effect of a single factor (linear or quadratic) or of a combination of them. Note that Eq. (17) is linear in the parameters α , and it can be rewritten as:   α0      £ ¤ α1  f = 1 x1 · · · x2p +ε (18) ..     .   α pp having arranged the parameters in a vector α . A different equation can be written for each observation by varying the factors (x1 , . . . , x p ) as indicated by CCD. By arranging the experimental responses in a vector f, a linear relationship between f and α can be expressed in matrix notation as: f = Xα + ε where X is a

(2 p + 2p + 1) × (2 p + p)

(19)

matrix. The least square estimate of α is:

αˆ = (XT X)−1 XT f

⇒

ˆf = Xαˆ

(20)

where fˆ is the fitted regression model. The difference between the actual observations vector f and the corresponding ˆ The residuals account both for the fitted model fˆ is the vector of residuals e = f − f. modelling error ε and for the fitting error due to the least square estimation.

References 1. Bussow, R., Petersson, B.: Path sensitivity and uncertainty propagation in SEA. Journal of Sound and Vibration 300(3-5), 479–489 (2007) 2. Culla, A., Carcaterra, A., Sestieri, A.: Energy flow uncertainties in vibrating systems: Definition of a statistical confidence factor. Mechanical Systems and Signal Processing 17(3), 635–663 (2003) 3. Culla, A., D’'Ambrogio, W., Fregolent, A.: Parametric approaches for uncertainty propagation in SEA. Mechanical Systems and Signal Processing (2010), doi:10.1016/j.ymssp.2010.05.001 4. D’'Ambrogio, W., Fregolent, A.: Reducing variability of a set of structures assembled from uncertain substructures. In: Proceeding of 26th IMAC. Orlando (U.S.A.) (2008) 5. de Langhe, R.: Statistical analysis of the power injection method. Journal of the Acoustical Society of America 100(1), 294–304 (1996) 6. Langley, R., Cotoni, V.: Response variance prediction in the statistical energy analysis of builtup systems. Journal of the Acoustical Society of America 115(2), 706–718 (2004)

14 7. Langley, R., Cotoni, V.: Response variance prediction for uncertain vibro-acoustic systems using a hybrid deterministic-statistical method. Journal of the Acoustical Society of America 122(6), 3445–3463 (2007) 8. Lyon, R.: Statistical analysis of power injection and response in structures and rooms. Journal of the Acoustical Society of America 45(3), 545–565 (1969) 9. Lyon, R., De Jong, R.: Theory and Applications of Statistical Energy Analysis. The MIT Press, Cambridge (U.S.A.) (1995) 10. Montgomery, D.: Design and Analysis of Experiments, 6th edn. Wiley, New York (2005) 11. Radcliffe, C.J., Huang, X.: Putting statistics into the statistical energy analysis of automotive vehicles. Journal of Vibration and Acoustics 119(4), 629–634 (1997) 12. Weaver, R.: Spectral statistics in elastodynamics. Journal of the Acoustical Society of America 85, 1005–1013 (1989)

Benefit of Acoustic Particle Velocity Based Reverberant Room Testing of Spacecraft Ing. E.H.G Tijs1, Ir. J.J. Wijker2, Dr. Ing. A. Grillenbeck3 1

Microflown Technologies, PO Box 2205, 6802 CE Arnhem, The Netherlands

2

Dutch Space BV, PO Box 32070, 2303 DB Leiden, The Netherlands

3

IABG mbH, Einsteinstrasse 20, 85521 Ottobrunn, Germany

Abstract

The vibro-acoustic load during launch takes a big toll on space structures. In order to simulate the dynamic loading as encountered during launch, both shaker facilities and high sound pressure reverberant rooms are used. Acoustic particle velocity sensors offer interesting new opportunities, for measuring both the applied noise field as well as the structural responses. Single particle velocity sensors in a so-called U probe can be used for very near field vibration measurements. When they are combined with a microphone in a PU probe the full sound vector can be measured. The novel perspectives of using PU probes for reverberant room testing comprise:
The classical control of the noise field and the measurement of the sound pressure level, and of acoustic quantities like the reverberation time may be complemented by making reference to the total acoustic energy.
The input power of the sound source can be measured directly using PU probes.
Arrays of U probes can be used to measure contactless surface vibrations at multiple measurement points simultaneously as an alternative to accelerometers or laser vibrometers.
PU probes can be used to measure local acoustic quantities near the structure like sound radiation, acoustic impedance and energy.
3D sound fields around structures can be visualized.
The degree of diffusion in a reverberation room can be better characterized. In this paper, the first experience with practical implementation and the results of recent measurements using the PU probe will be presented.

Introduction Nowadays, particle velocity sensors are used for many applications in industry such as automotive and aerospace, complementing traditional sensor technology, or are even offering entirely novel measurement capabilities. The most important features for some of these applications are the small size of the sensors, their intrinsic directional and broad band behavior, and their low influence to background noise and reflections under certain conditions. Particle velocity sensors are sometimes used alone, to measure structural vibration, or in combination with a sound pressure microphone to measure the sound intensity, impedance and energy. Here, the perspective of measuring in reverberant chambers under high sound levels is being investigated. Today, typically only two types of sensors are used for the acoustic noise tests for spaceflight: accelerometers are used to measure the vibration; microphones are used to measure the sound pressure. Accelerometers are relatively cheap, but although they are available in various types their mass can influence the response of the lightweight object under test. Laser vibrometers can be used for non-contact measurements instead, but their application is difficult because their mountings are subjected to vibrations, the number of measurement points is restricted due to the limited test time, and more importantly, it is extremely complicated to reach all outer/inner surfaces or measurement points. Velocity sensors placed close to the surface could be used for non-contact measurements. In the

T. Proulx (ed.), Advanced Aerospace Applications, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 4, DOI 10.1007/978-1-4419-9302-1_2, © The Society for Experimental Mechanics, Inc. 2011

15

16 (very) near field of a surface the structural vibration almost equals the particle velocity because the air layer can be considered to be almost incompressible [2-5]. Microphones of are used to measure the sound pressure level in the reverberant chamber. But the sound pressure level only represents the potential energy of the reverberant noise field. Using particle velocity sensors, also the kinetic energy could be measured complementing the measurement to record the total acoustic energy at any arbitrary point. Depending on the location in the reverberant chamber the standing waves result in different sound pressure readings from point to point. To reduce this effect, the mean sound pressure of microphone measurements on multiple locations is used. Sound pressure measured in the vicinity of a structure is also used to estimate the local acoustic load as input for simulation. However, some principles based on sound pressure measurements cannot be used here. E.g. P-P sound intensity probes cannot be used because of the reverberant conditions and because the relation between pressure and velocity cannot be assumed constant at these high sound levels. This paper will describe the applications of PU probes that are envisaged at the moment. To demonstrate some of these applications the first results of a test on the FRED/ATV solar array wing of Dutch Space [1] will be shown. Many accelerometers and sound pressure microphones were also installed on the panel during this test, and a comparison with these sensors will be made. These activities were made in the frame if the ESA funded project VAATLMDS conducted by Dutch Space and partners. The acoustic test and the measurements were carried out in the reverberant chamber of IABG [6].

1. Applications of PU probes in reverberant conditions at high sound levels Several applications of PU probes for reverberant room testing of space structures can be envisaged and will be discussed here. The applications involve enhanced or new methods to improve testing on test objects or the performance of the reverberant room.

1.1 Measurement of the full energy density By measuring the sound pressure only the potential energy is measured. The energy density is approximated with the potential energy alone via [7-10]:

E ≈ 2 *U = 2 *

p2 [W ] 2 ρc 2

(1)

With E being the sound energy, U the potential energy, p pressure, ρ air density and c speed of sound From the particle velocity vector u the kinetic energy K in one direction can be calculated:

r ρvr 2 K= [W ] 2

(2)

By measuring both the potential and kinetic energy in three directions the full energy density E can be calculated: 2

2

2

ux + u y + uz p2 E = U + K = U + Kx + Ky + Kz = +ρ [W ] 2 2 2 ρc

(3)

While the conventional sound pressure based energy measurements by using intensity probes are place dependent, actually the full energy density is a place independent quantity. Accordingly, sound energy measurements could be per-

17 formed in addition or could even replace all control microphones in a reverberant noise field (typically 5 to 10 control microphones are normally used). Also for measurements of the acoustic characteristics like the reverberation time and equivalent absorption area, full energy density measurements could be done with fewer probes than with microphones only.

1.2 Direct measurement of the input power A reverberant room for testing of space structures generally consists of several noise sources to cover a broad frequency range. The noise level inside the reverberation chamber is influenced by the presence of test structures and measurement equipment. The adjustment of the sound level is not so straightforward because the energy is measured indirectly via several control microphones. Because of the long reverberation time it can take a certain amount of time to obtain a stable sound pressure level after adjustment of the sound source level. A direct measurement of the input power with a PU probe could save time for such an adjustment. The input power could be measured with particle velocity alone at low frequencies, and with sound intensity probes at high frequencies. In the near field of a sound source almost all direct energy is kinetic energy. The potential energy near the source is much more influenced by the reflections. The assumption here is that the amount of energy being absorbed by the sound source is low. By measuring the kinetic energy with a particle velocity sensor the source can then be characterized at low frequencies [2-5]. This direct measurement of the sound source could also be used as a reference for calculation of the impulse response of all the other sensors, which could be used for measurement of the magnitude and delay of each reflection. At high frequencies the near field conditions do no not hold anymore. This means the sound intensity should be measured instead [2-5, 11]. However, sound intensity measurements with traditional P-P probes are impossible in these reverberant conditions because of the amount of reflections, or in other terms the p/I (pressure/intensity) index is too high. PU intensity probes are not affected by this p/I index problem.

1.3 Non contact vibration measurements Close to the surface, in the so called very near field, the particle velocity is almost equal to the surface velocity, due to the almost incompressible characteristic of air. For this relation to become valid two conditions have to be met: the distance rn to the surface should be much smaller its typical size L, and the wavelength λ should be larger than the vibration surface L [3]:

rn <<

λ L << 2π 2π

(4)

Practically the maximum frequency is often 1-2 kHz because it is often not possible to position the sensor very close to the surface and because the typical size of the surface cannot be regarded to be small anymore (at high frequencies there are many material modes). Accordingly, when placed on a separate mounting, particle velocity sensors can be used for non-contact vibration measurements. Also arrays of particle velocity sensors can be used. Because of the small size of the sensors also measurements inside structures are possible.

1.4 Acoustic quantities near the structure: sound radiation, impedance and energy Apart from vibration measurements with a velocity sensor only, it also possible to measure other quantities in the vicinity of the structure when these sensors are combined with a sound pressure microphone. Sound intensity, impedance

18 and energy are vector quantities that can be calculated in one or three directions, depending on the number of velocity sensors inside the probe [4-5, 7, 11]. Sound intensity I and impedance Z can be calculated directly with sound pressure p and particle velocity u:

I=

1 pˆ uˆ cos ϕ W / m 2 ; 2

[

]

(5)

Z=

p Ns / m 3 u

[

]

(6)

Where φ is the phase between p and u. The small sensor size allows detailed testing of the surface. Impedance measurements have been done with millimeter accuracy [16]. Many pressure microphone based methods are not valid because of the reverberant conditions and because the relation between pressure and velocity is not linear anymore at high sound levels. Although the measurement of these properties should be possible with PU probes more verifications at high sound levels is still required.

1.5 Visualization of 3D sound fields around structures The structure under test will influence the sound field inside the reverberant room. Their impact could be measured with 3D sound intensity and energy measurements. In a reverberation chamber, the noise sources are located at a certain part of the chamber. In consequence the noise field may not be ideally homogeneous. Hence it might be useful to measure the distribution of the acoustic load in the empty chamber or around the test object.

1.6 Diffusion The reverberation time is a scalar quantity for the room absorption and the objects inside chamber. With PU probes it is also possible to characterize the degree of diffusion via the ratio of intensity and energy [8,9]. In [8] a metric called the field index F was developed. Assuming plane waves the following ratio is a metric for the amount of diffusion (ranging from 0 to 1):

F=

2I

cE

[−]

(7)

Where I is intensity, c is the speed of sound and E is the sound energy. In an anechoic environment the intensity would be high and this ratio would approach unity. In a perfect reverberant room the intensity would be zero.

2. High sound level PU probes Although it is possible to measure with regular PU probes in reverberant field (figure 1) it was necessary to develop special low sensitive sensors to measure at these high sound levels. The development of these probes is an ongoing business, and previous versions are described in [14-15]. In figure 1d the 3D PU probe is shown that is used for the measurements in this paper. It consists of three low sensitive particle velocity sensors and one pressure transducer. To prevent that there are vibrations via the support the probe is suspended in springs. Some mass was added to the suspension to reduce the resonance frequency to 15-20Hz. The outer dimensions of the probe are 7x7cm, while the sensors themselves are almost at the same spot within a 7x7mm area. The total weight is 17 gram, of which 6 gram is rigidly fixed to the surface, and 11 gram is suspended in spring which can be regarded as dynamic weight that doesn’t influence the structure.

19 a

b

c

d

Fig. 1. (a) Regular 1D PU probe, (b) Regular 3D PU probe, (c) High sound level velocity and pressure sensor, (d) The 3D high sound level PU probe that is used here.

3. Description of the test on a folded solar array wing The objective of Dutch Space and their partners in the VAATMLDS project was to measure the response and acoustic environment characteristics of a bread board wing when exposed to a diffuse sound field [1]. Like during launch, in this test the 4 solar panel layers are folded into a package with outer dimensions of 1805 x 1156mm x139mm, see figure 2 and 3. Most sensors were installed in the 12mm gaps in between the panels.

Fig. 2. Left: Deployed solar array, front view. Right: 3D PU probe next to an accelerometer and a surface microphone. Pictures taken at the premises of Dutch Space.

20

Fig. 3. Right: Stowed solar array (picture taken at Dutch Space). 3D PU probe inside an air gap. (picture taken at IABG).

81 accelerometers and 8 B&K surface microphones were installed on the panel. Also eight 3D velocity probes were installed. A sketch of the sensor arrangement on top and in between the panels is shown in figure 4. Two surface microphones and velocity probes were installed on the top, the bottom panel and on each side of panel number 2. No microphones and velocity sensors were installed on the third panel.

Y

X

Z

Fig. 4. Exploded 3D view of the sensor arrangement on the solar array wing: 81 accelerometers (small black spheres), eight surface microphones (yellow) and eight 3D velocity probes were installed.

The stowed solar array wing was tested at the reverberant chamber of IABG. This chamber has a volume of 1378m3 and is capable to generate 156dB OASPL. Five control microphones and 1 monitor microphone were used during the tests. In figure 5 the test set-up is shown. During this test, first empty chamber calibration runs were performed at 140.2dB, 137.2dB and 133.2dB OASPL. After the integration of the test article, a low level test at 133.5dB was done, followed by an intermediate level test at 137.5dB, an acceptance level test at 140.5dB and finally a post low level test at 133.6dB OASPL. Unfortunately, something went wrong during the manufacturing process of the 3D PU probes, resulting in the fact that the integrated sound pressure microphones did not work during these measurements. In consequence, the sound pressure signals from the nearby surface microphones were used instead. Also there are several accelerometers relatively close to the probes to compare with. In this paper the sensor names are the same as in [17].

21

Fig. 5. The stowed FRED/ATV solar array wing surrounded by microphones inside the reverberant chamber. Photo was made at the premises of IABG

4. Measurements Results Since this was only a first reverberant noise test using the particle velocity sensors, not all the applications that were mentioned in chapter 1 are investigated. In this first step, a comparison could be made with accelerometer and a surface microphone very near to the probe.

4.1 Linearity of the probes First the linearity of the particle velocity probes is checked. This is done by comparing the spectra of the pre- and the post low level run [1]. There was no visible damage of the probes and the higher level responses are very similar to the lower ones. This means the high level run did not affect the integrity of the probes. As example the responses of probe 4 and the nearby surface microphone are shown: 1st low level test (133.5dB) intermediate level test (137.5dB) acceptance level test (140.5dB) 2nd low level test (133.6dB)

3

10

-2

10 2

10

-3

Velocity [(m/s)2/Hz]

Pressure [Pa2/Hz]

10

1

10

-4

10

0

10

-5

10

-1

10

-6

2

3

10

10 Frequency [Hz]

10

2

3

10

10 Frequency [Hz]

Fig. 6. Power spectral density during several measurement runs. Left: surface microphone SP3-2. Right: velocity element in direction X of PU probe 4. [17]

22

4.2 Velocity vs. acceleration The acceleration can be calculated from the velocity sensors that were placed normal to the panels. The results are compared with accelerometers that were positioned nearby. Some results are in reasonable agreement, others not so much. This might be due to the fact that both sensors do not measure at the same position. Two examples are shown in figure 7. In most figures the result of the velocity sensor is higher than the accelerometer. This might be caused by the PU probe suspension system that is used. The resonance frequency of the used suspension spring system is between 15 and 20Hz. Accelerometer 31 PU probe 4Y

Accelerometer 57 PU probe 5Y

0

10

-1

10

-1

Acceleration [g2/Hz]

Acceleration [g2/Hz]

10 -2

10

-3

10

-2

10

-3

10

-4

10

-4

10 10

2

3

2

10

3

10

10

Frequency [Hz]

Frequency [Hz]

Fig. 7. Acceleration measured with an accelerometer and a velocity sensor. Left: 1st low level run. Right: acceptance level run. [17]

4.3 Energy vs. sound pressure As mentioned earlier, the potential energy U can be calculated from the sound pressure which is often used to estimate the energy density inside the reverberant chamber. The total energy density E is the sum of potential energy and the kinetic energy K in all three directions. In the following graphs is shown that the 2*the potential energy U is often in quite good agreement with the energy density. In some cases the potential energy is a bit lower and it seems to be affected by standing waves at some frequencies. The kinetic energy component normal to the surface seems to be lowest in all cases. Kx Ky Kz 2*U E

-2

10 -3

10

-3

-4

10

Energy [W/Hz]

Energy [W/Hz]

10

-5

-4

10

-5

10

10

-6

10 -6

10

-7

10 10

2

3

10 Frequency [Hz]

10

2

3

10 Frequency [Hz]

Fig. 8. Kinetic-, potential- and full energy density during acceptance level run. Left: Probe2 and SP1. Right Probe4 and SP3-2. [17]

23

4.4 Diffusion From the particle velocity in three directions and the nearby sound pressure the sound intensity and energy can be calculated. In an anechoic environment the field index ratio 2I/cE will approach unity. In a reverberant environment this quantity shall approach zero. As expected this value is very near zero here (figure 9, left): 1

1 Probe 2, 1st low level (133.5dB) Probe 4, intermediate level (137.5dB) Probe5, acceptance level (140.5dB) Probe 7, 2nd low level (133.6dB)

0.9

0.8

0.7

0.7

0.6

0.6 2I/cE [-]

Density 2I/cE [-]

0.8

0.5

Sources from all directions One source at 30cm

0.9

0.5 0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0

2

10

3

10

10

2

10

3

Frequency [Hz]

Frequency [Hz]

Fig. 9. Left: Diffusion calculated at four different locations and measurement runs. [17]. Right: diffusion in a gym.

A remark has to be made here that for an accurate calculation of the intensity and energy the sensors should be as close as possible to each other (especially at high frequencies). Here the spacing between the surface microphones and the velocity sensors was relatively large (around 15cm in most cases). In order to demonstrate that the field index is not always near zero the diffusion was also measured in a gym. Two situations were measured; one with sources from all directions, and one with a single source relatively close to the sensor (figure 9, right). With sources from all directions the ratio 2I/cE is low because the intensity is low, while the pressure and energy are high, figure 10 left. The direct signal strength of a single nearby sound source is much stronger than its reflections which have a long path. Therefore the intensity is much higher and 2I is almost equal to the p2/ρc and E, figure 10 right. -2

10

10

-2

2I 2

p /(ρc) E

-3

10

-4

10

Sound levels

Sound levels

10

-5

10

-6

10

10

10

10

-7

10

10

2

3

10 Frequency [Hz]

10

-3

-4

-5

2I

-6

2

p /(ρc) E -7 2

10

10

3

Frequency [Hz]

Fig. 10. Scaled intensity, pressure and energy values. Left: Sound sources from all directions. Right: Single sound source at 30cm.

24

Conclusion & Outlook PU probes offer interesting opportunities for reverberant room testing of space structures. Here some of these applications were demonstrated with the data obtained from a measurement with a stowed solar array wing. Several sensors were placed on, and inside the array wing. The linearity and structural integrity of the probes have been tested, showing a similar behavior before and after a high level run (140.5dB OASPL). Velocity sensors can be used for non-contact vibration measurements. From the velocity normal to the surface the acceleration is calculated and compared to nearby accelerometers. The results show some differences which might be explained by the variation between different measurement positions and by the suspension system of the velocity sensors. The potential energy (sound pressure based), the kinetic energy (velocity based), and the full energy density (pressure & velocity) are compared. In some cases the potential energy deviates slightly from the full energy as the potential energy can be affected by standing waves. Also the principle of quantifying diffusion is tested with the ratio of intensity and energy. This field index factor shows to be almost zero in the reverberant room while it increases to a maximum of one in an anechoic environment. There are several ideas that should be tested in the future. Energy, intensity and acoustic impedance could be measured inside, around, or on the surface of structures, or can be used to measure the quality of the reverberant room. With PU probes the input power of the sound sources might be measured directly while being much less affected by reverberations, with the potential of saving time and money.

Acknowledgements This work was partially funded by the Netherlands Space Office under the MIFOSPA project.

References: [1] J.J. Wijker, G. Vermij, VAA-TM-LDS-500-002, VAATMLDS FRED/ATV Wing Acoustic Noise Test Plan/Procedure, 2010 [2] H-E. de Bree, W.F. Druyvesteyn, A particle velocity sensor to measure the sound from a structure in the presence of background noise, Forum Acousticum, 2005 [3] H-E de Bree, V. Svetovoy, R. Raangs, R. Visser, The very near field; theory, simulations and measurements, ICSV, 2004 [4] E. Tijs, S. Steltenpool, H-E de Bree, A novel acoustic pressure-velocity based method to assess acoustic leakages of an acoustic enclosure in non anechoic environments, Euronoise 2009 [5] E. Tijs,H-E de Bree, S. Steltenpool, Scan & Paint: a novel sound visualization technique, Internoise 2010 [6] L. Freyberg, A. Grillenbeck, New acoustic test facility, 4th International Symposium Environmental Testing for Space Programmes, 2001 [7] H-E de Bree, The Microflown E-Book, www.microflown.com, 2010 [8] T.E. Vigran, Aoustic intensity – energy density ratio: an index for detecting deviations from ideal field conditions, JSV, 1988 [9] J. Botts, E. Tijs, H-E de Bree, E. Arato, Acoustic particle velocity enabled methods to assess room acoustics, Euronoise 2009 [10] F. Jacobsen, A. Rodriguez Molares , Statistical properties of kinetic and total energy densities in reverberant spaces, JASA 2010 [11] F. Jacobsen, HE de Bree, A Comparison of two different sound intensity measurement principles, JASA, 2005 [12] S. Weyna , Experimental 3D visualization of power flow around obstacles in real acoustic fields ICSV11, 2004 [13] E. Tijs, H-E de Bree, Mapping 3D sound intensity streamlines in a car interior, SAE NVH 2009 [14] H-E de Bree, E. Tijs, A PU sound probe for high sound levels, DAGA 2009 [15] H-E de Bree, E. Tijs, Calibration of a particle velocity sensor for high noise levels, DAGA 2008 [16] E. Tijs, H-E de Bree, E. Brandao, High resolution absorption mapping with a pu surface impedance method, ASA NoiseCon, 2010 [17] H. Kreutzer, A. Grillenbeck, B-TR60-0440, VAATMLDS ATV STM-B Wing acoustic test report, 2010

Ultrasonic Vibration Modal Analysis Technique (UMAT) for Defect Detection

J. L. Rose, Paul Morrow Professor of Engineering Science and Mechanics, 411E EES Bldg., University Park, PA 16802, Chief Scientist, FBS, Inc., 3340 West College Avenue, State College, PA 16801 F. Yan, Applications Engineer, FBS, Inc., 3340 West College Avenue, State College, PA 16801 C. Borigo, Research Assistant, Pennsylvania State University, 212 EES Blg., University Park, PA 16802 Y. Liang, Research Assistant, Pennsylvania State University, 212 EES Blg., University Park, PA 16802

ABSTRACT Vibration modal analysis and ultrasonic guided wave tests are two commonly used defect detection techniques for Nondestructive Evaluation (NDE) and Structural Health Monitoring (SHM). Each technique has its unique technical merits and also limitations. In this paper, we discuss a new high frequency ultrasonic vibration test method that is developed to bridge the conventional low frequency vibration analysis together with ultrasonic guided wave techniques. The new ultrasonic vibration modal analysis technique (UMAT) takes advantage of the high defect detection sensitivity of ultrasonic guided waves, and at the same time, requires only a minimum number of testing points to cover a large structure, as can be achieved in low frequency vibration analysis. Time delay annular array actuators that are capable of tuning guided wave modes and frequencies are applied to introduce controlled high frequency vibrations to structures being tested. The defect detection sensitivity of the controlled vibrations depends on the sensitivity of the input guided wave modes and frequencies, which is associated with the displacement and stress wave structures of the guided waves. By looking at the vibration patterns and the resonant frequencies under guided wave mode and frequency tuning based on wave structure considerations, high defect detection sensitivity is achieved. Introduction Vibration modal analysis techniques have been widely adopted in various industries as an effective nondestructive defect detection method. The vibration techniques rely on vibration characteristics of structures including natural frequencies, modal shapes, as well as frequency response functions (FRFs) [1]. Traditional vibration modal analysis typically takes place at a rather low frequency range which includes only a limited number of lower order structural resonant frequencies. Due to the frequency limitation, the size of the defects that can be detected by the modal analysis techniques is relatively large. As a fast growing NDE and SHM technique, the ultrasonic guided wave method has superb defect detection sensitivity [2-5]. Besides the much higher operating frequency range compared to the traditional vibration techniques, ultrasonic guided wave techniques also have the flexibility of using different guided wave modes and frequencies for optimum sensitivity to different defect types [4, 5]. However, in a guided wave test, it is usually desired to send and receive transient guided waves in many different positions such that a reliable inspection on the whole structure can be achieved. This is due to the fact that the guided wave signals are affected by a defect only if the defect lies in the guided wave propagation path. Whereas in a vibration test, the complete defect detection coverage can be easily achieved since the structural vibrations response to defects located at any positions within the structure. As an effort to merge the vibration modal analysis techniques with guided wave techniques, a high frequency ultrasonic modal analysis technique (UMAT) has been proposed [6, 7]. The development of UMAT seeks the feasibility of applying modal analysis to high frequency vibrations introduced by controlled ultrasonic guided wave inputs. In a UMAT test, modal analyses are applied to the long time structural responses to a specifically defined ultrasonic guided wave input. Thanks to the fact that the long time structural responses result from multiple reflections and scatterings of the input guided wave energy, an overall coverage of the structure can be reached from

T. Proulx (ed.), Advanced Aerospace Applications, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 4, DOI 10.1007/978-1-4419-9302-1_3, © The Society for Experimental Mechanics, Inc. 2011

25

26

a very small number of test positions. In our previous work, we have demonstrated that the forced vibration of a structure is a function of the ultrasonic guided wave mode and frequency that is applied as the vibration loading function [6, 7]. By selecting an appropriate mode and frequency based on the guided wave sensitivity to a specific defect type, we have shown that excellent sensitivity can be achieved for the defect type. It is therefore possible to obtain good sensitivities to all different kinds of defects by varying the input guided wave modes and frequencies. Various guided wave transducers including angle beam transducers, electromagnetic acoustic transducers (EMATs), and piezoelectric comb transducers are available for guided wave mode and frequency control [2, 8]. In the work presented in this paper, a time delay annular array transducer is employed as the UMAT actuator. The main advantages of using such an actuator are its omni-directional guided wave excitation and the capability of automatic mode and frequency tuning through electrical time delay circuits. The next section discusses the guided wave excitations of annular array actuators. The applications of the time delay annular array actuator for both transient guided wave tests and UMAT tests are then presented in the following sections. Guided Wave Excitations of Annular Array Actuators A basic hypothesis of UMAT is that the long time structural response depends on the initial ultrasonic loading functions (guided wave modes and frequencies). A good control on the guided wave excitation is therefore a key to success of a UMAT test. Piezoelectric comb transducers have been proven as an effective means of guided wave mode control [8]. Through guided wave mechanics study, we have shown that the guided wave excitations of comb type transducers can be considered as a combination of two separated parameters. One parameter is the guided wave excitability which is a value determined purely by the wave structures of the guided waves, and thus is independent of the comb transducer geometry. In contrast, the other parameter, often calculated as the excitation spectrum, is determined by the geometry of the transducer only. It has been shown that the wavenumber domain excitation spectrum of a comb transducer can be evaluated through a spatial Fourier transform of the transducer geometry [8]. Obviously, different comb geometries such as different element spacing produce different guided wave excitations. To vary the guided wave excitations without physically changing the comb geometry, time delays can be applied to the comb elements to simulate element spacing variations [8]. Since the time delays can be easily changed using a delay circuit, the time delay comb transducers offer great flexibility for guide wave mode control. For UMAT testing, besides guided wave mode control, it is also desired to achieve high frequency vibrations with complete structural coverage for defect detections. Annular array transducers, due to their omni-directional wave excitations, have therefore been adopted to replace regular rectangular-shaped comb type transducers for UMAT applications. The circularcrested waves excited by an annular array can be well approximated as plane waves generated by a comb transducer under a far field assumption. In fact, it is quite common in today’s guided wave research community to use comb models for approximate studies of guided wave excitations by annular arrays. In our study, however, we found that there are differences between a comb approximation and an exact annular array solution for the circular-crested guided waves excited by annular arrays. By combining the point source solution for guided wave excitation [9] with the Huygen’s principle, we were able to show that the far field guided wave excitations of any axisymmetric sources can be written as:

[

 2π (3 D ) T uν ≈ 2 Eν • (t r eˆ r , tθ eˆ θ , t z eˆ z )  kν ri

]

T

∞  exp(ikν r )∫ rf (r )J 0 (kν r )dr  ,   0 (3 D )

Where uν is the guided wave particle displacement field of the ν th guided wave mode, Eν

(1)

is a 3D excitability matrix for

point sources, t denotes the surface traction introduced by the point source, r, θ, and z are the three orthogonal directions in a cylinder coordinate system whose origin is located in the center of the plate, d is the plate thickness, kν is the wave number,

ω is the radian frequency , J 0 denotes zero order first type Bessel function, and f (r ) is the spatial distribution of the axisymmetric source in r direction. It is apparent that the excitations can still be separated into two parameters, as in the comb case. What different is that the wave excitation spectrum governed by the array geometry, as given in the second curly brackets in Equation (1), is in a format of a first order Hankel transform rather than a Fourier transform. A comparison of the wavenumber domain spectrums of a comb transducer and an annular array is given in Figure 1. The annular array geometry is shown in Figure 2a. Note that excitation spectra are all normalized to 1. The wavelength x-axis is normalized by the element spacing s. A clear difference in the excitation spectra can be observed for the two transducers. Based on the Hankel transform given in Equation (1), it can be shown that the excitation spectrum of a time delay annular array illustrated in Figure 2b can be calculated as:

F (k , dt ) =

2π k

N

∑ {[(r n =0

0

+ ns + w)J 1 (k (r0 + ns + w)) − (r0 + ns )J 1 (k (r0 + ns ))]exp(iωndt )}

(2)

27

Different from the time delay comb transducer cases in which two distinct excitation spectrums can be calculated representatively for forward propagating and backward propagating waves, the excitation spectrum described by Equation (2) includes both inward and outward propagating waves of an annular array. This is due to the fact that for the derivation of Equation (2), the time delays were directly applied to the point source solution for the omni-directional circular-crested wave excitation. The final equation was a combination of the point source solutions based on the Huygen’s principle.

Figure 1 Comparison of annular array excitation spectrum and comb excitation spectrum. For the annular array, r0=2mm. Number of elements N=3 elements, element width w=2 mm, and spacing s=4 mm.

Figure 2 Axisymmetric geometry of an annular array and the illustration of a time delay annular array. Equation (2) was validated through the comparisons with finite element (FE) simulations. To calculate time domain signals, Eq. (2) was applied to the different frequency components of the driving signal for the annular array. The final signal can be reconstructed from an inverse Fourier transform of the frequency domain calculation results [9]. Example comparisons are shown in Figure 3. The time delay annular array used in both the analytical study and the FE simulation was a 4 element array. The element spacing was 3 mm and the element width was 1.5 mm. A time delay of 833 ns was applied to each adjacent element pair to shift the 300 kHz input signal for 0.25 period. Surface out-of-plane displacements at a far field location were used for the comparisons. As demonstrated, excellent agreements were obtained. Notice that the amplitudes of the signals were not normalized. Such a comparison therefore validates our time delay annular array guided wave excitation model.

28

Figure 3 Comparison between FEM result and analytical prediction for out-of-plane displacements excited by a 4 element time delay annular array. Excellent agreements were obtained. This validates our analytical model. Transient Guided Wave Experiment Prior to high frequency vibration modal analysis tests, a transient guided wave test was conducted first to validate the appropriate time delays for the excitations of different guided wave modes. A frequency of 200 kHz was used in the transient wave test. The test was carried out on a ¼” thick aluminum plate. Since the first cut-off frequency for the guided waves in the aluminum plate is much higher than 200 kHz, only two Lamb wave modes: A0 and S0 wave modes can be excited. The annular array transducer used in the experiment had 4 electrode rings. The electrode element spacing of the transducer was 2 mm. The active piezoelectric component of the actuator was made from piezo-composite materials. The transducer also had a copper housing and backing material made from mixture of tungsten powder and epoxy. The application of the backing layer was to reduce possible ringing signals of the piezo-composite element and gain better frequency control. Figure 4 shows the design and the finish of the annular array transducer. The transducer was used as the actuator for both the transient guided wave tests and the UMAT tests that will be discussed later.

Figure 4 Design and finish of the annular array actuator. Based on Equation (2), the excitation spectrum of the annular array shown in Figure 4 was calculated for a time delay dt = 500 ns. The result is shown in Figure 5 together with the dispersion curves of the ¼” thick aluminum plate. Notice that the wavenumber domain spectrum was combined with the frequency spectrum of the driving signals applied to the annular array to create a 2D spectrum. For ease of comparison to dispersion curves, the 2D spectrum was transferred into a phase velocity dispersion curve space [8], as shown in Figure 5. It is clearly shown that with the 500 ns time delay the annular array transducer can efficiently excite the 200 kHz A0 mode. The experimental signal for the 500 ns time delay is given in Figure 6. For comparison, the signal under a 4000 ns time delay for an efficient S0 excitation is also shown in Figure 6.

29

Figure 5 Excitation spectrum of the annular array transducer under a 500 ns time delay schedule.

Figure 6 Transient guided wave signals for the time delays 4000 ns and 500 ns. Based on the time-of-flights (TOFs) calculated from the group velocities of the two wave modes, the A0 and the S0 modes were identified from the guided wave signals. As expected from the excitation spectrum calculations, when the time delay 500 ns was used, the A0 mode was dominant in the signal. While for the time delay 4000 ns, the S0 mode was dominant. Those two time delays were then used for further ultrasonic vibration tests as two different guided wave modes can be selected by the time delays as the ultrasonic vibration loading function.

30

UMAT Experiments In the UMAT tests, 12 piezoelectric disk sensors were attached to 12 randomly selected positions on the surface of the aluminum plate using instant adhesives. The annular array actuator was also glued to the surface of the plate. In such a way, the measurement inconsistency due to sensor/actuator coupling conditions can be greatly suppressed. To take a baseline data, the plate was simply placed onto a flat table. Continuous sine wave outputs from the multichannel hardware system were applied to the 4 elements of the annular array actuator. Phase shifts among different elements that were corresponding to the time delays 4000 ns and 500 ns were applied to obtain the baseline data for the two different vibration loading functions. The peak-to-peak amplitudes of the forced vibrations under the two loading functions were recorded as the baseline. Previous to the damage detection tests, boundary condition variations were introduced to the plate by clamping the plate to the table surface at different clamping locations. For each boundary condition change, the forced vibration amplitudes at the 12 disk sensor locations were collected under the two different time delay annular array loading functions. For damage detection, a simulated corrosion area was introduced to the aluminum plate followed by a saw cut defect. Forced vibration data were recorded after each defect was introduced. Figure 7 shows the plate with the actuator, sensors, and the defects. The two pictures also demonstrate how the boundary condition variations were introduced. However, it should be mentioned that the boundary conditions for taking the damage detection data were the same as the baseline. In other words, when the boundary condition variation study was carried out, there were no defects on the plate. Whereas for the damage detection tests, there were no clamping boundary conditions. Based on the wave structure characteristics of the two guided wave inputs, it is expected that the 200 kHz A0 mode is more sensitive to surface conditions than the 200 kHz S0 mode. Since the boundary condition changes and the simulated corrosion defect are all located on the plate surface, the ultrasonic vibrations introduced by the A0 input was expected to yield larger variations than the ones introduced by the S0 input.

Figure 7 Actuator/sensor setup and demonstration of boundary condition variations used in the experiments. (Left) Boundary condition change A, (Right) Boundary condition change B. Note that the boundary conditions for taking the damage detection data were the same as the baseline, i.e., no clamping boundary conditions were applied. To quantify the ultrasonic vibration variations due to the boundary condition changes and the introduction of defects, relative vibration amplitude changes from the baseline data were calculated for every sensor under every test condition. The calculation results for all 12 sensor positions were then averaged to yield an overall relative change for each test condition. The overall relative changes of the ultrasonic vibrations under the S0 guided wave loading function and the A0 loading function are shown in Figure 8. As can be seen, rather obvious vibration amplitude changes were observed for the boundary condition changes and the damage. Between the two different guided wave vibration loading functions, the A0 mode input yielded larger changes compared to the S0 input, which agreed well with the sensitivity prediction based on wave structures. The variations of vibration amplitudes, for both loading functions, were higher for the damage than for the boundary condition changes. Figure 9 shows the relative changes of the vibration amplitude ratios for the two different loading conditions (two different time delays). Basically, an A0/S0 ratio was calculated for each sensor position under each test conditions by dividing the

31

vibration amplitude obtained with the 500 ns time delay by the corresponding amplitude obtained with the 4000 ns time delay. The results shown in Figure 9 are overall relative changes of the A0/S0 ratios for the boundary condition changes and the two defects. The results suggested that the A0/S0 ratio can be a good feature for damage detection as it is very sensitive to the plate damage, but does not react on boundary condition changes much. It may therefore be possible to use such a feature to distinguish damage from boundary condition changes.

Figure 8 Relative ultrasonic vibration amplitude changes introduced by boundary condition changes and damage. The results for both the S0 loading function and the A0 loading function are included.

Figure 9 Relative changes in the A0/S0 vibration amplitude ratios for different test conditions. The A0/S0 ratios were calculated by dividing the vibration amplitudes obtained under the A0 loading function by the corresponding amplitudes obtained under the S0 loading function. For nondestructive evaluation (NDE) and structural health monitoring (SHM) of a structural component within a mechanical system, it is not only desired to detect any damage to the component itself, but also necessary to reveal the integrity of the connections from the component under inspection to other system components. For example, it is not only important to inspect an aircraft wing section for corrosion and cracks, but also critical to evaluate the condition of the joints between the wing section and the aircraft fuselage. From UMAT test of view, the changes in the condition of the joints can be considered as a boundary condition change. The UMAT experimental results presented here demonstrate that the high frequency ultrasonic vibration patterns are subject to changes due to both damage and boundary condition variations. It has also been shown that by applying different guided wave loading functions and using different signal features, different sensitivity can be achieved for damage and boundary condition changes.

32

Concluding Remarks An ultrasonic modal ananlysis technique (UMAT) that combines the advantages of both ultrasonic guided wave methods and vibration modal analysis techniques is proposed for defect detection. Key to success of UMAT is to use special ultrasonic vibration loading functions provided by different guided wave modes and frequencies to achive optimum defect detection sensitivity. Time delay annular array transducers were proposed as the actuators for the UMAT tests to provide great flexibility in guided wave mode and freuqency selection. A theoretical solution for the guided wave excitation spectrums of time delay annular array actuators was developed and validated. Such a solution will be used for further annular array actuator design and the selection of appropriate time delays to achieve any desired guided wave loading functions. Example UMAT experiments were carried out. Two different guided wave loading functions that were associated with two different wave modes: A0 mode and S0 mode were used in the experiments for damage detection in an aluminum plate. The boundary condition changes for the vibration tests were also studied. It was demonstrated that the ultrasonic vibrations under both guided wave loading functions have excellent sensitivity to damage. The boundary condition changes affected the vibrations as well. By using the amplitude ratios calculated from the amplitudes obtained under two different guided wave loading functions, we were able to show the potential of differentiating damage and boundary condition changes. Acknowledgement This work is supported by AFOSR under STTR Grant #FA9550-09-C-0105 through program manager Dr. David Stargel. References [1] Ewins, D.J., (2000), Modal Testing: Theory, Practice and Application, second ed., Research Studies Press LTD., Baldock, Hertfordshire, England. [2] Rose, J.L., (1999), Ultrasonic Waves in Solid Media, Cambridge University Press, New York, NY. [3] Chimenti, D.E. (1997), “Guided Waves in Plates and Their Use in Materials Characterization,” Appl. Mech. Rev. vol. 50(5), 247-284. [4] Rose, J.L. (2004), “Ultrasonic Guided Waves in Structural Health Monitoring,” Key Engineering Materials Vol. 270-273, 14-21. [5] Rose, J.L., Pilarski, A., and Ditri, J.J., “An approach to guided wave mode selection for inspection of laminated plate,” J. Reinforced Plastics and Composites, Vol. 12(5), 536-544 (1993). [6] Rose, J.L., Yan, F., and Zhu, Y., “Ultrasonic Guided Wave Modal Analysis Technique(UMAT) for Defect Detection,” Proceedings of IMAC XVIII, Jacksonville, FL, Feb. 2010. [7] Fei Yan and Joseph L. Rose, (2010), Defect detection using a new ultrasonic guided wave modal analysis technique (UMAT), Proc. SPIE, Vol. 7650, 76500R. [8] Fei Yan and J.L. Rose, Time delay comb transducers for aircraft inspection, The Aeronautical Journal, Vol. 113 (1144), 2009: 417-427. [9] Yan, F., Ultrasonic guided wave phased array for isotropic and anisotropic plates, The Pennsylvania State Ph.D. Thesis, 2008.

Acoustic Testing and Response Prediction of the CASSIOPE Spacecraft Viresh Wickramasinghe, Anant Grewal and David Zimcik National Research Council Canada, Institute for Aerospace Research 1200 Montreal Road, Ottawa, Ontario, Canada K1A 0R6 Andrew Woronko, Patrick Le Rossignol and Vincent-Olivier Philie MDA Space Missions 13800 Commerce Parkway, Richmond, British Columbia, Canada V6V 2J3 Mark O’Grady and Raj Singhal Canadian Space Agency, David Florida Laboratory 3701 Carling Avenue, Ottawa, Ontario, Canada K2H 8S2

ABSTRACT A high intensity acoustic test in a reverberant chamber was conducted on the CASSIOPE spacecraft in the final stages of integration and test campaign to ensure that it would survive the acoustic loads during launch. This paper describes the acoustic test methodology, the details of the model used for analytical prediction of the structural response for acoustic excitation and discussion of the predicted response comparison with test results that provided confidence in the spacecraft structural design for acoustic loads. The objective of the spacecraft acoustic test was to demonstrate the ability of the structure and avionics to withstand the broadband random acoustic environment experienced within the launch vehicle payload fairing. The CASSIOPE spacecraft was tested in the reverberant chamber at overall sound pressure level up to 142.1 dB. The automatic spectral control system of the acoustic test facility, which used six control microphones, was able to achieve and the maintain target spectrum levels around the spacecraft within tolerances without manual adjustments to the noise generators’ controls. The dynamic response of the CASSIOPE spacecraft during the test was measured using a large number of accelerometers installed on critical locations of the structure. Low level pre-test and post-test structural response signatures as well as electrical integrity checks performed after the exposure to the proto-flight acoustic environment demonstrated the ability of the spacecraft to survive the launch. The acoustic response of the spacecraft was also predicted based on a finite element model analysis to identify the critical components, evaluate structural margins and assess the risks in proceeding with a proto-flight acoustic test based on the specified launch vehicle spectrum. The analysis method used to predict the responses combines the NX/NASTRAN solver and RAYON, a vibro-acoustic simulation software. The RAYON software functionality is based on a boundary element model that enables the creation of an accurate fluid loading on the structure, with consideration of fluid mass and damping effects. The study used a finite element model of the structure that was correlated through an experimental modal survey test and actual spectrum levels achieved during the acoustic test. Responses of most locations compared favourably with the predictions in critical locations such as the solar arrays. Due to the limited availability of the satellite as well as time and cost constraints in a spacecraft development program, it is important to perform both qualification tests as well as analytical predictions in an efficient and timely manner to validate structural designs of spacecraft.

1.0

INTRODUCTION

CASSIOPE is a Canadian small satellite mission combining multiple payloads [1]. The prime contractor for the mission is MacDonald Dettwiler and Associates (MDA). The mission is enabled by contributions from the Canadian Space Agency (CSA) and Technology Partnerships Canada (TPC). The mission objectives are to demonstrate the advanced communications technologies of the Cascade (CX) payload, to investigate the topside ionosphere using the ePOP science payload, and to develop a Canadian small satellite bus for future CSA missions. Bristol Aerospace Ltd. developed the bus as part of the CSA SmallSAT Bus initiative. The Cascade payload contributed by MDA provides a high speed Ka-Band store and forward capability [2]. The ePOP payload known as Enhanced Polar Outflow Probe is a suite of eight science instruments provided by the University of Calgary.

T. Proulx (ed.), Advanced Aerospace Applications, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 4, DOI 10.1007/978-1-4419-9302-1_4, © The Society for Experimental Mechanics, Inc. 2011

33

34

Figure 1 CASSIOPE spacecraft in the stowed launch configuration As part of the spacecraft environmental test campaign, a high intensity acoustic test was conducted on the proto-flight model of the CASSIOPE spacecraft, which included all of the payload, solar panels, bus instruments and electronics boxes. The proto-flight model (PFM) level acoustic testing was performed using the large reverberant chamber at the Institute for Aerospace Research of the National Research Council Canada (NRCC) in Ottawa, Ontario, Canada in June 2009. This test exposed the integrated spacecraft to reverberant acoustic loading to ensure that it would survive the acoustic loads during launch. MDA held overall responsibility for the test planning and execution, with facility and instrumentation support from NRCC and CSA, respectively. The reverberant acoustic test chamber facilities used for qualification of spacecraft are required to generate high levels of random noise and maintain high spectral accuracy for short durations, typically 30 to 90 seconds, which preclude manual adjustments of the noise generators’ controls within this very limited test durations [3]. The generation of target acoustic fields that represent launch vehicle loads is challenging due to the highly non-linear noise generation process and the effect of the spacecraft within the acoustic filed [4]. Interaction between the structural and the fluid medium is particularly important for the design of satellites such as CASSIOPE integrated with components that are susceptible to acoustic loads. Unaccounted dynamic behaviour for acoustic loading of the satellite structure may lead to catastrophic failure in structural components or damage to the payload during launch. Therefore, successful design and deployment of precision systems such as satellites require highly accurate analytical models that need to be validated using experimental data. Prior to the test campaign, analytical predictions of the acoustic response were performed by MDA Space Missions to verify structural margins of safety and to ensure predicted equipment interface responses were within qualification limits. The response of the spacecraft under acoustic excitation was predicted using an analysis method that combines the NX/NASTRAN finite element method (FEM) solver and the RAYON vibro-acoustic simulation software. The structural model developed using FEM was updated through a full-scale modal test of the spacecraft, which experimentally extracted the modal parameters up to 120 Hz for correlation. Details of this ground vibration modal survey test and FEM model correlation process have been published previously [5]. In addition, these structural response predictions uses the actual 1/3-octave acoustic spectrum levels achieved during the test as input. This finite element methodology was selected due to its capability to predict the responses of major structural panels, where peak responses are expected to be below 250 Hz. Other prediction methods, such as the statistical energy analysis, have also been studied for acoustic response prediction applications for spacecraft [6, 7]. The objective of this paper is to describe the details of the acoustic testing methodology and analytical prediction technique used to achieve successful qualification of the CASSIOPE spacecraft under launch acoustic loads. This paper provides details of the acoustic test procedure including the spacecraft configuration, test setup, instrumentation, data acquisition and spectrum analysis techniques to characterise the diffused acoustic environment to compare the achieved noise levels to the target spectrum. More importantly, this paper includes the details of the analysis that was used to predict the response of the spacecraft structure under acoustic loading using an experimentally correlated finite element model and the actual 1/3-octave spectrum levels achieved during the acoustic test. Experimentally observed responses at critical locations of the structure are compared with the analytical predictions in order to assess the fidelity of the tool for spacecraft response prediction for acoustic excitation. Evaluating the responses of the major elements of the spacecraft was particularly important to validate their structural designs for launch acoustic loads.

35

2.0

SPACECRAFT DESCRIPTION

The CASSIOPE spacecraft in the stowed launch configuration is shown Figure 1. The 490 kg spacecraft is a hexagonal structure of nominal side dimension of 1.6 m with 1.4 m in height. The interface to the launch vehicle consists of six discrete mounts which mate with nonexplosive actuators on the launch vehicle. The spacecraft core structure consists of a hexagonal vertical frame constructed of aluminum honeycomb sandwich panels and machined corner posts. There are three horizontal aluminum honeycomb sandwich panels: the ePOP payload panel, the internal Mid-Deck panel, and the CX payload panel. All panels of the structure support spacecraft avionics and harnessing. Five solar arrays, constructed of composite facesheet and aluminum core sandwich, are mounted to the external panels via titanium flexures to thermally decouple the arrays from the spacecraft. The solar arrays are especially susceptible to launch acoustic excitation due to their low mass to area ratio.

3.0

ACOUSTIC TEST SETUP

The objectives of the spacecraft proto-flight acoustic test were; (i) to demonstrate that the spacecraft secondary structure could survive the proto-flight level acoustic environment, (ii) to demonstrate that the spacecraft structural responses do not exceed the unit interface specifications, and (iii) to expose the spacecraft avionics to the proto-flight level acoustic environment such that subsequent electrical integrity checks provide confidence in the overall quality of the component integration and their Figure 2 CASSIOPE spacecraft in the test chamber ability to survive the launch loads. In order to perform the test, the fully integrated CASSIOPE spacecraft was placed in the center of the reverberant acoustic test chamber as shown in Figure 2. The spacecraft was installed on a fixture stand using launch vehicle interfaces. The fixture was isolated from the chamber floor using pneumatic isolators. The isolation mounts were inflated such that the resonance of the spacecraft and fixture assembly was below the minimum frequency of acoustic excitation. 3.1 Reverberant Chamber The reverberant acoustic chamber at NRCC is a specialized high-intensity noise testing facility designed to test full-size aerospace components at high levels of sound pressure field. This reverberant chamber has dimensions of 6.9m x 9.75m x 8.0m and encloses a test volume of ~540 m3. This test facility is capable of generating overall sound pressure levels greater than 157 dB with accurate acoustic spectrum shaping between 25 Hz and 10,000 Hz. Four horns with lower cut off frequencies of 25 Hz, 32 Hz, 100 Hz and 200 Hz are installed through the walls in order to generate the shaped noise spectrum. Test chamber walls were constructed with reinforced concrete to withstand high intensity noise generated within the chamber. 3.2 Measurement Microphones The acoustic environment around the spacecraft was measured by seven high intensity condenser microphones placed around the spacecraft as shown in Figure 3. The microphones were located approximately 0.6 m from the spacecraft external surfaces. These precision microphones feature a wide frequency range as well as a high dynamic range in order to accurately measure the far field noise. The signals from these microphones were amplified and conditioned and the amplified output of microphones labelled Mic1 through Mic6 were multiplexed in the time domain at a rate of 0.2 Hz and used as the control input to the automatic spectral control system of the test facility. However, microphone labelled Mic7 located underneath the ePOP deck was only used for monitoring purposes because the cavity between the spacecraft and support structure did not represent the diffused acoustic environment.

36

Figure 3 Tope view (left) and side view (right) of microphone positions 3.3 Noise Generation The reverberant noise field during for this acoustic test was generated by two Wyle Laboratories WAS-3000 airstream modulators, one mounted on the 25 Hz horn and the other on the 100 Hz horn located in the chamber wall. A supply of dry compressed air at a pressure of 25 psig was used to drive the generators. The WAS-3000 is an electro-pneumatic noise source rated at 30 kW acoustic power, with either sine or random wave input, over a frequency range of 25 to 10,000 Hz. The specifications for WAS-3000 show that the nominal controllable frequency range extending up to 1250 Hz and it can generate very high overall sound pressure levels. The actual controllable amplitude and frequency of these airstream modulators can be varied depending on the air pressure, drive current and spectral shape. It should be noted that the effective control of the noise spectrum is possible only between the lower cut-off frequency of the drive horn and the upper cut-off frequency of the noise generators, whereas the required test spectrum generally covers a much wider frequency range. Therefore, spectral content above the 1250 Hz is controlled only indirectly through non-linear distortion from lower frequency noise. During this acoustic test, two impingement corner jets were also used to supplement the high frequency content of the spectrum with controllable input. These impingement jets were operated at 19 psig and 13 psig in order to generate the required high frequency noise spectrum for this acoustic test. 3.4 Automatic Spectral Control System A customized automatic spectral control system (ASCS) developed at NRCC automatically analyzes and controls the random noise spectrum within the chamber during acoustic tests, ensuring the accuracy of the simulated environment is maintained throughout the duration of the test. The ASCS is able to match the acoustic specifications of the space shuttle, rocket launches, aircraft structural excitation, engine nacelle noise, unsteady turbulent airflows and other acoustic test specifications. The acoustic control system analyzes the multiplexed output from the measurement microphones using a spectrum analyzer to generate real-time 1/3-octave band levels. The control software compares the spectrum of the measured microphone signals with the target spectrum and updates the band levels of a bank of 1/3-octave filters within a noise shaper via an array of closed loop controllers. This results in an accurate and robust shaping of the drive current of the noise generators to maintain the noise environment within the specified tolerances without manual adjustments during the test. Prior to placing the test article in the chamber, room empty trials were carried out in order to ensure that the target acoustic spectrum could be achieved and maintained during the specified test duration. The controller parameters, including the steady state values of the 1/3-octave band shaper, from the room empty trials were used as the initial estimate values for the ASCS during the acoustic test with the spacecraft. Using these initial values, the ASCS is able to rapidly produce accurate acoustic levels in this reverberant chamber by overcoming challenges due to highly non-linear acoustic generation process and acoustic absorption or insertion effects of large spacecraft placed within the acoustic environment. 3.5 Response Accelerometers The response of the critical locations of the CASSIOPE spacecraft were measured using 99 high sensitivity miniature accelerometers provided by the David Florida Laboratory of the Canadian Space Agency, which were 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 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. These include locations to

37

extract fundamental modal frequencies of major panels and vibration response of critical equipment installed in the satellite. Response data from the accelerometers were subsequently used to compare to predicted responses for acoustic loads. 3.6 Data Acquisition System Two LMS SCADAS III digital data acquisition systems were used for simultaneous data recording of the accelerometers and microphone channels during the acoustic test [8]. The sampling frequency of the data acquisition system was set at about 10 times above the maximum frequency of interest in order to capture high quality data at high frequencies. The frequency resolution was set to 4 Hz for the test and the maximum number of linear averages was taken during the length of the test in order to mitigate the noise in the measured data.

4.0

ACOUSTIC TEST RESULTS

The proto-flight model (PFM) acoustic test levels for the CASSIOPE spacecraft were based on the acoustic spectrum analysis for the launch vehicle. The test sequence included pre-test low level at –10 dB for 30 seconds duration, acceptance level test at –3 dB for 30 seconds duration, PFM level test for 60 seconds duration and post-test low level at –10 dB for 30 seconds duration. Since the lower level tests were performed specifically to ensure safety of the spacecraft and/or validate the subsequent structural condition of the spacecraft, data pertaining only to the PFM level test is discussed in this paper. 4.1 Comparison of Achieved Spectrum to the Target The achieved acoustic level measured using the multiplexed signal of the six control microphones is compared with the target test levels and tolerances using OASPL and 1/3-octave bands as shown in Figure 4. The reverberant acoustic test facility was able to generate the required acoustic environment within tolerances to meet the OASPL and all of the 1/3 octave narrow bands except for two bands. The 40 Hz and 80 Hz 1/3-octave band were out of tolerance by 0.4 dB and 0.2 dB, respectively. However, both these bands, including all other bands, were within tolerance during room empty trials that were conducted to identify the control parameters for the ASCS setup. The accuracy of the acoustic spectrum achieved with the spacecraft in the chamber was very good and the small exceedance of tolerances in the two low frequency bands were considered minor.

Figure 4 Target and Achieved spectrum 4.2 Characterization of Acoustic Environment Comparison of the 1/3 octave bands measured by the six control microphones placed around the spacecraft shown in Figure 5 recognizes the variation in the surrounding acoustic environment in the presence of the CASSIOPE spacecraft. The data show larger variation in the lower frequencies. This variation in low frequencies may be attributed to two primary reasons; (i) presence of a large spacecraft within the acoustic filed with absorption or insertion effects (ii) presence of a relatively small number of acoustic modes due to the physical size of the chamber. The ability of a reverberant acoustic facility to produce a diffused environment at lower frequencies directly depends on the volume of the test chamber [9]. Multiplexing the signals from the six control microphones spatially distributed around the spacecraft as well as the time domain averaging

38

of the multiplexed signal during the test is used to account for the variation in the input spectrum during this challenging broadband random noise generation process.

Figure 5 Spatial variation acoustic environment around the spacecraft 4.3 Structural Responses The structural responses of the spacecraft measured using accelerometers were analyzed in the form of Power Spectral Density (PSD) plots and overall acceleration was measured in root-mean-square (RMS) values in units of g(rms). With the exception of a few locations that showed minor peak response exceedance, most of the responses measured during the PFM level acoustic test were well within unit qualification levels. Typical structural responses of five different panels during the PFM level acoustic test are shown Figure 6. These particular locations were selected for this present study because these panels exhibited a variety of dynamic behaviour due to acoustic excitation. Characteristics of these curves will be discussed in detailed in Section 6.0 by comparing with the response predictions performed using the analytical method introduced in Section 5.0. It is important to note that low level pre-test and post-test structural response signatures as well as electrical integrity checks performed after the exposure to the proto-flight test level confirmed the structural integrity of the spacecraft to survive the launch acoustic loads.

Figure 6 Responses of different panels during the PFM level acoustic test

39

5.0

RESPONSE PREDICTION METHODOLOGY

Successful design and deployment of precision systems such as satellites require very accurate analytical tools that must be validated using experiments to be used for prediction. Unaccounted dynamic behaviour in the satellite structure may lead to failures during the test campaign or prone for catastrophic damage during the launch. Therefore it was important to perform analytical predictions of the acoustic response of the CASSIOPE spacecraft prior the test in order to verify structural margins of safety and to ensure responses of critical components were within qualification limits. This current study enhanced the confidence of these prediction methodologies to be used in an efficient and timely manner to validate spacecraft designs. 5.1 Analysis Objectives This response prediction analysis focused on the areas of the spacecraft which were most susceptible to the acoustic excitation. These areas of concern are; (i) external items with low mass to area ratio such as the body mounted solar arrays, (ii) external payload panel responses, and (ii) the spacecraft electronics equipment interfaces. The objectives of the analysis was to predict the acoustic responses at these locations, in terms of power spectral density (PSD) and root-mean-square (RMS) values, and to compare the levels obtained to structural allowables and equipment qualification levels. In the current study the frequency range as been limited to 250 Hz. This low frequency analysis enveloped most of the primary modes of critical spacecraft components while reducing complexity of the FEM and improving computational time. 5.2 Analytical Method The CASSIOPE spacecraft features lightweight solar panels mounted adjacent to the primary structure honeycomb panels. In this configuration, the fluid acoustic loads need to be adequately modeled. Thus a vibro-acoustic simulation software, RAYON, was selected for this application. This simulation software uses the boundary element method (BEM) model that creates the fluid sound pressure loading on the structure. The RAYON software is a fluid-structure analysis software dedicated to linear acoustic and vibration analysis in the frequency domain. Although there are several options available for application of mechanical and acoustic loads to the FEM model, planar waves are the most suitable to represent acoustic excitation of the CASSIOPE spacecraft. A diffuse acoustic field that exists in the test chamber was simulated by means of uncorrelated planar waves uniformly distributed around a sphere. The diffuse acoustic field was then modeled using 26 planar waves with 45º separation between each wave. The boundary of the fluid model was defined using a 2D surface model. The appropriate side of the BEM surface was exposed to acoustic sound pressure excitation to represent the current application. The test correlated CASSIOPE spacecraft FEM model was generated by Bristol Aerospace Ltd. and MAYA Heat Transfer Technologies, while the fluid BEM surfaces were generated by MDA. These models are presented in the Figure 7. The FEM model correlation was performed for modes up to 120 Hz based on a full-scale modal test of the flight model spacecraft performed by NRCC. The spacecraft modal survey was conducted in a flight-like configuration with a high level of equipment integration to experimentally obtain the most realistic modal information to update the FE model. The multi-input multi-output modal test which used two independently driven shakers simultaneously, generated multi-referenced Frequency Response Functions (FRFs) to resolve closely spaced modes [10]. Furthermore, the advanced curve fitting algorithm used for the modal analysis generated clear stability diagrams using FRFs so that the modal parameters could be extracted without much difficulty. The details of the modal survey ground vibration test have been published previously [5]. The purpose of the present study is to evaluate the accuracy of the acoustic analytic method within the frequency range of 25 Hz to 120 Hz for which the FEM model was correlated based on the modal test, and also for a higher frequency range, up to 250 Hz, that becomes critical for bus mounted units. The RAYON software was used in conjunction with the NX/NASTRAN software to obtain the acceleration responses in PSD and RMS as well as loads and stresses of the spacecraft under acoustic excitation. 5.3 Modal Damping Modal damping is an important parameter that needs to be properly quantified in order to accurately predict the response of structures subjected to excitation loads. Within the present study, the structural modal damping was defined using the standard amplification parameter known as Q. The parameter Q = 1/2ζ, where ζ is the critical damping ratio. For the present predictions of acoustic responses, the Q values were defined as follows for two distinct frequency ranges; a Q of 10 was set for the frequency range from 25 Hz to 150 Hz while a Q of 25 was set for the frequency range from 150 Hz to 250 Hz. These Q values were applied for the entire spacecraft FEM model. The correlation between the response predicted through this analysis and the experimental test results are shown to have reasonable agreement using these amplification factors. However, the experimental Q values extracted from the base input vibration test performed on the CASSIOPE spacecraft were different than the above mentioned nominal values assumed for the FEM predictions. The Q values derived for the frequency range of 25 Hz to 120 Hz from the low level sine sweeps vibration test for several response locations, namely, Top

40

Figure 7 CASSIOPE spacecraft FEM (left) and BEM (right) representation Solar Array, ePOP Panel, and the CX Panel were 33, 10 and 18, respectively. These Q value amplification factors are all higher than those values used to predict the acoustic responses of the structure. Typically the Q values obtained from a low magnitude test are higher than Q obtained from a high magnitude test because the level of damping increases for higher magnitude responses. Therefore, using Q values derived from low level vibration test provides more conservatism if used for the analytic prediction of acoustic responses.

6.0

COMPARISON OF TEST DATA WITH PREDICTIONS

In this section, the predicted structural responses from the analysis are compared to the acceleration obtained during the CASSIOPE spacecraft PFM level acoustic test. The power spectrum density (PSD) and the root mean square (RMS) acceleration are presented for the frequency range of 25 Hz to 250 Hz. All responses presented are the panel out-of-plane responses, which are significantly higher than the in-plane responses. The input excitation level used for the analysis was the averaged 1/3-octave acoustic sound pressure levels measured by the multiplexed signal of the six control microphones. The frequency range for the analysis has been limited to 250 Hz, which is the frequency range of interest for the major structural panel responses. The frequency uncertainty needs to be considered for the test data because the frequency resolution was set at 4 Hz. Results are presented for five response locations in order to cover the dynamic behaviour of different panel configurations. These include a lightweight-large area of the Top Solar Array, a heavy and populated payload panels known as CX and ePOP panels, an internal panel known as the Mid-Deck panel, and an external avionics support panel known as the +X RAM panel shown in Figure 7.

Figure 8 PSD responses for Top Solar Array

6.1 Top Solar Array The PSD of the predicted and test responses at the center of the large hexagonal top solar array, which is mounted to the CX panel via flexures, is shown in Figure 8. The dominant natural frequencies of the panel are predicted well and the PSD response curve closely matches the test data observed during the acoustic excitation. The predicted RMS acceleration was 17 g(rms) while the measured response was 9.2 g(rms). The predicted response was 5 dB higher than that measured during the test, however this is considered acceptable given the uncertainty in modal damping of the lightweight-large structure during acoustic test. The natural frequencies and PSD peak values are summarized in Table 1.

41

Table 1: Peak Frequency and PSD Level comparison for Top Solar Array FEM Predictions Peak Frequency PSD Level (Hz) (g2/Hz) 68 5.3 150 2.4 200 7.5

Test measurements Peak Frequency PSD Level (Hz) (g2/Hz) 72 0.7 156 1.2 220 2.6

Table 2: Peak Frequency and PSD Level comparison for CX payload panel FEM Predictions Peak Frequency PSD Level (Hz) (g2/Hz) 98 0.37 200 0.044

Test measurements Peak Frequency PSD Level (Hz) (g2/Hz) 116 0.14 224 0.17 6.2 CX Payload Panel The honeycomb CX payload panel located directly under the top solar array is populated with heavy electronic units. The PSD of the predicted and test responses for this panel are shown in Figure 9. The natural frequencies and PSD amplitude levels for the prediction and test are compared in Table 2. The FEM predicted RMS acceleration response was 2.5 g(rms) while the measured response was 2.4 g(rms) and this difference is considered negligible. The primary panel natural frequency at 116 Hz is predicted well, while the second peak is not captured by the FEM analysis. The poor prediction of the main peak around 224 Hz may be attributed to the level of fidelity of the FEM model at higher frequency for the CX panel, since the correlation of the FEM model with spacecraft experimental modal test data was limited to 120 Hz. One possibility is that this main peak occurs above 250 Hz in the FEM predicted responses, thus invisible since it is beyond the frequency range of 25 Hz to 250 Hz considered in this study.

Figure 9 PSD responses for CX panel

6.3

ePOP Payload Panel

The ePOP panel is a honeycomb deck which closes out the spacecraft and supports a number of payload units. The predicted and test PSD responses for the ePOP panel are shown in Figure 10. The FEM predicted RMS acceleration response was 5.8 g(rms) while the measured response was 2.5 g(rms). The predicted acceleration response was 7 dB higher than the test. As shown in Figure 10, the first main peak predicted at 88 Hz was not measured during the acoustic test, nevertheless this mode was correlated with the FEM model following the spacecraft ground vibration modal test. Further investigation showed that the acoustic field in the proximity of this spacecraft panel, recorded by a microphone labelled Mic7 in Figure 3 for monitoring only, shows relatively higher variation amplitudes at frequencies near the ePOP panel resonance frequency range of 70 Hz to 100 Hz. This is likely due to the presence of cavities between the mounting fixture and the ePOP panel as well as its close proximity to the top floor of the support stand. These interface stand boundaries were not modeled in the Figure 10 PSD responses for ePOP panel

42

acoustic prediction FEM analysis. At higher frequencies, from 150 Hz to 250 Hz, the predicted PSD has a reasonable match with the test measurement. The predicted response PSD peak at 196 Hz was 0.6 g2/Hz while the measured PSD peak at 188 Hz was 0.14 g2/Hz. In this frequency range the Mic7 microphone showed stable noise measurement which correlated well with the far field noise environment measured by control microphones.

Figure 11 PSD responses Mid-Deck panel

6.4 Internal Mid-Deck Panel The Mid-Deck panel is positioned inside the bus and is therefore not directly exposed to the external acoustic sound pressure field. The response of this interior panel to the acoustic field is of interest as lightweight sensitive equipment, with low frequency vibration modes, are mounted onto it. The PSD amplitude level predicted by analysis and measured during test, in the out-of-plane direction, are presented in Figure 11. In the present study the Mid-Deck panel was not part of the BEM model and the spacecraft internal cavity was not defined in the RAYON model. The responses predicted for this panel are then only the indirect consequence of the other panel responses excited by the external acoustic pressure. However, in reality the encapsulated air inside the spacecraft bus may interact with the dynamic motion of the external bus panels and result in a small acoustic excitation of the Mid-Deck panel. This effect is not taken into account in the present study and could explain the difference shown in the comparison plot. The frequencies of the predicted PSD peaks show a good match with the test peaks, but the overall magnitude of the predicted PSD is lower than measured data.

6.5 +X RAM Panel The +X RAM panel is located on the front external face of the CASSIOPE spacecraft. This response location is adjacent to the spacecraft computer equipment interface. Although the response at this location was very low, the predicted PSD response showed a relative good match with the test data. The FEM predicted overall RMS acceleration response was 0.49 g(rms) while the measured response was 0.50 g(rms). The difference between the predicted RMS Figure 12 PSD responses +X RAM panel response and the measured response is negligible. As shown in the Figure 12, the acoustic test PSD response reveals a spacecraft first mode of 48 Hz while the FEM modal analysis and acoustic prediction revels a first mode at 64 Hz. This phenomenon is attributed to the presence of the pneumatic isolators between the fixture and the chamber floor. The isolator flexibility was not modeled for the analytical predictions based on FEM.

7.0

CONCLUSIONS

The high intensity acoustic test was performed on the proto-flight model of the CASSIOPE spacecraft and the acoustic response of the spacecraft was predicted using a derived analytical model. The CASSIOPE spacecraft was tested in a reverberant acoustic chamber at overall sound pressure levels up to 142.1 dB and the automatic spectral control system of the test facility was able to achieve and the maintain target spectrum levels around the spacecraft within tolerances without manual adjustments during the test. The acoustic test setup in the chamber provided a stable acoustic environment to simulate the acoustic excitation expected during launch although the generation of a diffused random noise filed is challenging due to the highly non-linear noise generation process and the effect of the spacecraft within the acoustic filed.

43

Low level pre-test and post-test structural response signatures as well as electrical integrity checks performed after the exposure to the proto-flight acoustic environment demonstrated the ability of the spacecraft to survive the launch. The acoustic responses of the spacecraft were predicted using an analytic method based on the RAYON software in conjunction with the NX/NASTRAN software. This simulation software used a BEM model to that creates the fluid sound pressure loading on the structure to perform linear acoustic and vibration analysis in the frequency domain. This analysis tool predicted the response of the spacecraft structure under acoustic loading condition using an experimentally correlated structural FEM model and the actual spectrum achieved during the PFM level acoustic test. The responses in most critical locations of the major structural elements such as the Top Solar Array, CX panel and +X RAM panel compared favourably with the predictions from the FEM analysis. This analytical tool is useful for spacecraft response prediction for acoustic excitation experienced during qualification test as well as launch. The approach presented here provided an efficient and timely means to validate structural integrity of spacecraft design.

ACKNOWLEDGEMENTS The authors would like to gratefully acknowledge the invaluable contribution from Harold Dahl and Ian Walkty from the Bristol Aerospace Ltd, and Phillippe Tremblay from MAYA Heat Transfer Technologies for the generation and correlation of the CASSIOPE spacecraft Finite Element Model used in this study. In addition, Yong Chen, Luc Hurtubise and Brent Lawrie of Aeroacoustics and Structural Dynamics Group significantly contributed for the successful performance of the acoustic test on the CASSIOPE spacecraft. Finally, the authors wish to thank Lubomir Djambazov and Gaetan Forget of MDA for their excellent contributions to the test planning and setup activities.

REFRERENCES [1] Giffin, G.B., Ressl, V., Yau, A., King, P., “CASSIOPE: A Canadian SmallSAT-Based Space Science and Advanced Satcom Demonstration Mission,” Proceedings of the 18th AIAA/USU Conference on Small Satellite, Paper SSC04-VI-5, August 2004. [2] Giffin, G.B., Magnussen, K., Wlodyka, M., Duffield, D., Poller, B., Bravman, J. “CASCADE: A Smallsat System Providing Global, High Quality Movement of Very Large Data Files”, Proceedings of the IAC 2004 Conference, Paper IAC04-M.4.06, September 2004. [3] Hieken, M. H., Levo, R. W., “A High Intensity Reverberant Acoustic Test Facility,” Proceedings of the 34th Annual Technical Meeting of the Institute of Environment Sciences, pp. 131-135, May 1998. [4] Westley, R., Nguyen, K., and Westley, M. S., “Non-Linear Generation of Acoustic Noise in the I. A. R Spacecraft Chamber by Manual or Automatic Control, Proceedings of the 16th Space Simulation Conference, pp. 195-210, November 1990. [5] Wickramasinghe, V. K., Zimcik, D. G., Chen, Y., Tremblay, P. Dahl, H., and Walkty, I., “Modal Survey Test and Model Correlation of the CASSIOPE Spacecraft,” Proceedings of the International Modal Analysis Conference - IMAC XXVIII, February, 2010. [6] Manning, J. E., “Statistical Energy Analysis – An Overview of its Development and Engineering Applications,” Proceedings of the 59th Shock and Vibration Symposium, October 1988. [7] Lyon, R.H. and DeJong, R.G., “Theory and Application of Statistical Energy Analysis,” 2nd Edition, ButterworthHeinemann, Boston, 1995. [8] LMS Instruments, LMS SCADAS III Data Acquisition Front-End, Breda, The Netherlands, www.lmsintl.com, 2009. [9] Beranek, L., “Noise and Vibration Control,” Institute of Noise Control Engineering, 1988. [10] Ewins, D. J., “Modal Testing: Theory, Practice and Application,” Research Studies Press, 2000.

Force Limited Vibration Testing Applied to the JWST FGS OA Yvan Soucy

Peter Klimas

Senior Structural Dynamics Engineer Engineering Development Canadian Space Agency 6767 route de l'Aéroport St-Hubert, Qc, J3Y 8Y9, Canada [email protected]

Advanced Member of Technical Staff Space Systems COM DEV Canada 303 Terry Fox Drive, Suite 100 Ottawa, ON, K2K 3J1, Canada [email protected] ABSTRACT

The Fine Guidance Sensor (FGS) is the Canadian contribution to the James Webb Space Telescope (JWST). The optical assembly (OA) is one of the components of the FGS. In October 2009, the Engineering Test Unit (ETU) of the FGS OA was tested in preparation for testing of the flight hardware planned for the Spring of 2011. The presence of a required interface ring having about 25% of the test item mass (80 kg) complicated the planning and performance of the Force Limited Vibration (FLV) testing. Modifications to our standard FLV procedure were developed and successfully applied during testing of the ETU. Such modifications were deemed necessary based on pre-test analysis showing that the FGS OA apparent or dynamic mass was significantly changed due to the presence of the interface ring. The paper first main part presents results of the pre-test analysis that demonstrated that the proposed simple modifications to the standard FLV procedure would ensure conservatism during testing (when compared to testing without the interface ring). The other main part of the paper presents details and results of the successful ETU vibration test. For one of the lateral axes, force limiting was combined with response limiting applied in the higher frequency range. The use of force limiting resulted in significant reduction of overtesting for all three axes of excitation. For the two lateral axes, the input acceleration PSD was notched by more than 20 dB; for the vertical axis, it was notched by 17 dB. 1.

Introduction

The James Webb Space Telescope (JWST) will be the successor of the very successful Hubble space telescope. It will be a large infrared telescope with a 6.5-meter primary mirror [1]. The JWST is an international collaboration between NASA, the European Space Agency (ESA) and the Canadian Space Agency (CSA). The NASA‟s Goddard Space Flight Center is managing the development effort. Launch of the JWST is planned no sooner than 2014. The Fine Guidance Sensor (FGS) is the Canadian contribution to the JWST [2]. The two functions of the FGS are (i) for the guider to provide very precise pointing information (< 5 milli arc sec) for the control of the observatory and (ii) for the Tunable Filter Instrument to provide a narrow-band imaging capability. The prime contractor of the FGS is COM DEV Canada. The main and largest component of the FGS is the Optical Assembly (OA) [2]. Two versions of the FGS OA hardware were built: the Engineering Test Unit (ETU) and the Flight Unit. The ETU is a combination of fully flight representative hardware and of high fidelity mass/thermal „dummies‟ with a measured mass of 80 kg. In 2009, the ETU has been subjected to several types of testing in order to reduce risk for the flight model. This test campaign of the FGS ETU OA included vibration testing, at the CSA David Florida Laboratory (DFL). The vibration testing consisted of qualification sine, qualification random and sine burst runs. This paper focuses the results of the random vibration test. In order to reduce the overtesting inherent with conventional vibration testing and to protect the sensitive optical equipment, it was decided in the planning phase that the Force Limited Vibration (FLV) approach, [3] and [4], would be used for both the ETU and flight unit vibration testing. However, it became apparent a while before the ETU testing that the presence of a required massive interface ring would likely affect sufficiently the test data used by FLV to warrant some modifications to

T. Proulx (ed.), Advanced Aerospace Applications, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 4, DOI 10.1007/978-1-4419-9302-1_5, © The Society for Experimental Mechanics, Inc. 2011

45

46 the our standard force limited procedure. Pre-test analysis led to development of simple procedure modifications and showed that these changes would ensure conservatism during testing. Force limiting is excellent for notching the input acceleration at the main modes of a test item. However, it cannot be used efficiently for notching the input for higher order localized modes hardly „visible‟ at the test item interface with its mounting structure. Consequently, based on the pre-test analysis, it was decided to complement force limiting with the more traditional response limiting approach for one of the lateral axes in order to reduce the influence of fixture ringing at higher frequencies. The paper first presents an overview of the JWST, FGS and FGS OA. This is followed by a discussion on the OA interface ring requirement, design and characteristics. The next section provides some background on the estimation of force limits. The paper then describes the test configuration and sequence. This is followed by a section on test issues caused by the presence of the interface ring. The first main section of the paper follows and presents a discussion on the pre-test analysis, including prediction results with their leading planned procedure modifications for the testing phase. The paper continues with the other main section presenting the key FLV-related results of the ETU random vibration testing. 2.

Overview of James Webb Space Telescope

The James Webb Space Telescope (JWST) is a passively cooled deployable telescope that will orbit the L2 earth-sun point. The JWST Observatory consists of an optical telescope element (OTE), a spacecraft element, and an Integrated Science Instrument Module (ISIM) element (Fig. 1). The ISIM contains four science instruments: the Near-Infra-Red Camera (NIRCam); the Near Infra-Red Spectrometer (NIRSpec); the Mid-Infra-Red Imager (MIRI); and the Fine Guidance Sensor (FGS), which also serves as part of the Observatory line-of-sight control system (Fig. 2).

Fig. 1 JWST observatory with ISIM components (http://www.jwst.nasa.gov/instruments.html) 3.

Fig. 2 ISIM structure with science instruments (http://www.jwst.nasa.gov/instruments.html)

Fine Guidance Sensor (FGS)

The FGS consists of an Optical Assembly, three sets of Electronics and software that run on the ISIM Command and Data Handling Units. The primary function of the FGS is to provide continuous pointing information to the Observatory, which is used to stabilise the line-of-sight, allowing JWST to obtain the required image quality. Once in fine guide mode, the FGS will provide pointing information to a precision of  5 milli-arcseconds. The secondary function of the FGS is to provide JWST with a narrow-band imaging capability through the Tuneable filter Instrument (TFI). With a spectral resolution of between 70 and 100 and the ability to continuously tune this pass band over the wavelength ranges 1.6 to 2.5 and 3.2 to 4.9 microns, the FGS will provide JWST with a very powerful capability, not only for surveys of deep fields and star formation regions but also for extra-solar planet characterization. 4.

FGS Optical Assembly (OA)

The FGS Optical Assembly consists of an aluminum optical bench supported by three titanium kinematic mounts. The optics that support the FGS-Guider imaging function are mounted on one side of the optical bench, while the optics that support the FGS Tuneable Filter Imager (TFI) functionality are mounted on the other side. The Guider module images two separate

47 regions of the JWST field of view onto two independent 2k x 2k focal plane arrays: these channels are dedicated to the guider function. The TFI module images a third field of view with a 2k x 2k focal plane array. It contains a filter wheel and tuneable filter (Fabry-Perot etalon) that allow flexible narrow-band imaging to be performed. The FGS OA is a high precision optical instrument aligned at the kinematic mount (KM) interface to the ISIM structure. Therefore, the FGS OA coordinate system is defined at the ISIM level using the kinematic mount interface. The coordinate system axes are labeled V1, V2, and V3. 5.

FGS OA Interface Ring

Due to the strict alignment requirements of the FGS OA, the kinematic mounts mate to a very high precision interface on the ISIM structure. However, to meet the vibration test objectives without damaging the hardware from over-excitation at resonance frequencies (caused by the high impedance of the vibration table), force limited vibration was the selected notching methodology. To permit FLV while still maintaining the required position tolerance on the kinematic mount (KM) interfaces, the FGS OA was mounted on an interface ring. The interface ring allows for precise placement of the FGS OA KMs, while simultaneously interfacing with the force sensors. As seen in Fig. 3, the force sensors are sandwiched between the vibration fixture lift plate (used to interface the shaker) and the interface ring to which the FGS OA is mounted.

Fig. 3 Model of FGS OA on Vibration Fixture Prior to the vibration test, the vibration fixture underwent vibration characterization tests (Fig. 4). The fixture characterization was accomplished by performing low-level sine-sweeps in each of the FGS primary axes. As shown in Fig. 5, to further verify the fixture response in the final test configuration (with the FGS OA), low level sine-sweeps were also performed with an FGS OA simulation mass mounted on the vibration fixture. The first observed natural frequency of the vibration fixture occurred at 588 Hz (at the longest span of the interface ring) and was visible during V2 excitation. Modes 2 and 3, seen at 634 and 677 Hz, coincide with a similar V2 excitation of the beams between the remaining KM pairs. The addition of the simulation mass did not affect the response of the interface ring.

Fig. 4 Vibration fixture characterization

Fig. 5 Vibration fixture with OA simulation mass

48 6.

Estimation of Force Limits

The most widely used procedure for estimating the force limits is the so-called semi-empirical method. The basic idea behind semi-empirical force limits is to properly envelope the input force at the fundamental frequency of the test item. The random vibration form of the semi-empirical method is given by the following two relations [3]:

Sff (f )  C 2 M 02 Saa (f )

f  f0

(1)

Sff (f )  C 2 M 02 Saa (f ) (f / f 0 ) n

f  f0

(2)

where Sff is the sought limit to the force spectral density (or power spectral density, PSD), S aa is the defined input acceleration spectral density, M0 is the physical mass of the test item, f0 is the fundamental frequency (or frequency of the first significant mode) of the test item in the test axis being considered, C 2 is a dimensionless constant which depends on the configuration, and „n‟ is the roll-off ratio to account for the decrease in residual mass with frequency. In Equations (1) and (2), the mass M 0 is known and Saa is defined (usually from launch environments or standards). The values of f0 and 'n' may be obtained from finite-element (FE) analysis and confirmed by a low-level sine or random run performed just prior the full-level vibration testing. The constant C2 is a key parameter of the semi-empirical method which sets the force limits throughout the complete frequency bandwidth of excitation. However, C2 is the only parameter of Equations (1) and (2) that cannot be obtained directly or from a low level sine or random run. Normally, some engineering judgment must be exercised to choose the value of C2. In fact, the level of conservatism of the force limits depends on the selected value for C2. In addition to the information provided in Ref. [3], detailed discussion on the criteria that have been used for selecting C2 and on the range of values it is normally expected to take can be found in Ref. [4]. 7.

Test Configuration and Sequence

The FGS ETU OA vibration test was performed on the 178 kN shaker at the CSA DFL. Input notching was employed through force limiting with the aid of six tri-axial force sensors placed between the fixture plate and interface ring. Due to fixture effects, force limited notching was only employed up to 300 Hz, or approximately half the fundamental frequency of the vibration fixture. Above 300 Hz, response limiting was used at the base of the KMs to supplement force limiting. The vibration test campaign of the FGS OA consisted of a suite of sine and random tests for each of the defined V2, V1 and V3 axes respectively. The axis order was selected to minimize reconfiguration time while allowing the tests with the highest input levels to be performed last. As shown in Table 1, the test suite included -12 dB, -6 dB and full qualification levels for both (sine and random) tests. Lastly, a sine burst strength test was performed in the V3 axis. Table 1 FGS ETU OA Vibration Test Sequence Run 1

Test Description Low Level Sine Sweep

2 3

-12 dB Qualification Level Sine Sweep -6 dB Qualification Level Sine Sweep

Comments Measurement of FRFs (including apparent mass), natural frequencies and associated Q factors Force limit verification test Force limit verification test

4 5 6 7 8 9

Qualification Level Sine Sweep Verification Low Level Sine Sweep -12 dB Qualification Level Random Test -6 dB Qualification Level Random Test Qualification Level Random Test Verification Low Level Sine Sweep

Qualification level with force limiting Verification of natural frequencies Force limit verification test Force limit verification test Qualification level with force limiting Verification of natural frequencies

10 11

Sine burst test Verification Low Level Sine Sweep

V3 axis only V3 axis only

49 The low level sine sweep is required to verify natural frequencies, determine FGS integrity after full level testing and to calculate the apparent mass FRF (total interface force/input acceleration), which was used to derive the predicted force PSD under the input spec acceleration. The apparent mass further served to verify that the force sensor set-up was working properly and to perform an in-situs calibration of the force sensors. The purpose of the qualification level sine test is to validate the FGS OA‟s ability to survive the dynamic environment occurring during launch. To minimize the possibility of overtesting the hardware, force limited notching was employed at the design limit load (DLLs) derived through a coupled loads analysis on the ISIM level FE model. The lower level random runs provide confidence in the FLV parameters selected earlier. If the apparent mass has significantly changed from the previous run (due to nonlinearity of the test item with increasing input levels) and the input notching profile is significantly different than the expected notching level, the FLV parameters are modified accordingly for the full level run. The FGS OA was mounted to the shaker head expander in the V2 configuration and the shaker slip table in the V3 configuration as shown in Fig. 6 and Fig. 7 respectively. Due to stringent contamination control plan implemented on the program, the FGS OA was double bagged throughout all testing activities.

Fig. 6 V2 test configuration of the FGS ETU OA 8.

Fig. 7 V3 test configuration of the FGS ETU OA

Test Issues and Selected Approach

Once it was decided that force limiting would be applied for testing the FGS OA models to reduce the overtesting, it became apparent that the presence of the massive interface ring would modify the measured interface forces. This is due to the fact that the mass of the interface ring is about 25% the mass of the OA test item. Also, the measured apparent mass frequency response function or FRF (total interface force / input acceleration) in the direction of excitation would be affected. The apparent mass from a low-level run is used (i) for deriving the parameter „n‟ of Equation 2, and (ii) for predicting the unclipped interface force PSD when force limiting has been applied during the run, the unclipped interface force serves in the selection process of C2. These assumptions were confirmed by comparing the data obtained from FE analysis with and without the interface ring. A solution proposed earlier (for a much more massive planned interface plate) consisted in measuring the apparent mass with the force sensors inserted between the vibration fixture lift plate and the interface ring, and subsequently removing the effect of the interface ring analytically, resulting in a prediction for the apparent mass of the OA without the interface ring. The latter apparent mass was then to be used for computing the FLV parameters C2, f0, and n. This solution was abandoned for the following reasons: (1) Although pretty straightforward in theory, analytical removal of the effects of the interface ring was considered to likely be much more complicated in practice over a frequency band covering several OA modes, due the phase angle reversal for the force response contribution of the OA beyond its fundamental frequency. (2) Due to the complexity and possible errors/uncertainties that could occur with the approach when applied with imperfect and possibly noisy test data, it was deemed required to verify this approach with analytical perfect data and subsequently with some test data before applying it on the OA hardware. (3) Programmatic schedule did not provide the time that would have been required to properly verify the approach before

50 using it on the FGS ETU OA. (4) Through pre-test FLV analysis, a simpler solution was shown to be acceptable for the present test item and the frequency bandwidth in which force limiting was to be applied. The retained approach, described in Section 9, consists in modifying our standard FLV procedure to account for the presence of the interface ring. This approach was chosen because: (1) The pre-test FLV analysis showed that it ensures conservatism, i.e. no under-testing is caused by the presence of the ring. Results of this analysis are also presented in Section 9. (2) The approach allows the use of softflv program that does not account for the presence of massive interface hardware between the force sensors and the test item. The Matlab-based softflv program was developed within the CSA Space Technologies Sector to simplify calculations related to FLV and to plot the results allowing to quickly see the effects of different values of C2, f0, and „n‟ on the test limits. (3) The approach was applicable in a timely fashion for the ETU vibration testing. 9.

Pre-Test Analysis for FGS ETU OA

The main objectives of the pre-test analysis were to confirm that (i) the interface ring had a significant enough effect on the measured apparent mass and force PSD to warrant some actions before computing the FLV parameters, and (ii) testing could be performed by simply modifying our force limiting procedure without risking of under-testing the test item because of the presence of the ring. A secondary objective was to derive preliminary values for the key C2 constant; such values subsequently were either to be confirmed or modified based on processing test data from low-level vibration runs, performed just prior the full-level runs. This pre-test analysis was performed only weeks prior the FGS ETU OA vibration testing 9.1

Input Specification for the FGS ETU OA

Random vibration testing of the FGS ETU OA was to be performed at qualification level. The input level is given in Table 2 and was applied during the test for a duration of 120 seconds. The same input level was applied to the finite element (FE) models for the pre-test analysis. This test specification was derived using a coupled loads analysis performed at the ISIM level at NASA. Table 2: Random vibration level for the FGS ETU OA

9.2

Frequency (Hz) 20

PSD (g2/Hz) 0.03

30 50 120 190 2000 Overall grms

0.03 0.01 0.01 0.004 0.004 3.01

Finite-Element Modeling

The FGS ETU OA finite element model used as part of the pre-test analysis was composed of a composite model consisting of a NASTRAN FGS OA structural model (shown in Fig. 8) and a NASTRAN vibration fixture model (shown in Fig. 9). The FE model of the FGS optical assembly was based on the critical design review (CDR) structural model. The pre-test FE model mass was adjusted to be the FGS OA mass allocation of 92 kg. To verify the integrity of the FGS OA NASTRAN finite element model, a set of validity checks were completed including: unit enforced displacement and rotation, free-free dynamics with a stiffness equilibrium check, unit gravity loading and unit temperature increase as recommended by [5]. The unit enforced displacement and rotation check confirms that no artificial grounding exits in the model. The free-free dynamics with stiffness equilibrium check validates that the model responds as a rigid body when unconstrained and confirms that the stiffness matrix does not have any malformed single point constraints or rigid elements. The unit gravity loading check confirms that the model returns appropriate reaction forces when subjected to gravity loading. Lastly, the unit temperature increase validity check affirms that the model return appropriate displacements when subjected to temperature loading.

51

Fig. 9 FE model of the vibration fixture

Fig. 8 FE model of the FGS OA with interface ring

The vibration fixture finite element model was correlated to reflect the observed fixture response upon completion of the vibration fixture characterization test. The resulting maximum discrepancy of the updated fixture FE model compared to the experimental results is a 3.2% frequency deviation at 633.9 Hz (mode 2). To assess the impact of the vibration fixture on the response of the FGS OA, a random analysis was performed on the structural FGS OA model (without the interface ring). The force response PSD was recorded for each axis, converted into the apparent mass and analyzed using softflv. Following an identical approach, a random analysis of the FGS OA with the interface ring was performed. The two results were compared and it became evident that at the fundamental frequency of the FGS OA, the interface ring has minimal impact. Based on the lowest measured fundamental frequency of the interface ring at about 588 Hz, it was decided to apply force limiting to frequencies up to 300 Hz. This upper limit was set for all three orthogonal axes. This upper limit selection, based on transmissibility curves, is in line with recommendation made in Reference [6]. In a different context of designing/selecting a fixture, this Recommended Practice document states that (i) it is desirable for the lowest fixture natural frequency to be at least 1.7 times the highest vibration test frequency and (ii) as a rule of thumb, the transmissibility at the upper test frequency should not exceed 2. In the context of the maximum frequency for which notching is applied, these recommendations are respected regarding the interface ring since the lowest f0 of the ring is about twice the maximum frequency for which force limited notching is applied. 9.3

Equivalent C2 Value in the Presence of Interface Ring

For a specified C2 value, Equations (1) and (2) compute the limits to be imposed to the interface force measured right underneath the OA test item. However, for the present configuration with the force sensors being located below the interface ring (instead of right below the test item), the measured force is somewhat different than the required interface force since the measured force includes the force generated by the ring. Since the softflv program has not been developed to account for significant interface mass as in the present case, it is required to modify the procedure for computing the force limits to properly apply these equations. This is done by deriving an equivalent C2 (called C2eq) at the location of the force sensors that will correspond to a specified C2 at the interface of, or right underneath, the test item. Before deriving the expression for an equivalent C2, let us consider a relation for estimating the measured force PSD Sff,meas as a function of other test parameters. The force PSD measured by the force sensors can be written as 2

Sff ,meas  M OA Saa  M 2ring Saa

(3)

52 where M OA is the apparent mass of the OA, M ring is the physical mass of the interface ring, S aa is the input acceleration PSD acting underneath the force sensors, and the define the amplitude. The following observations may be made regarding this equation: (1) The second term may be considered as being exact for frequencies well below f0 of the interface ring (the lowest f0 being 588 Hz). This implies that the ring may be considered as acting as a rigid body. (2) The + sign (scalar sum) is exact below f0 of the OA which is well below f0 of the ring. Consequently, Sff,meas is exact in this frequency range. (3) For frequencies near and beyond f0 of the OA, Equation (3) is not exact anymore since the scalar sum must be replaced by a vector sum due to phase angle reversal of the OA forces. However, due to difference in phase, the effect of the ring is less with the vector sum compared to the scalar sum. Consequently, as long as the frequency is still well below f0 of the ring, the equation gives conservative (higher) values, thus eliminating the risk of under-testing the OA because of the presence of the interface ring. In fact, Equation (3) and its derivative Equation (4) for force limits only need to be applied at f0 of the OA, in order to derive the equivalent C2 to account for the presence of the interface ring. The highest frequency taken as f0 of the FGS ETU OA occurred during test and was measured at 138 Hz. Based on the above observations, these two equations thus produces conservative values in that frequency bandwidth of interest. In order to compute C2eq, the first term of Equation (3) is replaced by the expression in Equation (1) for deriving the force limits. This results in an expression for computing the force limits in the presence of the ring. These force limits, Sff,lim, are thus obtained from 2 Sff ,lim  C 2 M OA Saa  M 2ring Saa

(4)

Now, for softflv implementing Equations (1) and (2) with C2eq, the physical mass M0 has to be replaced by the total mass, Mtotal, of both the OA and the ring since both of them are above the force sensors and thus contribute to the force they measure. So, from Equation (1), the force limits can be written as

Sff ,lim  C 2 eq M 2total Saa

(5)

Equating Equations (4) and (5) and isolating C2eq, one gets





2 C 2 eq  C 2 M OA  M 2ring / M 2total

(6)

For the pre-test analysis and subsequent testing of the FGS ETU OA, C2 was selected to be 2 for the reasons provided in the next sub-section. The other variables are Moa = 92 kg (203 lbm), Mring = 23 (50 lbm). Using these parameters, Equation (6) result in C2eq = 1.3. 9.4

Selection of C2 Parameter for FGS ETU OA

The selected value of the C2 constant was 2 for all three excitation axes for both pre-test analysis and vibration testing of the FGS ETU OA. This value is for the OA by itself and leads to an equivalent C2, C2eq, of 1.3 with respect to the force underneath the interface ring, as shown previously. This selection of a value of 2 was made on the basis of three of the criteria discussed in details by Ref. [3]. A description of these criteria and a discussion of their application for the FGS OA follow. (1) Extrapolation of interface force data for similar mounting structures and test items. Based on limited number of flight data [7] [8], it has been observed that in normal conditions C2 value as low as 2 might be chosen for complete spacecraft or strut-mounted heavier equipment, C2 value as high as 5 might be considered for directly mounted lightweight test items [9]. Also, based on results of R&D activities on complex structures, the C2 values were found to be below 5 for most cases and below 10 for the large majority of cases [10][11]. With the three pairs of kinematic mounts attaching it to its mounting structure and its mass of over 90 kg, the OA clearly

53 fits in the category of strut-mounted heavier equipment. This fact by itself leads to a selection of a C2 of 2. (2) Comparison with quasi-static loads. The maximum expected acceleration at the center-of-gravity (CG), the quasi-static limit load or Design Limit Load (DLL), is a key conservative parameter in the design, analysis and testing of space structures. One means of obtaining the limit loads (expressed in g‟s) is from coupled loads analysis. For random vibration testing, it is common to multiply the measured rms force by a peak factor of 3 (3-sigma) for comparison with the limit loads. Because of its conservative nature, the DLL is used to provide an upper bound for C2. Consequently, the selection of C2 should normally ensure that the peak CG acceleration (derived from the clipped force PSD) does not exceed the limit load [3][9]. For the FGS ETU OA, a qualification factor of 1.25 is applied to the DLL to obtain the qualification level DLL to be used as the upper bound. It should be noted that the sine qualification testing of the FGS ETU OA (not discussed in the present paper) was also limited to the same DLL, and resulted in input notching at resonances based on measurements made by the force sensors. In a mathematical form, the DLL and rms of the clipped force PSD are related together through the following simple relation: DLL  3 x rms force / M0 (7) where the expression at the right of the inequality represent the peak CG acceleration. For the ETU, the qualification DLL values are 4.4, 4.4 and 8.1 for the V1, V2 and V3 axes, respectively. The DLL is larger for V3 since this axis corresponds to the launch direction. In comparison, using a C2 value of 2 and the modified procedure of the next two sub-sections results in peak CG accelerations of 4.7, 4.9 and 4.7. For the V1 and V2 directions, the peak CG accelerations are slightly above the DLL. This exceedance of about 10% is acceptable. This thus confirms that a C2 value of 2 is not too low, since higher C2 values would have resulted into even higher CG acceleration, thus significantly violating the criterion of the peak CG acceleration not exceeding the DLL. Consequently, this confirms the selected value for C2 based on the first criterion. The OA being a fairly symmetric structure, it is reasonable to select the C2 value for the V3 lateral direction as being the same as for the other lateral direction, V1. It should be mentioned that, to avoid flight hardware damage, it has been proposed to apply larger peak factors in light of the frequent observance of 5-sigma test peaks in time histories of responses in random vibration tests [12]. For this project, this recommendation was not applied since it would have led to C2 values lower than 2. Such very low C2 values have historically been used only for large spacecraft, as for MER [13]. Another reason for not implementing a larger peak factor was that the analysis did not indicate any risk of hardware damage, in the event of larger than 3-sigma spikes. (3) Assessment of the amount of force clipping and input notching This criterion is related to the fact that force limiting should not result in excessive clipping of the force PSD or notching of the input acceleration PSD at f0. Even if the interface force and the input acceleration are two different parameters, their assessment are included in a single criterion since they are normally evaluated together. Although this criterion is the one requiring the largest amount of engineering judgment, it is still often extremely useful for finalizing the C 2 value selection, especially when no pertinent value for comparison purposes (e.g. Design Limit Load) is readily available. For testing, this criterion should be implemented first at a low-level run (e.g. -12 dB) and the decision should be continuously re-assessed, and modified if pertinent, at the various intermediate levels before reaching the full level. 9.5

Excitation in the V2 Vertical Direction

The apparent mass of the OA predicted at location of the force sensors was estimated analytically for the two configurations with and without the ring. Fig. 10 presents an overlap of these two FRFs for excitation along the V2 vertical direction. The parameter f0 of Equation (2) is taken as being the frequency of the first significant mode occurring at about 104 Hz. The overlap shows that the two curves are very similar in the range of f 0 to 300 Hz, the maximum frequency for applying force limiting. Consequently, it is reasonable to say that the roll-off ratio „n‟ of Equation 2 derived from the apparent mass with the ring would be basically the same as the one for the „unknown‟ apparent mass without the ring. This statement is even more reasonable since notching is limited to only 300 Hz: as the distance between f0 and the notching frequency increases, any difference between the „n‟ of these two apparent masses would result in larger difference in the force limits.

54

The apparent mass with the ring is shown again in blue in Fig. 11. The low-frequency asymptotic value of the apparent mass corresponds to the total physical mass of the OA and the ring. The horizontal mass line in red is inclined for frequencies above f 0 and its inclination is selected so that it follows the rolling-off of the apparent mass, up to 300 Hz. In fact, using the portion of the apparent mass above 300 Hz would be meaningless in the present case, since the two apparent masses are completely different beyond that frequency. The slope of the line is 0.75. Since the apparent mass is shown on a log-log plot, the value of the exponent 'n' in Equation (2) is twice this slope, i.e. n = 1.5. Apparent Mass and Curve Fit

Apparent mass comparison - V2 App mass without ring

App mass with ring

Corrected Amplitude (lbm)

Amplitude (lbm)

10000

1000

100

2

10

10

1 10

100

1000

10000 10

frequency (Hz)

Fig. 10 Comparison of apparent masses with and without ring, for V2 direction (1 lbm = 0.454 kg)

20

30

40 50 60 708090100 Frequency (Hz)

1000

2000

Fig. 11 Apparent mass with ring, overlapped with mass line for V2 direction

In order to apply the second and third criteria discussed earlier, softflv derives the force limits and the clipped force PSD. Once the FLV parameters f0 and „n‟ are selected based on the apparent mass M with or without the interface ring, the program computes the force limits using Equations 1 and 2, as well as C2 or C2eq. Then, using the same apparent mass and the input spectrum Saa presented in Table 2, softflv computes the unclipped force PSD S 0ff . The unclipped force PSD is derived using the following relation [14]: 2

S0ff (f )  M (f ) Saa (f )

(8)

For the case of the OA without the ring, Fig. 12 presents an overlap of the force limits and the unclipped force PSD. By applying FLV, all portions of the force PSD above the force limits (red part) is eliminated and one is left with the clipped force PSD (blue part). Similarly, Fig. 13 shows an overlap of the force limits and the unclipped force PSD for the case of the OA with the ring. Looking at these plots, one has to remember that for testing, no force limits is to be implemented by the control system beyond 300 Hz; force clipping appearing at these higher frequencies is solely due to the fact that the version of softflv used did not allow selecting the frequency bands for which force limiting is applied. Comparing these two figures, one notices that more of the force PSD is eliminated in the presence of the ring. This apparent difference is caused by the force PSDs being extracted at two different locations (right underneath the OA versus underneath the interface ring) leading to the use of a C2 of 2 versus a lower C2eq of 1.3. Also, looking at Fig. 13, it appears as if the PSD peak around f0 at about 104 Hz is completely removed. This is caused by the fact that using a C2eq of 1.3 is effectively a very low value resulting in most of the PSD peak being eliminated. However, having in the present case the left-hand side of the peak in the same frequency band as a downward slope of the input acceleration specification makes the remaining beginning (in blue) of the peak less apparent. This results in the force clipping to appear even „worse‟.

55 Sff flv VS Sff std

4

10

Sff flv VS Sff std

3

10

3

10

Amplitude (lb²/Hz)

Amplitude (lb²/Hz)

2

10

1

10

2

10

0

10

1

10

-1

10

20

100

1000

Frequency (Hz)

20

2000

100

Frequency (Hz)

1000

2000

Fig. 13 Force limits and unclipped force with ring, for V2 direction

Fig. 12 Force limits and unclipped force PSD without ring, for V2 direction (1 lb2 = 20 N2)

The input notching for the two configurations are shown in Fig. 14 and Fig. 15. One observes that the depths of the notches are fairly similar up to 300 Hz, above which frequency no force limit is to be implemented during testing. Looking at the depth of the notch at f0 for both figures, one might be surprised that a C2 (or C2eq) of 1.3 results in a notch of only 9 dB, as opposed to notch of 11 dB for a larger C2 of 2; this is simply due to the unclipped force PSD peak being slightly higher in the absence of the interface ring. Also, the larger width of the notched region below f0 in the presence of the ring is, here again, the result of a lower C2 value. Saa flv VS Saa std

-2

10

-3

10

Amplitude (g²/Hz)

Amplitude (g²/Hz)

Saa flv VS Saa std

-2

10

-3

10

-11 dB @ 105

-9 dB @ 104

-8 dB @ 225

-8 dB @230 20

100

Frequency (Hz)

1000

Fig. 14 Input specification with notched input without ring, for V2 direction 9.6

2000

20

100

Frequency (Hz)

1000

2000

Fig. 15 Input specification with notched input with ring, for V2 direction

Excitation in the V1 Lateral Direction

As for the V2 direction, the apparent mass of the OA predicted at location of the force sensors was estimated analytically for the two configurations with and without the interface ring for the V1 lateral axis. Fig. 16 presents an overlap of these two FRFs. The parameter f0 of Equation (2) is taken as being the frequency of the most significant mode occurring at 83 Hz.

56

Amplitude (lbm)

Preliminary analysis of the apparent mass without the Apparent mass comparison - V1 ring (blue curve) have resulted in a parameter „n‟ App mass without ring App mass with ring equal to 3, using the high peaks between 500 and 700 1.00E+04 Hz for finding the slope of inclined mass line. This 1.00E+03 analysis has also shown that, with notching frequency being limited to 300 Hz, no significant notching (i.e. 1.00E+02 more than 3 dB) would occur above 122 Hz. Now, in order to get the „n‟ parameter for the apparent mass 1.00E+01 with the ring (pink curve), one would have to use the higher frequency modes beyond 1000 Hz. Looking at 1.00E+00 10 100 1000 10000 the overlap of the two plots of the figure, the inclined 1.00E-01 mass line of the apparent mass without the ring frequency (Hz) (unknown during testing) would be quite different than the measured one with the ring. Consequently, Fig. 16 Comparison of apparent masses with and the „n‟ parameter of the apparent mass without the without ring, for V1 direction ring cannot be estimated from the measured one as the higher frequencies are contaminated with fixture resonances. However, as no significant notching should occur beyond 122 Hz, the issue was eliminated by using zero for the value of „n‟. This provides conservative (higher) force limits and implies that there is no reduction in the force limits beyond f0 to account for the reduction of residual mass above that frequency. In fact, it means that only Equation (1) is relevant for computation of the force limits. However, the fact that notching should not occur for frequencies higher than f0 by more than 40 Hz implies that this decision should not result in being overly FGS ETU conservative. In fact, comparison of peak CG accelerations (for criterion 2) for the configuration without the ring showed that V1 axis of 3 by less than 7%. its value with an „n‟ of zero is exceeding its counterpart with an „n‟ value Saa flv VS Saa std Analysis

The input notching for the two configurations are shown in Fig. 17 and Fig. 18. One observes that the depths of the notches with I/F ring are basically the same at f0, and no more than 3 dB different for theFGS other large notch around 94 Hz. As for the first axis, one 04-Oct-2009 observes a larger width of the notched region below f0 in the presence of the ring. Again, this is caused by a lower C2 value. Saa flv VS Saa std

-2

10

-2

Amplitude (g²/Hz)

Amplitude (g²/Hz)

10

Saa flv VS Saa std

10

-7 dB @ 93

-10 dB @ 95

-3

10

-3

-17 dB @ 83 20

-16 dB 80 100

Frequency (Hz)

1000

Fig. 17 Input specification with notched input without ring, for V1 direction

2000

20

100

Frequency (Hz)

1000

2000

Fig. 18 Input specification with notched input with ring, for V1 direction

The results for V3, the other lateral axis, are basically the same as for V1 and are thus not included in this paper. Comparison of the two apparent masses also led to selection of a value of zero for the „n‟ parameter. These results may be seen in [15]. 10. Random Vibration Testing of ETU OA 10.1

Introductory Remarks

In October 2009, shortly after completion of the pre-test analysis that defined the modified FLV procedure, the vibration test campaign of the FGS ETU OA was successfully performed. Vibration testing encompassed qualification sine, random vibration and sine burst (V3 axis only).

57

As outlined in Table 1, for each of the three axes, the random vibration sequence consisted of the following runs: - Low-level sine sweep, 0.05 g, 10 to 2000 Hz, 2 octave/min - Random run, -12 dB down from full level, force (and response for V3 only) limiting enabled - Random run, -6 dB down from full level, force (and response for V3 only) limiting enabled - Random run, 0 dB or full level, force (and response for V3) limiting enabled, as per spec of Table 2, duration: 120 sec - Low-level sine sweep, 0.05 g, 10 to 2000 Hz, 2 octave/min After each run, force related data were processed with softflv in order to select f0 and „n‟ (for V2 only), and to confirm the pre-selected C2 and C2eq. As mentioned in the previous sub-section, „n‟ was taken as zero for both V1 and V3 directions. Data from the pre- and post-random runs were compared to ensure no significant deviation appeared that could point out to possible hardware damage occurring during the full-level random vibration run. The measured masses were slightly different than those used in the pre-test analysis, namely MOA = 80 kg (177 lbm), Mring = 18 kg (40 lbm) as the FGS OA was not scheduled for the mass properties test until days before the vibration test. Therefore, the values used for the pre-test analysis were estimates based on CAD models. Applying Equation 6 with these new masses and a C2 of 2 gives a slightly different value of 1.4 for C2eq used by softflv. 10.2

Excitation in the V2 Vertical Direction

The first axis for which the FGS ETU OA was tested is the vertical direction, V2. The apparent mass obtained from -12 dB run is shown in blue in Fig. 19. The FLV parameter f0 is selected as the frequency of the most significant mode at 138 Hz. The low-frequency asymptotic value of the apparent mass corresponds to the total physical mass of the OA and the ring; this mass is 98 kg (217 lbm). The horizontal mass line in red is inclined for frequencies above f0 and its inclination is selected so that it follows the rolling-off of the apparent mass, up to 300 Hz. The slope of the line is unity, leading to a value of 2 for „n‟.

3

10

Corrected Amplitude (lbm)

For the full-level run, Fig. 20 presents the input specification of Table 2 with the measured notched input (bottom), and the measured force PSD with the derived unclipped force PSD (top). The unclipped PSD, which would have occurred without the use of force limiting, was computed by bringing up the measured notched input to the specification level in the selected frequency band delimited by two continuous vertical lines. The band starts at 40 Hz which is the frequency at which the notching begins; it ends at 300 Hz, the maximum frequency for which the control system was programmed to implement force limiting. It should be noted that vertical scaling is different for both plots; this tends to distort the fact that the input notching has the same depth as the height of the unclipped portion of the corresponding force PSD peak.

2

10

1

10

20

Regarding Criterion 3 discussed earlier, the 10 dB input notch at f0 and the maximum 17 dB notch at a higher frequency are acceptable levels of notching that do not create any concerns of possible under-testing.

40

60

80

100

200 Frequency (Hz)

400

600

800

1000

2000

Fig. 19 Test-based apparent mass, overlapped with mass line for V2 direction

The peak CG acceleration derived from the rms of the clipped force PSD is 4.7. The pre-test analysis showed that, for this axis, this value with the ring was slightly lower (10%) than the value obtained without the ring (the real value we are interested in). We can thus estimate the peak CG acceleration without the ring to be a bit more than 5. Since this value should not normally exceed the DLL of 4.4, we want at least to minimize the exceedance when it appears as in the present case. As discussed before we did not want a C2 value lower than 2 for such test item. This comparison confirms, as for the pre-test analysis, that the selected C2 of 2 was indeed an appropriate choice.

58 Force PSD

4

Amplitude (lb²/Hz)

10

:Unclipped :Clipped

2

10

0

10

2

3

10

10

Frequency (Hz) Acceleration PSD

:Reference :Unnotched :Notched

Amplitude (g²/Hz)

-2

10

-3

10

-10 dB @ 138

-17 dB @ 208

-4

10

-12 dB @ 248

2

10

3

Frequency (Hz)

10

Fig. 20 Input spec with notched input (bottom), and measured force PSD with unclipped force PSD (top), from full-level run in V2 direction 10.3

Excitation in the V1 Lateral Direction

After the successful testing in the vertical axis, the OA was tested laterally along the V1 direction. For the full-level run, Fig. 21 shows the input specification with the measured notched input (bottom), and the measured force PSD with the derived unclipped force PSD (top). The only significant input notch occurs at the f0 of 90 Hz and has a significant depth of 23 dB. In fact, this large notching has triggered some investigation to ensure that there was no risk of under-testing the hardware. This investigation reassured us that this was the case. Also, the peak CG acceleration derived from the rms of the clipped force PSD is 4.5. The pre-test analysis showed that, for this axis, this value with the ring was exactly the same as the value obtained without the ring; the higher force rms was divided by a higher total mass. Consequently, the selected C2 value of 2 lead to a peak CG acceleration basically equal to the DLL of 4.4. This comparison thus confirms the validity of the selected C2. It should be noted that, as discussed in the pre-test analysis, the quite large width of the notched region below f0 is caused by the very low C2eq value of 1.4. Without the ring, the C2 of 2 would have resulted in a narrower width.

59 Force PSD :Unclipped :Clipped

4

Amplitude (lb²/Hz)

10

2

10

0

10

-2

10

2

10

3

Frequency (Hz)

10

Acceleration PSD :Reference :Unnotched :Notched

-2

Amplitude (g²/Hz)

10

-3

10

-4

10

-23 dB @ 90 2

10

3

Frequency (Hz)

10

Fig. 21 Input spec with notched input (bottom), and measured force PSD with unclipped force PSD (top), from full-level run in V1 direction 10.4

Excitation in the V3 Lateral Direction

The last test axis for the OA was V3, the other lateral direction. Fig. 22 shows, for the full-level run, the same results as before. In the force limiting region, below 300 Hz, the only significant input notch is at the f0 of 86 Hz and has a fairly large depth of 21 dB. The quite large notching occurring at frequencies above 800 Hz was triggered by the response limiting strategy implemented to protect some units against overtesting. The response limiting was implemented at the kinematic mount interface with vibration fixture, negating the fixture amplification caused by its resonances at higher frequencies. The peak CG acceleration derived from the rms of the clipped force PSD is 4.2. The peak CG acceleration is quite lower than the DLL of 8.1. Consequently, Criterion 2 was not a factor in the C2 selection for the V3 axis. The main factor was Criterion 1 and the fact that there was no reason from a structural or dynamic point of view to have the selected C2 for this axis to be any different than the one for the other lateral direction.

60 Force PSD :Unclipped :Clipped

4

Amplitude (lb²/Hz)

10

2

10

0

10

2

10

3

10

Frequency (Hz) Acceleration PSD

:Reference :Unnotched :Notched

-2

Amplitude (g²/Hz)

10

-3

10

-4

10

-5

10

-23 dB @ 1184

-21 dB @ 86

-30 dB @ 1404 2

10

3

Frequency (Hz)

10

Fig. 22 Input spec with notched input (bottom), and measured force PSD with unclipped force PSD (top), from full-level run in V3 direction 11. Conclusion The FGS ETU OA vibration test consisted of multiple sine and random runs, as well as a sine-burst test at the qualification vibration environment. The FGS ETU OA vibration test objectives included the verification of the FGS OA fundamental frequency (low level sine sweep), qualification of the main structure while being subjected to an equivalent dynamic launch environment (sine test), survival of qualification level workmanship test (random test), and verification of the primary structure and critical structural interfaces strength adequacy (sine burst test). Throughout the testing, force limited notching was successfully implemented, and in the case of random testing, a unique implementation of the semi-empirical force limited vibration approach to account for presence of a significant mass interface ring was implemented. While testing at the qualification levels, no structural failures were observed through low-level sine sweep verification methods as well as by close visual inspection. Furthermore, pre/post vibration alignment checks showed that alignment stability was maintained. Due to the success of the modified FLV approach during the random test of the FGS ETU OA, this method will be employed for the FGS proto-flight model (PFM) OA vibration test scheduled in spring of 2011. For the PFM test, the analysis will be performed using an updated FGS OA structural FE model (based on the ETU test) allowing for more confidence in the selected FLV parameters prior to initiating the first run.

61 12. Acknowledgements The authors acknowledge Dr. Johanne Heald of the CSA, Structures Research Scientist, for recommending the Force Limited Vibration approach to the team. We would like to thank Karl Saad, CSA JWST FGS Project Manager, for supporting the implementation of the approach within the project. We would also like to thank the technical personnel of the vibration laboratory of the CSA David Florida Laboratory for their support during the test program. The authors would like to further acknowledge the team at COM DEV Canada for their hard work and determination allowing FGS OA test to be successfully performed ahead of schedule while simultaneously supporting the build of the PFM unit. Special thanks goes to James Gallagher, Roman Gawlik, Ian Clark and Keith Fancy, for preparing FGS ETU OA for the vibration test as well as skillfully recognizing and solving many issues before they became problems. Lastly, the authors would like to acknowledge Ashley McColgan, COM DEV Canada Mechanical Engineering Lead, for his insight and guidance into structural matters relating to the project. References [1]

Clampin, M., “Overview of the James Webb Space Telescope Observatory”, Proc. SPIE 7731, 773106, 2010.

[2]

Rowlands, N., Saad, K., Hutchings, J.B., and Doyon, R., “James Webb Space Telescope (JWST) Fine Guidance Sensor: overview”, Can. Aeronaut. Space J. 55, 69-78, 2009.

[3]

Anon., “Force Limited Vibration Testing”, Jet Propulsion Laboratory, NASA Technical Handbook, NASA-HDBK7004B, Pasadena, CA, 29 pages, 2003.

[4]

Soucy, Y., “On Force Limited Vibration for Testing Space Hardware”, Proceedings of the IMAC XXIX Conference, Jacksonville, Florida, 2011.

[5]

Simmons, Ryan (ed), “The FEMCI Book”, National Aeronautics and Space Administration. Greenbelt. Goddard Space Flight Center, Washington, DC : NASA, 1998

[6]

Anon., “Vibration and Shock Test Fixturing”, IEST-RP-DTE013.1, Institute of Environmental Sciences and Technology, p. 11, 1998.

[7]

Scharton, T.D., “In-Flight Measurements of Dynamic Force and Comparison with Methods Used to Derived Force Limits for Ground Vibration Tests”. Proceedings of the European Conference on Spacecraft Structures, Materials and Mechanical Testing, Braunschweig, Germany, 1998.

[8]

Scharton, T.D., “Force Limits Measured on a Space Shuttle flight”, Journal of the IEST, Vol. 45, pp. 144-148, 2002.

[9]

Chang, K.Y., “Force Limit Specifications vs. Design Limit Loads in Vibration Testing”, Proceedings of the European Conference on Spacecraft Structures, Materials and Mechanical Testing, ESA Publications Division, Noordwijk, The Netherlands, pp. 295-300, 2001.

[10] Soucy, Y., Dharanipathi, V.R. and Sedaghati, R., “Investigation of Force-Limited Vibration for Reduction of Overtesting”, Journal of Spacecraft and Rockets, Vol. 43, No. 4, pp. 866-876, 2006. [11] Soucy, Y. and Montminy, S., “Investigation of Force Limited Vibration Based on Measurements During Quicksat Satellite Vibration Testing”, Proceedings of the 14th Canadian Astronautics Conference - ASTRO 2008, Montreal, Qc, April-May 2008. [12] Scharton, T.D., Pankow, D. and Sholl, M. “Extreme Peaks in Random Vibration Testing”, The 2006 S/C and L/V Dynamic Environments Workshop, Hawthorne, CA, http://www.aero.org/conferences/ sclv/2006proceedings.html, 2006. [13] Scharton, T.D. and Lee, D., “Random Vibration Test of Mars Exploration Rover (MER) Flight Spacecraft”, The 2003 S/C and L/V Dynamic Environments Workshop, El Segundo, CA, www.aero.org/conferences/sclv/2003proceedings.html, 2003. [14] Scharton, T.D., “Vibration-Test Force Limits Derived from Frequency-Shift Method”, Journal of Spacecraft and Rockets, Vol. 32, No. 2, pp. 312-316, 1995. [15] Soucy, Y. and Klimas, P., “Force Limited Vibration Testing of the JWST FGS OA”, The 2010 S/C and L/V Dynamic Environments Workshop, El Segundo, CA, www.aero.org/conferences/sclv/2010proceedings.html, 2010.

On Force Limited Vibration for Testing Space Hardware Yvan Soucy Senior Structural Dynamics Engineer Engineering Development Canadian Space Agency 6767 route de l'Aéroport St-Hubert, Qc, J3Y 8Y9, Canada [email protected] ABSTRACT The Force Limited Vibration approach was developed in the nineties to reduce the overtesting associated with conventional vibration testing of aerospace hardware. Several methods have been considered for the estimation of the force limits. Because of its numerous advantages, the semi-empirical method is the most widely used technique for deriving these force limits. The paper first presents the mathematical relations of the semi-empirical method. The so-called C2 constant is the only parameter of the method that cannot be obtained directly or from low-level preliminary runs. The paper thus continues with a detailed discussion on criteria normally used for selecting the C2 constant and on the range in which it is normally expected to fall. The next section of the paper discusses the advantages of force limiting over the more traditional response limiting for notching vibration input to space hardware. 1.

Introduction

Force Limited Vibration (FLV) testing is an improved test approach developed in the nineties at the NASA Jet Propulsion Laboratory (JPL) to reduce the overtesting associated with conventional vibration testing of aerospace hardware [1]. In addition to controlling the input acceleration as in conventional vibration testing, the FLV approach measures and limits the reaction force between the test item and the shaker. Estimation of the force limits is the most critical step of the FLV approach. The semi-empirical method is presently the technique most widely applied to derive the force limits between the test item and the shaker. The advantages of the semiempirical method over previously-developed more analytical techniques (e.g. simple and complex TDFS methods) are significant and have greatly contributed to the widespread acceptance of the FLV approach taking place in the space community. The two main advantages of the semi-empirical method are: (i) the simplicity of the technique and (ii) the fact that there is no need of impedance information for the mounting structure. The FLV approach has been used successfully at JPL since the mid-nineties on numerous pieces of flight hardware at all levels-of-assembly, including several complete large spacecraft [2]. The FLV testing has also been implemented at several other NASA centers and U.S. companies on flight missions. The interest for and application of this approach has spread to several other countries [3, 4]. In particular, FLV testing has been applied successfully in Canada on flight hardware for seven different missions, at various levels-of-assembly (from unit to complete spacecraft) [5, 6]. In addition, for the Canadian Space Agency (CSA) Quicksat project, FLV has been used to test the spacecraft and several units as well as for a R&D activity on the C2 parameter [7]. The paper first presents the mathematical relations of the semi-empirical method and explains how the various parameters are obtained. The paper then continues with a detailed discussion on criteria normally used for selecting the key C2 constant and on the range in which it is normally expected to fall. The advantages of force limiting over the more traditional response limiting for notching vibration input to space hardware are presented afterwards.

T. Proulx (ed.), Advanced Aerospace Applications, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 4, DOI 10.1007/978-1-4419-9302-1_6, © The Society for Experimental Mechanics, Inc. 2011

63

64 2.

Impedance Background

Impedance characteristics are related to input and response at single points on a structure. Strictly speaking, mechanical impedance refers to the ratio of input force divided by response velocity. However, in the literature, the name mechanical impedance is also given to the ratios force/displacement and force/acceleration. One important impedance relationship used in FLV testing is the apparent mass, defined as the mechanical impedance-type frequency response function (FRF) consisting of input force divided by acceleration. The force and acceleration of the apparent mass refer to the same d.o.f. implying that it is a drive point FRF. The apparent mass can vary greatly with frequency and peaks at the resonance frequencies of the structure, as seen in Fig. 1 for the case of the single oscillator shown in Fig. 2. The oscillator has a mass m of unity and a dynamic amplification or quality factor Q of 20. The abscissa of Fig. 1 is the frequency, normalized by the oscillator resonance frequency fo.

Fig. 2 Single oscillator in test configuration

Fig. 1 Apparent mass of a single oscillator

For both the flight configuration with the test item coupled with the flexible mounting structure and the vibration test configuration with an isolated test item, the interface force spectral density (or power spectral density, PSD) S0ff is related to the interface acceleration spectral density Saa and the apparent mass M of the test item as follows [1]: 2

S 0ff (f )  M (f ) S aa (f )

(1)

Equation (1) is really the second law of Newton applied to random vibration. The 'f' is used to emphasize the frequency dependence of both the spectral densities and the apparent mass. Unfortunately, the precise analytical approaches to obtain the parameters defined in the above equation are not practical, due to structural complexities of space vehicles. The primary difficulty in using Eq. (1) to derive force limits is that the equation has to be evaluated at the coupled system resonance frequencies, which are generally unknown. Consequently, the difficulty in deriving the force limits from the above equation has led to the development of more practical techniques, including the semi-empirical method which is the most widely used approach for the reasons provided in the introduction. 3.

Mathematical Relations of Semi-Empirical Method

The basic idea behind the method is to properly envelop the input force spectrum at the fundamental frequency of the test item. The random vibration form of the semi-empirical method is given by the following two relations [1]:

Sff (f )  C 2 M 02 Saa (f )

f  f0

(2)

Sff (f )  C 2 M 02 Saa (f ) (f / f 0 ) n

f  f0

(3)

65 where Sff is the sought limit to the force spectral density, Saa is the defined input acceleration spectral density, M0 is the physical mass of the test item, f0 is the fundamental frequency (or frequency of the first significant mode) of the test item in direction of excitation, C (or C2) is a configuration-dependent constant, and „n‟ is the roll-off ratio. The C parameter is included in the equations to account for the vibration absorber effect taking place at the primary mode frequencies of the test item when attached to a flexible mounting structure, as during the launch or higher assembly-level testing, instead of to a rigid fixture. As discussed in [1], a physical interpretation of C may be obtained by comparing Eq. (2) with the figure of normalized force limits derived from the simple TDFS model, in which an oscillator representing a mode of vibration of the test item is coupled with another oscillator representing a mode of vibration of the mounting structure. It is then shown that the quality factor Q of the isolated test item is replaced by C. A lower C value results in lower force limits, implying a more significant vibration absorber effect at the coupled system level. The factor (f/f0) n in Eq. (3) is introduced to reflect the decrease in the test item residual mass with frequency. This factor is associated to the asymptotic apparent mass of the test item which is an approximation of the residual mass plot [8]. The asymptotic apparent mass is the critically damped apparent mass (i.e. its derived function having a critical damping ratio equal to unity). For a plate excited in bending or a rod excited longitudinally, the magnitude of the asymptotic apparent mass fall off as one over frequency at frequencies above f0, corresponding on a log-log plot to a straight line having a negative slope of –1. Because of the power nature of the PSD, the exponent 'n' in Eq. (3) is twice the slope of the asymptotic apparent mass. As a consequence, the value of 'n' equals to 2 in [1, 8] is based on the above assumption of the asymptotic apparent mass falling off as one over frequency. One method of calculating the asymptotic apparent mass is to take a geometric average of the apparent mass FRF over frequency, so that there is equal area above and below the curve on a log-log plot [9, 10]. This method is very useful in defining the exponent 'n'. As an example, Fig. 3 presents the test-based apparent mass of an optical instrument, together with its „asymptotic‟ counterpart [5]. In fact, the „asymptotic‟ apparent mass made up of two lines is really an approximation of the asymptotic apparent mass which is more complex around the fundamental frequency. The inclined line of the „asymptotic‟ apparent mass has a slope of 1.5, leading to an exponent 'n' of 3.

Fig. 3 Optical instrument apparent mass and „asymptotic‟ apparent mass (1 lbm =0.454 kg) In Eqs. (2) and (3), the physical mass M0 is known, Saa is defined or imposed. The value of f0 may be obtained from finiteelement (FE) analysis and finalized from a low level sine or random run performed just prior to the full level vibration testing. The value of the exponent 'n' may be obtained from FE analysis or simply assumed initially to have the nominal value of 2, and finalized from the low level sine or random run performed just prior to the full level vibration testing. This parameter is normally estimated to be between 1 and 3.

66 The constant C2 is a key parameter of the semi-empirical method which sets the force limits throughout the complete frequency bandwidth of excitation. However, C2 is the only parameter of Eqs. (2) and (3) that cannot be obtained directly or from a low level sine or random run. Some engineering judgment must be exercised to choose the value of C 2. In fact, the level of conservatism of the force limits depends on the selected value for C2. Another important point may be made from the observation of Equations (2) and (3): the force limits derived from the semi-empirical method are proportional to the input acceleration specification. Consequently, any conservatism in enveloping the interface acceleration spectra is carried over to the force specification. The aim of FLV is only to account for the input notching due to the vibration absorber effect, not to compensate for over-conservatism in defining the input acceleration envelope. The next section presents several criteria that may be considered for selecting a proper C2 value. The subsequent section discusses the range of values that has been either measured or applied for real-life applications, or derived from R&D projects. 4.

Criteria for Selection of C2 Parameter

In real-life applications, selection of the value for C2 may be based on any, or a combination, of the following criteria. (1) Extrapolation of interface force data for similar mounting structures and test items This criterion was the one invoked when the method was introduced by JPL and it is the one most often stated in the literature. The application of this criterion does require some engineering judgment and its strict application implies a clear similarity demonstration between the configuration under consideration and the reference configuration. In the case of large spacecraft with mass in the range of thousands of kg, this criteria would normally justify a C2 value of 2. For much smaller test items such as equipment and secondary structures, C2 may take a larger range of values as discussed in the next section and more caution should be used in selecting this key parameter, especially for small lightweight unit. (2) Comparison with force limits from TDFS methods The more analytical simple and complex TDFS methods result in force limits derived in typically one-third octave frequency bands. When the required input information is available, these TDFS methods may be used for selecting the C2 values. Application of Eq. (2) using as Sff the TDFS force limits of the frequency band encompassing the fundamental frequency f0 of the test item leads to estimation of C2 values. Each of the two TDFS methods generates one value of force limits in this frequency band of interest. It is recommended in Ref. [1] to compute the force limits with both TDFS methods and to select the largest of the two values. This recommendation is quite appropriate especially in light of a study based on 15 case configurations that showed the complex TDFS method always resulting in force limits higher than the reference coupled-system interface forces (over-testing) and the simple TDFS method computing sometimes force limits lower than their reference value (under-testing) [11]. One might wonder what are the advantages of using the semi-empirical method for deriving the force limits in cases for which enough information is available for using the more analytical TDFS methods. Two reasons that justify considering the semi-empirical method in such cases follow. First, force limits derived from the TDFS methods are just as accurate as the impedance information which are much dependent of the accuracy of the FE models (or test-based apparent masses) for both the test item and the mounting structure, up to the maximum frequency of excitation, often 2000 Hz for secondary structures and equipment. Although the impedance information related to interface force does not require FE models as precise as those needed for defining response limiting, it still needs reasonably accurate models. On the other hand, the semi-empirical approach using the TDFS methods for defining or confirming C 2 requires reasonably precise impedance information and FE models only up to the frequency band encompassing f 0, usually much lower than 2000 Hz. The force limits for the higher frequencies are derived using Eq. (3) and the parameters f 0 and 'n' measured from the apparent mass of the test item at the low-level runs. Second, even in the case of force limits derived from the TDFS methods expected to be accurate for the complete excitation bandwidth, one might want to estimate the corresponding C 2 of the semi-empirical method as a simple check of these 'analytically' derived force limits. For typical structures, any corresponding C2 values significantly different than the typical values discussed in the next section might suggest a verification of the FE models, procedures or tools used for deriving these limits. It should be noted that this comparison with force limits from the TDFS methods is normally not possible when the test item is the complete spacecraft, since the requested impedance information for the mounting structure (i.e. the launch

67 vehicle) is not readily available. However, the required information is more likely to be available when the test item is a secondary structure since the mounting structure becomes the remaining of the spacecraft (or part of it in the vicinity of the test item). (3) Comparison with quasi-static limit load The maximum expected acceleration at the center-of-gravity (CG), the quasi-static limit load or Limit Load Factor (LLF), is a key parameter in the design, analysis and testing of space structures. The typical spacecraft structural design approach includes an initial sizing of primary structural members based on conservative quasi-static design limit loads followed by more detailed coupled load analysis to determine component accelerations. The limit loads are provided by the launch organization, and are usually based on flight test data if available and previous analysis of similar payloads. Similarly, limit load values may be provided for secondary structures of a spacecraft. The limit loads are simply the interface force divided by the weight of the structure. When limit loads are available and applicable, it is common for random vibration test to multiply the measured rms force (the square root of the area under the PSD curve) by a peak factor of 3 (3 sigma) to get the maximum CG acceleration for comparison with the limit loads. It should be noted that it has been proposed recently to apply larger peak factors in light of the frequent observance of 5 sigma test peaks in time histories of responses in random vibration tests [12]. This comparison criterion is easily implemented by superposing the force limit curve obtained from Eqs. (2) and (3) over a force PSD curve (measured from a low-level run or derived from the analysis of a FE model), and by computing the rms value of the clipped force PSD resulting from the application of the force limits. Because of its conservative nature, the limit load is used to provide an upper bound for selecting a C 2 value. Since random vibration testing is considered as a workmanship test as opposed to a strength qualification test such as sine sweep excitation, the limit load should be considered as limit not to exceed, as opposed to a value to reach. Consequently, the limit load is really useful only if the application of the present criterion leads to a low enough corresponding C2 value. For example, suppose the limit load and the force PSD are such that this upper limit criterion results in a C2 value of 20, this is not much useful since the above Criterion No.1 of extrapolation of interface force data would normally dictate the selection of a much lower value for C2 as discussed in the next section. (4) Comparison with coupled system interface force When such information is available, the maximum interface force between the test item and the mounting structure for the launch configuration or for higher level-of-assembly test configuration may be used for assessing a proper value for C2. Such interface force may be obtained from coupled load analysis or from previous interface measurements during launch or during coupled system testing. The interface force of interest here is only the maximum of the total force PSD occurring at the peaks corresponding to the two system level modes that are adjacent to the fundamental frequency f 0 of the test item by itself (on a rigid base in its test configuration). This fundamental mode of the test item corresponds to an antiresonance in the interface acceleration PSDs in system configuration, caused by the vibration absorber effect. Reference [13] presents a good example of such comparison. It should be mentioned that the antiresonance in the acceleration PSDs at system level is likely to be slightly different than f0 due to off-axis boundary condition differences. Once this maximum interface force is obtained, it may be used as S ff in Eq. (2), implying that one is selecting the force limit at f0 so that it is equal to the maximum interface force in the vicinity of f0 at system level. This is in fact implementing the objective of force limited vibration. The application of Eq. (2) with the test specification as S aa results in computation of the C2 value. This approach is obviously applicable only for cases for which one has confidence that the maximum interface force provided has not been made too conservative (through the addition of very large margin or margin over margin). As for the comparison with force limits from the TDFS methods, one might wonder why the semi-empirical method should be applied if system level interface forces are known. A reason for still using the method follows. Provided interface forces are most likely to be reasonably precise (or simply known) only up to a maximum frequency not too far above f0, i.e. way below the maximum frequency of excitation likely to be 2000 Hz for secondary structures and equipment. On the other hand, the semi-empirical approach requires only one precise value of the force limits at system level as discussed previously. The force limits for the complete excitation bandwidth are then derived using Eqs. (2) and (3) and the parameters f0 and 'n' measured from the apparent mass of the test item at the low level run.

68 (5) Comparison with the mechanical impedance correction technique The mechanical impedance correction technique is a simple method introduced in 1987 by K.A. Sweitzer to reduce the input PSD at the major modes of a test item, in order to correct the impedance mismatch between test and flight configurations [14]. This method may be used as a criterion for selecting a value for C 2 which results in a level of notching at the fundamental frequency equivalent to the notching provided by the correction technique. In the context of the semi-empirical approach, this corresponds in equating the value of C2 to the amplification measured in the apparent mass at f0. Here again, the apparent mass is obtained from a low-level run performed prior the full level run. A recent investigation has shown that the use of the mechanical impedance correction technique may in fact lead to quite conservative C2 and force limits selection [15]. This analytical investigation based on 65 different cases using a Q of 20 for all modes resulted in all cases (except one) defining force limits that were overtesting with respect to the reference system level interface forces. Also, more than one third of theses cases turned out to produce force limits that are more than 6 dB higher than their reference values. Because of the conservative nature the mechanical impedance correction technique, the C2 value derived from this criterion should be taken as a value not to exceeded or as an upper bound. (6) Assessment of the amount of force clipping and input notching This criterion is related to the fact that force limiting should not result in excessive clipping of the force PSD or notching of the input acceleration PSD at f0. Even if the interface force and the input accelerations are two different parameters, their assessment are included in a single criterion since they are normally evaluated together. Although this criterion is the one requiring the largest amount of engineering judgment, it is still often extremely useful for finalizing the C2 value selection, especially when no pertinent value for comparison purposes (e.g. Limit Load Factor) is readily available or useful (being too high). This criterion should be implemented first at a low-level run (e.g. –12 dB) and the decision should be continuously re-assessed, and modified if pertinent, at the various intermediate levels before reaching the full level. It is recommended in [1] that input notches deeper than 14 dB be implemented only with appropriate peer review. As a minimum, significant assessment should be done before selecting C 2 values that will result in such deep notch. Also, input notching of the order of 14 dB for real structures can often result in significant rms reduction of the interface force of about 50%. As a general guideline for how much of the force PSD may be clipped, the force limits may not completely eliminate the force PSD peak associated with the selected f0 and there must still be some force amplification left. The amount of appropriate residual peak to be left is dependent on several factors and is decided on a case by case basis. For cases when the input acceleration profile has breakpoints in the vicinity the f0, it is sometimes difficult to assess the amount of residual force PSD that is left from the peak being clipped. Reference [16] presents an example of an extreme such case and how the original input profile was replaced by a much simpler profile for the sole purpose of allowing an easier assessment of the residual force. 5.

Range of Values for C2 Parameter

The literature contains several documents that globally define a range in which the C 2 value is likely to fall most of the time for typical space hardware. Based on interface acceleration and force data measured during a limited amount of flights and obtained in some R&D studies, C2 is most likely to fall in the range of 2 to 5. However, the R&D studies indicate that the broader range of 1 to 10 does encompass the large majority of the C 2 values that were observed. It should be mentioned that C2 values lower than 2 have historically been reserved only for large and massive spacecraft [2]. Also, it is important to note that in some particular cases values higher than 10 might be more appropriate, especially for very lightweight structures. The following paragraphs present the main references upon which the above range limits are based. This list includes only references that are based on interface data from flight measurements or analysis/testing of representative hardware or of relatively complex structures. The Advanced Composition Explorer (ACE) spacecraft was launched in 1997 with a Delta II launch vehicle. Flight data were measured at the interface of the Cosmic Ray Isotope Spectrometer (CRIS) instrument. Analysis of these data resulted in an estimated value of about 2 for C2 as reported in Ref. [17].

69 Two experiments were flown on two shuttle flights in 1998 and 1999. These experiments were mounted on the shuttle sidewall of the cargo bay. The measured acceleration and force data yields a C2 of about 2 [18]. In Ref. [19], Chang compared the CG acceleration from the semi-empirical method with limit loads predicted by the Mass Acceleration Curve (MAC) data. He suggested that normally C2 values as low as 2 might be chosen for complete spacecraft or strut-mounted heavier equipment, and that C2 values of 5 might be appropriate for some hardware such as direct-mounted lighter equipment. The results of an investigation of FLV with a test item representing an electronic box mounted on a honeycomb panel were reported in Ref. [20]. The paper shows that a C2 value of 1.4 was consistent with the maximum interface force. Reference [13] presents the main results and findings of an investigation of the semi-empirical method, especially the range of values taken by C2 and the parameters on which it depends. The paper includes the details and the results of an in-depth analytical sensitivity study with 134 different cases, and the results of the experimental validation of the analytical procedures. The C2 values are found to be below 5 for most cases and below 10 for the large majority of cases. The results of a R&D activity investigating the C2 coefficients for three different units of a spacecraft are presented in Ref. [7]. These C2 values were derived from unit interface data measured during random vibration runs of the spacecraft. All derived C2 values are less than 10, with most of them being less than 2. As part of an investigation focused on the overtesting during shock testing, Ref. [21] also includes some vibration results and presents the analytical estimation of C2 for 20 different configurations involving a reconfigurable prototype of an electronic box as a test item. The majority of the C2 values are found to be less than 5. 6.

Force Limiting Versus Response Limiting

In order to alleviate the overtesting problem, most organizations have turned to some form of response limiting for several decades at the spacecraft level and some of them also at the equipment level. Response limiting consists of (1) predicting, usually through couple load analysis or when available through flight data on similar structures, the in-flight response at critical locations on the test item, (2) measuring these responses during the test, and (3) reducing or notching the input acceleration at the critical resonance frequencies, so that measured responses do not exceed the predicted limits. The FLV approach was presented earlier as being an improved approach for notching the input acceleration during a vibration test and reducing the input load to a more realistic level. This section now presents the main advantages of force limiting over the more traditional response limiting approaches. The main advantages of the semi-empirical force limiting over response limiting follow. 

One of the key criteria for defining input notching is based on measuring the acceleration at the test item CG and comparing it with the maximum CG acceleration expected during the flight, the Limit Load Factor. A widely used approach for obtaining this measurement is to use an accelerometer located at the static CG of the structure. However, as discussed in Ref. [22], once a structure starts to deform under vibration, the CG moves away from the static CG and becomes a virtual point, rather than a point fixed to the structure. It thus becomes very difficult to measure the true CG acceleration during a vibration test, especially at frequencies above the fundamental resonance. On the other hand, since the CG acceleration is equal to the interface force divided by the mass of the structure as discussed earlier, it is straightforward to measure it using the force sensors inserted between the structure and the test fixture.



The FLV approach relies on directly measuring and limiting the input interface load or force. Consequently, unlike response limiting, force limiting does not rely on the knowledge of the relationship between the acceleration response of a critical point on the structure and the input load. Such knowledge is always subject to uncertainties created by the limitations of FE modeling, especially for higher frequency bandwidth and for random excitation. Force limiting thus almost eliminates reliance on accurate FE models for notching of the primary modes; however, it can take advantage of such tools if available, as discussed earlier for Criteria 2 and 4. In fact, a FLV vibration test can be done without performing any pre-test analysis, even if such analysis using FE-based apparent masses can be quite useful in predicting the outcomes of the tests and in performing some sensitivity analysis.

70 

In the case of response limiting, it may be sometimes complicated to take measurement at some critical locations. In fact, some of these locations may simply not be accessible. In contrast, the force sensors for FLV are incorporated at the interface attachments, locations normally easily accessible.



A test based on force limiting involves a significantly smaller number of limiting channels in the control strategy than for response limiting. This reduces the cost of test set-up and implementation, and simplify the definition and implementation of the control strategy for the tests. Indeed for equipment testing, one usually requires only a single limiting channel, the total interface force in the direction of excitation. Especially at spacecraft level but also sometimes at equipment level, the use of a limited number of accelerometers at critical locations might be required for monitoring and possibly limiting purposes. Based on their measurements during preliminary low-level runs, some of these accelerometers may be used for complementary response limiting of local modes, in addition to force limiting.



Force limiting results in automatic notching of input acceleration in real time for all global modes (those most significant from an interface point of view). Also, it is observed that for a majority of practical cases, the C2 values of the semiempirical method (and consequently the force limits) are independent of the damping values [1, 13]. This is important considering that damping may vary significantly from the preliminary lower test level runs (e.g. -12 dB, -6 dB) to the 0 dB full level run. This consequently can significantly simplify the decision process for defining and fine tuning the notching level of individual modes in order to obtain an acceptable rms input force.



The process of getting agreement by all parties involved on acceptable notched input values for response limits may sometimes be very tedious and may require lengthy analysis and discussions. On the other hand, because it relates to the physical vibration absorber effect and the dynamics of the test item at its interface, force limiting provides a theoretically sound basis for input notching which once understood is easier to justify and to be acceptable to all parties, and thus reduces the effort of coming to an agreement on this sometimes serious issue.



Because of the previous advantages, FLV is easier and faster to plan and to perform. It consequently results in a faster and more economic test campaign.

7.

Conclusion

This paper review the background behind the semi-empirical method of Force Limited Vibration (FLV) for reducing the overtesting associated with conventional vibration testing of aerospace hardware. It also includes a detailed discussion on various criteria applicable in real-life situations for selecting the key C2 parameter defining the force limits. It is shown that, although C2 is a configuration-dependent parameter, there are several criteria that may be applied, individually or in combination, in order to make a proper selection of this parameter. The paper also presents a list of significant advantages of force limiting over the more traditional response limiting for reducing the input acceleration or load at the primary modes of the test item, in order to bring it down to a more realistic level. Because of its numerous advantages, force limiting is a very powerful approach for reducing artificial hardware failure during vibration testing, especially for sensitive test equipment. References 1.

Anon., “Force Limited Vibration Testing”, Jet Propulsion Laboratory, NASA Technical Handbook, NASA-HDBK7004B, Pasadena, CA, 29 pages, 2003.

2. Scharton, T.D. and Lee, D., “Random Vibration Test of Mars Exploration Rover (MER) Flight Spacecraft”, The 2003 S/C and L/V Dynamic Environments Workshop, El Segundo, CA, www.aero.org/conferences/sclv/2003proceedings.html, 2003.

3.

Salvignola, J.-C., Laine, B., Nganc, I, Honnend, K. and Kommere, A., “Notching During Random Vibration Test Based on Interface Forces – the JWST NIRSPEC Experience”, Proceedings of the European Conference on Spacecraft Structures, Materials and Mechanical Testing, Toulouse, France, 2009.

4.

Qinzhong, S., “Introducing the Status and Topics of New JAXA Handbook for Spacecraft Mechanical Test”,

71 Proceedings of the Aerospace Testing Seminar, Manhattan Beach, CA, 2009. 5.

Soucy, Y., Singhal, R., Lévesque, D., Poirier, R., Scharton, T.D., “Force Limited Vibration Testing Applied to the Fourier Transform Spectrometer Instrument of SCISAT-1”, Canadian Aeronautics and Space Journal, Vol. 50, No. 3, pp. 189-197, 2004.

6.

Soucy, Y., Woronko, A, Tremblay, P., and O‟Grady, M. “Force Limited Vibration Testing Applied to the CASSIOPE Spacecraft”, Proceedings of the 15th CASI Astronautics Conference - ASTRO 2010, Toronto, Canada, 2010.

7.

Soucy, Y. and Montminy, S., “Investigation of Force Limited Vibration Based on Measurements During Quicksat Satellite Vibration Testing”, Proceedings of the 14th Canadian Astronautics Conference - ASTRO 2008, Montreal, Canada, 2008.

8.

Scharton, T.D., “Force Limited Vibration Testing Monograph”, Jet Propulsion Laboratory, NASA Reference Publication RP 1403, Pasadena, CA, 1997.

9.

Skudrzy, E., “Simple and Complex Vibratory Systems”, Pennsylvania State University Press, University Park, PA, 1968.

10. Skudrzy, E., “The Mean-value Method of Predicting the Dynamic Response of Complex Systems”, J. Acoust. Soc. Am., Vol. 67, No. 4, 1980. 11. Soucy, Y., Dharanipathi, V., and Sedaghati, R., “Comparison of Methods for Force Limited Vibration Testing”, Proceedings of the IMAC XXIII Conference, Paper No. 25, Orlando, FL, 2005. 12. Scharton, T.D., Pankow, D. and Sholl, M. “Extreme Peaks in Random Vibration Testing”, The S/C and L/V Dynamic Environments Workshop, Hawthorne, CA, USA, http://www.aero.org/conferences/sclv/2006proceedings.html, 2006. 13. Soucy, Y., Dharanipathi, V., and Sedaghati, R., “Investigation of Force-Limited Vibration for Reduction of Overtesting”, AIAA Journal of Spacecraft and Rockets, Vol. 43, No. 4, pp. 866-876, 2006. 14. Sweitzer, K.A., “A Mechanical Impedance Correction Technique for Vibration Tests”, Proceedings of the 33rd ATM, Institute of Environmental Sciences, San Jose, CA, pp. 73-76, 1987. 15. Soucy, Y., Dharanipathi, V., and Sedaghati, R., “Investigation of Limit Criteria for Force Limited Vibration”, The 2005 S/C and L/V Dynamic Environments Workshop, El Segundo, CA, http://www.aero.org/conferences/sclv/2005proceedings.html, 2005. 16. Soucy, Y. and Chesser H., “Force Limited Vibration Testing of the MOST Telescope", Proceedings of the 13th Canadian Astronautics Conference - ASTRO 2006, Paper No. 33, Montreal, Canada, 2006. 17. Scharton, T.D., “In-Flight Measurements of Dynamic Force and Comparison with Methods Used to Derive Force Limits For Ground Vibration Tests”, Proceedings of the European Conference on Spacecraft Structures, Materials and Mechanical Testing, Braunschweig, Germany, ESA SP-428, pp. 583-588, 1999. 18. Scharton, T.D., “Force Limits Measured on a Space Shuttle Flight”, Journal of the IEST, Vol. 45, No. 1, 2002. 19. Chang, K.Y., “Force Limit Specifications vs. Design Limit Loads in Vibration Testing”, Proceedings of the European Conference on Spacecraft Structures, Materials and Mechanical Testing, Braunschweig, Germany, ESA SP-468, 2001. 20. Ritzmann, S. and Jahn, H., “Comparison of Dynamic Loads to Space Instruments, Depending on the Stage of Development”, Proceedings of the 22nd Aerospace Testing Seminar, The Aerospace Corporation, El Segundo, CA, 2005. 21. Deblois, J.-P., “Examination of the Shock Overtesting in Assembly-Level Test”, M.A.Sc. Thesis, Department of Mechanical and Industrial Engineering, Concordia University, Montreal, Canada, 2009. 22. Vujcich, M. and Scharton, T., “Combined Loads, Vibration, and Modal Testing of the QuikSCAT Spacecraft”, 1999 World Aviation Conference, San Fransisco, CA, Paper 1999-01-5551, 1999.

Calculation of Rigid Body Mass Properties of Flexible Structures Kevin Napolitano, Martin Schlosser ATA Engineering, Inc., 11995 El Camino Real, San Diego, California 92130 ABSTRACT Experimental measurement of rigid body mass properties, specifically pitch inertia, of aircraft is becoming increasingly important for flutter analysis. This paper proposes a methodology for calculating rigid body inertia properties when the flexible modes of an aircraft are coupled to some degree with their corresponding rigid body modes, i.e. the rigid body modes are not fully rigid. The methodology will be demonstrated on a simple analysis model of a plate-like structure where the rigid body and primary flexible structural modes are coupled. INTRODUCTION Techniques using frequency response functions for measuring rigid body inertia properties are well established [1-3]. These methods use a rigid body coordinate transformation of both the measurement degrees of freedom and the applied loads to create transfer function matrices from rigid body forces to rigid body accelerations. These methods rely on using frequency response function data in the frequency range above the primary rigid body modes and below the primary structural modes. This frequency region contains the “mass line” of the rigid body modes that can be used to infer the rigid body mass properties. Methods have been developed to account for cases where the structural modes begin to couple with the rigid body modes. In these cases where the rigid body modes are close to but below the primary structural resonant frequencies, estimates of residual terms [4] or spatial filtering techniques [5-6] to remove the effects of the flexible body dynamics areay be used to remove the influence of the higher modes on the analysis. However, these methods rely on the mass line of the rigid body frequency response functions, and therefore are not valid for cases when the rigid body modes of vibration are higher in frequency than some flexible modes of the structure being tested. An example of one such structure is a commercial aircraft mounted on soft tire supports. Ideally, an aircraft will be mounted to a soft suspension system that will be able to separate the rigid body suspension modes from the structural modes by a frequency ratio of 5 to 10. However, it is oftentimes financially unfeasible to build suspension systems for structures with large flexible wings. As a point of reference, typical bungee cord suspensions usually provide suspension frequencies near or above 1 Hz. Typical specialized airbag supports provide suspension frequencies from 0.5 to 1 Hz. Rigid body frequencies using these types of suspension systems are usually within the primary wing bending frequency of a commercial airliner which is sometimes near 0.9 Hz. In these cases the primary structural modes, such as the wing bending mode, is coupled with the rigid body modes such as the rigid body bounce mode. Oftentimes, the aircraft is mounted on deflated tires, and in these cases what is identified as the rigid body bounce mode is over a factor of two higher in frequency than the primary structural modes of the aircraft. An example of rigid body modes coupling with structural modes is presented in Figure 1 below.

T. Proulx (ed.), Advanced Aerospace Applications, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 4, DOI 10.1007/978-1-4419-9302-1_7, © The Society for Experimental Mechanics, Inc. 2011

73

74

First Vertical Bounce Mode Near 0.9 Hz: 1st Structural Mode

Vertical Bounce Mode Above 2 Hz; 12th Mode Figure 1 – Example of flexible modes coupling with rigid body vertical bounce mode.

The objective of this paper is to develop the mathematical framework to measure rigid body inertia properties for flexible structures where the rigid body modes of vibration are highly coupled with the flexible structural modes. The paper assumes that the structure is connected to ground through a flexible suspension system and that the loads between the structure and the suspension system are measured. A method for verifying that the number and location of sensors is adequate using pre test analysis is also proposed. First a review of the rigid body mass matrix coordinate transformation is given. Next, a derivation of the method to solve for the rigid body mass matrix is presented. A system of constraint equations to enforce a given form of the rigid body mass matrix is derived. Cross-orthogonality and pseudo-orthogonality methods used in modal testing are used to verify sensor selection. Finally, an example of the methodology is then presented.

REVIEW OF RIGID BODY COORDINATE TRANSORMATION Assume that a linear structure has been adequately discretized such that the continuous displacement field X(x,y,z) can be estimated from a number of discrete degrees of freedom, q . The resulting well-known equation of motion can be derived from energy or variational principles and is

{}

[M ]{q&&} + [C ]{q&}+ [K ] {q} = { f } , qq

where

[M ] , [C ] , and [K ] qq

qq

of degrees of freedom,

qq

qq

qq

q

(1)

are the mass, damping, and stiffness matrices associated with a finite number

{q}, and { f q } is the applied force vector associated with these same degrees of freedom.

The accuracy and fidelity of this discrete set of equations is dependent on how well the continuous deformation of the structure can be estimated from the finite set of degrees of freedom. Rigid body mass properties about a point in space are defined by performing a specific rigid body coordinate transformation of the continuous system down to six discrete rigid body degrees of freedom. These degrees of

75 freedom are three perpendicular displacement vectors of unit length, and three unit rotations about these three displacement degrees of freedom. For a discrete system, the rigid motion can be defined by using a coordinate transformation matrix

{q} = [Rqr ] {r}, where

(2)

[R ] is the rigid body coordinate transformation matrix, and {r}is a 6 by 1 vector of rigid body degrees of qr

freedom. For a given Cartesian coordinate system where the degrees of freedom are defined as translations and rotations at a physical point in space, the coordinate transformation, {q i } = Ri {r }, is as follows:

[ ]

 x i  1  y  0  i   z i  0  = θx i  0 θy i  0    θz i  0

where

0 1 0 0 0 0

0 0 0 − ∆z i 1 ∆y i 0 1 0 0 0 0

∆z i 0 − ∆x i 0 1 0

− ∆y i   X  ∆x i   Y  0   Z    0  R X  0   RY    1   R Z 

(3)

,

( x i , y i , z i ) are the translational degrees of freedom at location “i”, (θx i , θy i , θz i ) are the rotational

(∆x i , ∆y i , ∆z i ) correspond to the distance between location “i” and the location of the rigid body degree of freedom “r”, and ( X , Y , Z , R X , RY , R Z ) are the six rigid body degrees of

degrees of freedom at location “i”, freedom.

The rigid body mass matrix is therefore defined as the mass matrix that results from the following coordinate transformation

[M RB ] = [Rqr ]T [M qq ][Rqr ].

(4)

The rigid body mass matrix has a particular form

0 0 0 mZcg − mYcg   m  0 m 0 − mZcg 0 mXcg    0 0 m mYcg − mXcg 0  [ M RB ] =   − mZcg mYcg Ixx Ixy Ixz   0  mZcg Ixy Iyy Iyz  0 − mXcg   0 Ixz Iyz Izz  , − mYcg mXcg

(5)

where m = mass, {Xcg,Ycg, and Zcg} are X,Y, and Z coordinates of the center of gravity with respect to the defined Cartesian coordinate system, and Ixx, Iyy, Ixz, Ixy, Ixz, and Izz are the rigid body inertia properties. If the structure behaves rigidly over the frequency range of interest during a test, applying the rigid body transformation matrix to the test data has been shown to be sufficient to extract reasonable inertia properties.

76

DERIVATION In order to calculate rigid body mass properties where the rigid body modes are coupled with the flexible body modes, the flexible motion of the structure must be taken into account. This is done by removing the assumption that the structure moves only as a rigid body. Assume that the structure has been discretized by a set of physical degrees of freedom {q} that are sufficient to describe the motion of the structure. Assume that the independent

degrees of freedom {q}are now associated with physical degrees of freedom (rotations and translations) that are directly measured from test with instruments such as accelerometers. Assume that all forces applied to a structure under test are measured. These forces include applied loads such as those due to shakers, and reaction forces due to the suspension system. Assume the structure itself does not store or dissipate energy if it is moved rigidly. Pre multiply Equation 1 by the transpose of the rigid body transformation matrix

([ ]

[ ] {q&} + [K ]{q} = { f }).

[ Rqr ]T M qq {q&&} + C qq

qq

q

Since there is no energy dissipated nor is there any energy stored, the equation of motion is reduced to

[

[ Rqr ]T M qq

] {q&&} = [M ]{q&&} = [ R

qr

[ Rqr ] such that (6)

[ R]T ([C ]{q&} + [K ]{q}) = {0} , and therefore

]T { f q } ,

(7)

where

[M ] = [ R ] [M ] . T

qr

Note that

{q&&} and { f q } are

known (measured), and

spatial locations and orientations. The entries in

[M ]

qq

(8)

[ Rqr ] is known because it is generated from measured are unknown.

This equation is valid in the time domain as well as the frequency domain. However, this paper will concentrate on frequency domain analysis. At a given frequency, Equation 7 becomes

[ M ]{q&&(ω K )} = [ Rqr ]T { f q (ω K )} ,

(9)

where ω K is a single frequency. Additionally, the equation holds for frequency response functions (FRF) using an arbitrary reference. In practice the reference is a load cell associated with the excitation force applied to the structure. Thus,

[ M ][H (ω K ) ] = [ Rqr ]T [F (ω K )],

(10)

[H (ω K )] is a frequency response function matrix whose rows correspond to measured degrees of &&}, and whose columns correspond to independent reference signals such as shaker drive point load freedom in {q cells, [F (ω K )] is a frequency response function matrix of measured loads whose rows and columns correspond to the same degrees of freedom as [H (ω K ) ]. The number of rows and columns in [H (ω K ) ] are defined as NDOF and NREF, respectively. Note that the majority of rows in [F (ω K )] will be empty (loads are measured at where

77 very few degrees of freedom) and therefore the operation

[ Rqr ]T [F (ω K )] can be condensed to only the applied

load degrees of freedom. Also note that the derivation assumes that the accelerations for all load degrees of freedom are measured. This assumption is likely not necessary if the deformation of the structure at the reaction force degrees of freedom can be approximated with the other measured accelerometers; i.e. there is not significant unmeasured local deformation at the loading location.

[ ]

In order to force the solution for M to be a real valued matrix, the real and imaginary part of Equation 10 can be set equal to each other and then assembled together as follows.

[ M ][Re([H (ω K )]) Im([H (ω K )])] = [ Rqr ]T [Re([F (ω K )]) Im([F (ω K )])] ,

(11)

or

[

] [ ] [F (ω )] .

[ M ] H (ω K ) = Rqr

T

(12)

K

In order to assemble the equation into the standard format for solving a linear set of equations, the transpose of Equation 12 is taken such that

[H (ω )] [M ] = [F (ω )] [ R T

T

T

K

K

qr

].

(13)

The acceleration and force FRF matrices can be stacked together in the following manner to create a system of equations that can be used to solve for

[ M ]T .

 [H (ω1 )]T   [F (ω1 ) ]T    T  T   [H (ω 2 )] [ M ]T =  [F (ω 2 )] [ R ] , or [ A][ M ]T = [Y ]     qr M M    T T  [H (ω N )]  [F (ω N )] 

(14)

[ ]

T

The matrix [ M ] contains NDOF rows. To obtain a unique solution the rank of the matrix A must therefore be at least NDOF. In most cases, however, a limited number of unique operating deflected shapes will dominate the response of the structure and therefore the matrix A will be ill-conditioned.

[ ]

[ ]

One way to improve the conditioning of A is to assume the structure deforms in a limited number operating shapes. This assumption is implemented with a coordinate transformation from the physical degrees of freedom {q}to a reduced set of generalized degrees of freedom {u}through the relationship

{q} = [U ]{u},

(15)

where [U ] is a matrix containing a finite number of assumed deformation shapes. These shapes can be anything that accurately represents the motion of the structure such as the modes derived from finite element analysis or perhaps extracted during vibration testing, The shapes can also be the dominant shapes extracted from a singular value decomposition from test data such as time history data or frequency response function data. In this paper,

[U ] was constructed in two steps. The first six shapes are the rigid body shapes of the test degrees of freedom, [ Rqr ] . The remaining shapes are the shapes needed to estimate the remaining shapes associated with the matrix

the rows of

[A] (which contain the operating deflected shapes of the FRF matrix). First, the contribution of the

78 rigid body shapes is estimated by solving for least squares sense.

[C~]

in the following equation to minimize the error matrix

[]

~ T [ E ] = [A] − [ Rqr ] C Then a singular value decomposition of the error matrix

(16)

[E ] is performed such that

[E ] = [U ][ S ][V ]T . The matrix

[E ] in a

(17)

[]

[U ] is then constructed using the first “n” columns of [U ] , called Uˆ , and the rigid body shapes

[ Rqr ] such that

[[ ] [Uˆ ]] .

[U ] = Rqr

(18)

Note that the inverse of Equation 22 can be solved in a least squares sense to estimate the motion of the generalized degrees of freedom from the physical degrees of freedom.

{u} = ([U ]T [U ])−1 [U ]T {q} = [U ]−1 {q}

(19)

The coordinate transformation to a limited number of degrees of freedom can be applied directly to the frequency response function matrix, or it can be applied by inserting the identity matrix between the Equation 14 and then noting that [U ]

−1

[A] matrix and [ M ]T

[U ] = [ I ] .

  [H (ω1 )]T     [H (ω1 )]T    [F (ω1 )]T       T  T  T    [H (ω 2 )]  [U ]−1  [U ][ M ]T =   [H (ω 2 )] [U ]−1 [ M ] =  [F (ω 2 )] [ R ], or A [ M ] = [Y ] .         qr M M M       T  T T   [H (ω N )]    [H (ω N )]   [F (ω N )] 

[

[]

]

In order to add constraint equations in the next section of this paper, the six columns of create one column vector of unknowns such that

[]

A        

{ { [A ] { [A ] { [A ] [A ] { []{

} } } } } }

matrix

(20)

[M ] are now stacked to

  M (:,1)   {Y (:,1)}       M (:,2)  {Y (:,2)}   M (:,3)  {Y (:,3)} )   , or Aˆ M = Yˆ . =   M (:,4)  {Y (:,4)}   M (:,5)  {Y (:,5)}     A   M (:,6)  {Y (:,6)}

[ ]{ } { }

(21)

)

The notation

in

{X (:, j )} corresponds to the jth column of the matrix [X ] . Once a solution for {M } is calculated, the

[M ] can be assembled. The matrix [ M ]T is then calculated from

79

[ M ]T = [U ] [ M ], or[ M ] = [ M ]T [U ] . −1

−T

(22)

The rigid body mass matrix can now be calculated substituting Equation 8 into Equation 4 such that

[ ] [M ][R ] = [M ][R ] = [ M ] [U ] [R ] = [M ] [R ] ,

[ M RB ] = Rqr

T

−1

T

qq

qr

qr

T

qr

(23)

where

[R ] = [U ] [R ]. −1

(24)

qr

Also note that since

[ M RB ] is symmetric. Therefore, [ M RB ] = [ M RB ]T = [R ] [ M ] . T

(25)

INCLUSION OF CONSTRAINT EQUATIONS The rigid body mass matrix has a particular form as defined in Equation 5. Constraint equations can be used to enforce this form by examining the rows and columns of Equation 25, and noting that

{

}

M RB (i, j ) = {R (:, i )} M (:, j ) . T

(26)

Terms in the rigid body mass matrix, shown in Equation 5, equal to zero can be specifically enforced in all cases. If specific terms in the rigid body mass matrix have already been measured, such as the mass of the structure using conventional scales, then those terms can be defined explicitly. If terms are not defined, then constraint equations with equivalent values can be enforced. For example, the first three diagonal terms of [ M RB ] are equal. Therefore,

{R (:,1)} {M (:,1)}− {R (:,2)} {M (:,2)}= 0 {R (:,1)} {M (:,1)}− {R (:,3)} {M (:,3)}= 0 T

T

T

T

Symmetry of the overall rigid body mass matrix can be enforced as well as anti-symmetry of the upper diagonal portion of the rigid body mass matrix. With these principles in place, the algorithm for generating constraint equations is as follows:

80 1. Enforce all zero entries in rigid body mass matrix

{R (:,1)} {M (:,2)}= 0 {R (:,1)} {M (:,3)}= 0 {R (:,1)} {M (:,4)}= 0 {R (:,2)} {M (:,1)}= 0 {R (:,2)} {M (:,3)}= 0 {R (:,2)} {M (:,5)}= 0 {R (:,3)} {M (:,1)}= 0 {R (:,3)} {M (:,2)}= 0 {R (:,3)} {M (:,6)}= 0 {R (:,4)} {M (:,1)}= 0 {R (:,5)} {M (:,2)}= 0 {R (:,6)} {M (:,3)}= 0 T

T

T

T

T

T

T

T

T

T

T

T

2. Mass terms: a. If mass has been measured

{R (:,1)} {M (:,1)}= m {R (:,2)} {M (:,2)}= m {R (:,3)} {M (:,3)}= m T T T

b. Otherwise

{R (:,1)} {M (:,1)}− {R (:,2)} {M (:,2)}= 0 {R (:,1)} {M (:,1)}− {R (:,3)} {M (:,3)}= 0 T

T

T

T

3. mZcg terms: a. If mZcg has been measured (both mass and Zcg)

{R (:,1)} {M (:,5)}= mZcg {R (:,2)} {M (:,4)}= −mZcg {R (:,4)} {M (:,2)}= −mZcg {R (:,5)} {M (:,1)}= mZcg T

T T

T

b. Otherwise

{R (:,1)} {M (:,5)}+ {R (:,2)} {M (:,4)}= 0 {R (:,1)} {M (:,5)}+ {R (:,4)} {M (:,2)}= 0 {R (:,1)} {M (:,5)}− {R (:,5)} {M (:,1)}= 0 T

T

T

T

T

T

4. mYcg terms: a. If mYcg has been measured (both mass and Ycg)

{R (:,1)} {M (:,6)}= −mYcg {R (:,3)} {M (:,4)}= mYcg {R (:,4)} {M (:,3)}= mYcg {R (:,6)} {M (:,1)}= −mYcg T

T

T

T

81 b. Otherwise

{R (:,1)} {M (:,6)}+ {R (:,3)} {M (:,4)}= 0 {R (:,1)} {M (:,6)}+ {R (:,4)} {M (:,3)}= 0 {R (:,1)} {M (:,6)}− {R (:,6)} {M (:,1)}= 0 T

T

T

T

T

T

5. mXcg terms: a. If mXcg has been measured (both mass and Ycg)

{R (:,2)} {M (:,6)}= mXcg {R (:,3)} {M (:,5)}= −mXcg {R (:,5)} {M (:,5)}= −mXcg {R (:,6)} {M (:,2)}= mXcg T

T

T

T

b. Otherwise

{R (:,2)} {M (:,6)}+ {R (:,3)} {M (:,5)}= 0 {R (:,2)} {M (:,6)}+ {R (:,5)} {M (:,3)}= 0 {R (:,2)} {M (:,6)}+ {R (:,6)} {M (:,2)}= 0 T

T

T

T

T

T

6. Ixy terms: a. If Ixy has been measured

{R (:,4)} {M (:,5)}= Ixy {R (:,5)} {M (:,4)}= Ixy T

T

b. Otherwise

{R (:,4)} {M (:,5)}− {R (:,5)} {M (:,4)}= 0 T

T

7. Ixz terms: a. If Ixz has been measured

{R (:,4)} {M (:,6)}= Ixz {R (:,6)} {M (:,4)}= Ixz T

T

b. Otherwise

{R (:,4)} {M (:,6)}− {R (:,6)} {M (:,4)}= 0 T

T

8. Iyz terms: a. If Iyz has been measured

{R (:,5)} {M (:,6)}= Iyz {R (:,6)} {M (:,5)}= Iyz T

T

b. Otherwise

{R (:,5)} {M (:,6)}− {R (:,6)} {M (:,5)}= 0 T

T

9. If Ixx has been measured:

{R (:,4)} {M (:,4)}= Ixx T

10. If Iyy has been measured:

{R (:,5)} {M (:,5)}= Iyy T

11. If Izz has been measured:

{R (:,6)} {M (:,6)}= Izz T

For example, assume that all of the terms in the rigid body mass matrix have not been measured by other means. Then the constraint equations can be assembled as follows

82

 0  0   0  T  {R (:,2)}  0   0 T   {R (:,3)}  0  0   {R (:,4)}T  0   0  T  {R (:,1)} T   {R (:,1)}  0  0   − {R (:,5)}T  0   0  T − {R (:,6)}  0   0  0   0  0   0 

{R (:,1)}

T

0 0 0 0 0 0 {R (:,3)}T 0 0 {R (:,5)}T 0 T − {R (:,2)} 0 {R (:,4)}T 0 0 0 0 0 0 0 T − {R (:,6)} 0 0 0

0 {R (:,1)}T 0 0 {R (:,2)}T 0 0 0 0 0 0 {R (:,6)}T 0

− {R (:,3)} 0 0 0 0 {R (:,4)}T 0 0 {R (:,5)}T 0 0 0 0

T

0 0 {R (:,1)}T 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 {R (:,2)}T 0 0 0 0 0 0 0

0 0 {R (:,2)}T 0 {R (:,3)}T 0 0 0 0 0 T − {R (:,5)} T − {R (:,6)} 0

0 {R (:,1)}T {R (:,1)}T {R (:,1)}T 0 0 0 {R (:,3)}T 0 0 {R (:,4)}T 0 T − {R (:,6)}

 0 0  0 0     0 0    0  0  0 0     0 0  0 0     0 0 T  0 {R (:,3)}     0 0    0   M (:,1)  0   M (:,2) 0 0       0  M (:,3)  0 =   M (:, 4 ) 0  0    M (:,5)  0 0       0 M (:, 6 ) 0  0  0    0 {R (:,1)}T  T  0 {R (:,1)}    0 {R (:,1)}T    T  {R (:,2)}  0 T 0 {R (:,2)}    {R (:,2)}T  0 0  0    {R (:,4)}T  0 T  0 {R (:,5)}   

(27)

or

[Cˆ ]{M) }= {Xˆ }. []

{}

Note that the form of the constraint matrix Cˆ and the right hand side Xˆ will change depending on the constraints to the solution defined by the user. The constraint matrix can be used to directly remove degrees of freedom from the final set of equations, or it can be appended to the original set of equations and solved for in a least squares sense. Also, the Equation 27 can be pre multiplied by a weighting matrix, de-emphasize the constraints over the test data.

[W ][Cˆ ]{M }= [W ]{Xˆ }

[W ] to help emphasize or

)

(28)

83 The final set of equations can then be assembled as

[] { } []

 Aˆ  )  Yˆ  ˆ ) ˆ , or  Aˆ  M = Yˆ  . M =      ˆ ˆ   [W ] X  [W ] C 

{}

{ }

(29)

The solution to this equation can then be used to create the rigid body mass matrix as discussed in Equations 21 through 25. SELECTION OF TEST MEASUREMENT DEGREES OF FREEDOM The last issue this paper will discuss is determining if enough degrees of freedom have been measured to adequately capture the rigid body mass. We suggest that this issue can be addressed if an analysis model is available by using pretest analysis techniques developed for ground vibration tests. First target modes are selected based on how well the modes approximate rigid body deformation. Then a pseudo-orthogonality calculation can be performed to determine whether or not the selected sensors have adequately captured the mass of the target modes. We propose that the target modes be selected by inspecting the cross-orthogonality matrix between the rigid body shapes and the finite element model shapes. In this case the model of the suspension system is added to the finite element model of the structure in order to estimate the interaction between the rigid body modes and the flexible modes. Elements of the cross-orthogonality matrix [ORG ] between the rigid body shapes, Rqr , with

[Φ ] , are calculated as follows: {Ri }T [M ]{Φ j } ORG (i, j ) = ({Ri }T [M ]{Ri })({Φ j }T [M ]{Φ j })

[ ]

respect to the analysis shapes,

where

,

(30)

{Ri } is the ith column of the rigid body transformation matrix and and {Φ j } is the jth column of the mode

shape matrix. The square root of the sum squared (RSS) of the first “N” columns of the cross-orthogonality matrix

[ORG ] can be

calculated for each row of the cross-orthogonality matrix to determine whether or not those modes can adequately represent the rigid body motion. If the first six modes of vibration are in fact rigid body modes, then RSS of each row of the first six columns of the cross-orthogonality matrix will be equal to one. As the rigid body modes start coupling with the flexible body modes, more modes will need to be included to ensure the RSS of the first “N” terms in each row approaches 1. Once the target modes have been selected, any number of algorithms can be used to select a sensor set such that they can capture the mass of the test article adequately. There have been numerous papers written over the years on optimal sensor selection techniques [7-8]. This paper, however, will concentrate on verifying that a selected set of sensors is adequate. Using modal testing as a guide, we propose using the pseudo-orthognality matrix, POGG , to verify a sensor set is adequate. Psuedo-orthogonality is defined as

[

]

[POGG ] = [Φ GA ]T [M AA ][Φ GA ] ,

(31)

[M AA ] is the statically condensed mass matrix down to the test degrees of freedom and [Φ GA ] are the mass normalized (with respect to the full mass matrix [M ] ) analysis shapes parsed down to the test degrees of where

freedom. For the terminology in this paper, the subscript “A” is equivalent to the subscript “q” and are the physical test measured degrees of freedom. Ideally, the terms in the pseudo-orthogonality matrix approach the identity matrix. At this time we do not have recommendations for minimum values of the diagonal terms, nor maximum

84 values for off-diagonal terms. For a modal test these values are ideally 98% and above for the on-diagonal values and less than 2% for off-diagonal values. EXAMPLE: Flat Plate A method for calculating rigid body mass properties of flexible structures has been outlined, and will be demonstrated in this section on an analysis model of a square flat plate. The demonstration model is a .08” by 100” by 100” steel plate constructed from 10,000 equivalently sized NASTRAN CQUAD elements. The rigid body mass properties about the origin and the center of gravity can be exported from NASTRAN and are presented in Table 1 below.

Table 1. Model rigid body mass matrix with respect to origin (left) and center of gravity (right).

0 0 0 0 − 29.27  0.585  0 0.585 0 0 0 29.27    0 0 0.585 29.27 − 29.27 0    0 29.27 1951 − 1463 0   0  0 0 − 29.27 − 1463 1951 0    0 0 0 3903  − 29.27 29.27

0 0 0 0 0  0.585  0 0.585 0 0 0 0    0 0 0.585 0 0 0    0 0 488 0 0   0  0 0 0 0 488 0    0 0 0 0 976  0

The model is constrained at three locations with springs restraining all six degrees of freedom as shown in Figure 2 below. The translational and rotation spring stiffness were set at 100 lb/in and 100 lb-in/radian, respectively. Simulated forces were applied at three locations.

Supports (3 Nodes, 6 DOF per node)

Z Y

X

Applied Loads (3 Nodes, 7 DOF total)

Figure 2 – Finite element model of plate showing support and applied load locations.

85 Three different sensor sets were studied; a 5x5 grid of vertical accelerometers, an 11x11 grid of accelerometers, and a 21x21 grid of accelerometers. Each of these sensor sets included measurement of the accelerations at all degrees of freedom that would impart a force on the plate which includes all spring degrees of freedom as well as all loading degrees of freedom. Figure 3 presents the accelerometer measurement locations for each case.

5x5 Accelerometer Locations

11x11 Accelerometer Locations

21x21 Accelerometer Locations Figure 3 – Accelerometer measurement locations. Accelerations were measured at all support and applied load degrees of freedom. The number of Z-direction accelerometers was varied to cover a 5x5, 11x11, and 21x21 evenly spaced grid. The first step in the analysis is to determine the target modes from the finite element model by performing the cross-orthogonality calculation in Equation 30. The results of the calculation are presented in Table 2. The inplane rigid body deformations (X,Y, and RZ) are coupled with the three in-plane finite element model mode numbers 5, 9, and 10 whose natural frequencies are 1.31, 3.60, and 3.63 Hz, respectively. The two out-of-plane rotational modes (RX, and RY) are mostly coupled with finite element model mode numbers 1 through 4. Finally, the out-of-plane translational motion has large contribution from finite element model modes 1, 3, 4, 11, and 16. These mode shapes are presented in Figure 4.

86 Table 2. Cross-orthogonality table between six rigid body shapes and finite element model mode shapes. Rigid Body/FEM Cross Orthogonality Table FEM shapes

Rigid Shapes

ORG

2

3

4

5

6

7

8

9

10

11

12

13

0.69

0.79

1.05

1.22

1.31

1.55

2.81

2.86

3.60

3.63

3.84

5.27

5.40

1X

0.00

0.00

0.00

0.00

0.07

0.00

0.00

0.00

0.86

0.50

0.00

0.00

0.00

2Y

0.00

0.00

0.00

0.00

0.11

0.00

0.00

0.00

0.50

0.86

0.00

0.00

0.00

3Z

0.34

0.03

0.73

0.30

0.00

0.03

0.00

0.00

0.00

0.00

0.36

0.01

0.00

4 RX

0.21

0.88

0.05

0.39

0.00

0.03

0.02

0.09

0.00

0.00

0.02

0.05

0.02

5 RY

0.87

0.31

0.24

0.26

0.00

0.07

0.07

0.02

0.00

0.00

0.04

0.04

0.02

6 RZ

0.00

0.00

0.00

0.00

0.99

0.00

0.00

0.00

0.00

0.13

0.00

0.00

0.00

14

15

16

17

18

19

20

21

22

23

24

25

26

ORG Rigid Shapes

1

5.43

6.37

7.02

8.77

8.94

9.26

10.10

10.32

10.39

12.42

12.63

13.64

15.71

1X

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

2Y

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

3Z

0.01

0.03

0.31

0.07

0.05

0.03

0.09

0.05

0.08

0.01

0.01

0.06

0.02

4 RX

0.05

0.05

0.03

0.01

0.05

0.01

0.00

0.01

0.01

0.03

0.00

0.00

0.01

5 RY

0.05

0.02

0.04

0.04

0.01

0.02

0.01

0.01

0.00

0.01

0.01

0.00

0.01

6 RZ

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

87

Figure 4 – Modes with significant contribution to vertical (Z) rigid body deformation. The RSS of the rows of the cross-orthogonality matrix

[ORG ] as a function of modal frequency is presented in

Figure 5. The in-plane rigid body deformations (X, Y, and RZ in red) are captured by the three rigid body modes of the finite element model below 4 Hz. The out-of-plane rigid body rotational deformations (RX, and RY) are mostly captured below 1.5 Hz, and there are other minor contributors up to approximately 10 Hz. The rigid body vertical deformation consists of several finite element model modes, and its deformation is captured by all modes below 14 Hz.

88

Rigid Body/FEM Cross Orthogonality RSS 1.00

0.95

RSS

0.90

0.85 Rigid Body X Rigid Body Y 0.80

Rigid Body Z Rigid Body RY Rigid Body RY

0.75

Rigid Body RZ 0.70 0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

Frequency, Hz

Figure 5 – RSS of rows of cross-orthogonality matrix

[ORG ] as a function of frequency. Almost all of the rigid

body motion is captured in all modes below 14 Hz. The next step in the analysis is to verify the adequacy of the sensor sets by calculating the pseudo-orthogonality matrix as defined in Equation 30. A plot of the diagonal terms for each of the sensor sets is presented in Figure 5. Values close to one indicate that the sensor set can adequately capture the mass of that mode. For a modal test, a value above 95% is desired for all target modes. As such, this plot indicates that the dense 21x21 sensor set is more than sufficient for capturing the mass of nearly all of the modes at or below 50 Hz. The ability of the 11x11 sensor set to capture mass of modes gracefully degrades to 50 Hz. The 5x5 sensor set cannot capture the mass of the out-of-plane modes. Note that all sensor sets are sufficient for capturing mass of the in-plane rigid body mode.

89

Pseudo-Orthogonality 125.00

Value, Percent

100.00

75.00

50.00

21x21 25.00

Mass of in-plane rigid body modes (at 1.31, 3.60, 3.63 Hz) are captured by all sensor sets.

0.00 0.00

5.00

10.00

15.00

11x11 5x5

20.00

25.00

30.00

35.00

40.00

45.00

50.00

Modal Frequency, Hz

Figure 6 – Diagonal of pseudo-orthogonality matrix for each sensor set. All plate modes below 100 Hz were calculated using the finite element model code NASTRAN. These shapes were used to calculate frequency response functions of the accelerations, H (ω K ) , and forces acting on the

[

[

]

]

structure, F (ω K ) , with respect to the seven force inputs as defined in Figure 2 from 0.1 to 50 Hz. The procedure derived in this paper was then applied to calculate the rigid body mass properties as a function of analysis frequency and sensor set. The rigid body coordinate transformation matrix

[R ] qr

was defined with respect to the center of the plate which is

also the center of gravity. Since the plate is also symmetric, the resulting rigid body mass matrix should only have terms on the diagonals. This was done to obtain a clear understanding of the results. The number of assumed shapes in the matrix [U ] was equal to the minimum of 1) the total number of sensor degrees of freedom and 2) the number of modes in the given analysis frequency band plus six. All of the constraint equations in Equation 27 were used with the exception of the equations associated with coupling between the in-plane and out-of-plane motion. This was done because errors associated with out-ofplane estimates of mass terms were corrupting the results of the in-plane results even though the in-plane and out-of-plane motion are decoupled. There is likely a very interesting debate on whether or not constraint equations should be applied to the solution, and that debate is left for future discussion. The results are presented in Figure 7 through Figure 10. The first three curves in each figure are the ratio between calculated and model mass terms as a function of maximum analysis band frequency. The minimum analysis band frequency in all cases was 0.1 Hz. The fourth curve is the RSS cross-orthogonality of the corresponding mass term as previously shown in Figure 5. Figure 7 presents results for the mass calculated in the out-of-plane direction. Both the 21x21 and 11x11 sensor sets are able to converge on a solution above 12 Hz, which is also near the frequency where the RSS crossorthogonality indicates that all of the mass in the Z direction is captured by the finite element model modes. While

90 the 21x21 sensor set results in more accurate estimates of the Z-direction mass, the 11x11 estimate is within one percent. Note that the error for both is very small as the analysis frequency band increases. Results for the 5x5 sensor set vary wildly as a function of frequency and fall off dramatically above 20 Hz.

Ratio: Calculated Mass (Z Direction)/ Model Mass 1.10

1.05

Ratio

1.00

0.95

Mass (Z): 21x 21

0.90

Mass (Z): 11x 11 Mass (Z): 5 x 5 'RSS Z'

0.85 0

5

10

15

20

25

30

35

40

45

50

Frequency, Hz

Figure 7 – Ratio of calculated mass (Z-direction) and model mass. Frequency signifies maximum frequency in analysis band.

Results for the out-of-plane inertia terms, Ixx, and Iyy are shown in Figure 8 and Figure 9, respectively. The solutions converge at a lower frequency than the out-of-plane mass term presented in Figure 7 which is consistent with the fact that the RSS cross-orthogonality terms approach 100 percent at a lower frequency. Note that the solution for the 5x5 sensor sets starts to converge but then diverges above 20 Hz.

91

Ratio: Calculated Ixx/ Model Ixx 1.10

1.05

Ratio

1.00

0.95

Ixx: 21x 21

0.90

Ixx: 11 x 11 Ixx: 5 x 5 'RSS RX'

0.85 0

5

10

15

20

25

30

35

40

45

50

Frequency, Hz

Figure 8 – Ratio of calculated Ixx and model Ixx with respect to center of gravity. Frequency signifies maximum frequency in analysis band.

Ratio: Calculated Iyy/ Model Iyy 1.10

1.05

Ratio

1.00

0.95

Iyy: 21x 21

0.90

Iyy: 11 x 11 Iyy: 5 x 5 'RSS RY'

0.85 0

5

10

15

20

25

30

35

40

45

50

Frequency, Hz

Figure 9 – Ratio of calculated Iyy and model Iyy with respect to center of gravity. Frequency signifies maximum frequency in analysis band.

92 Figure 10 presents results for the calculated mass for the in-plane direction X. The value for mass in the X direction was correctly calculated for all sensor cases and for all frequency bands. Since the operating deflected shapes below 3 Hz are a linear combination of the three rigid body modes, the solution is accurate even though the RSS of the X motion cross-orthogonality is near zero. The results are the same for the Y direction mass term as well as the in-plane rotational inertia term Izz.

Ratio: Calculated Mass (X) / Model Mass 1.10

1.05

Ratio

1.00

0.95

M(x): 21x 21

0.90

M(x): 11 x 11 Iyy: 5 x 5 'RSS X'

0.85 0

5

10

15

20

25

30

35

40

45

50

Frequency, Hz

Figure 10 – Ratio of calculated mass (X direction) and model mass with respect to center of gravity. Frequency signifies maximum frequency in analysis band. VERIFICATION OF OFF DIAGONAL TERMS

As a final check, the rigid body shapes,

[R ] , were calculated with respect to the origin at the corner of the plate qr

as shown in Figure 2. In this case, the analysis frequency band was 0.1 to 16 Hz, and the 11x11 sensor set was used. The calculated mass matrix is presented in Table 3 and the ratio between calculated values and values exported by the finite element model are presented in Table 4. These results are consistent with the answers when the rigid body deformations were assumed to be about the center of gravity. The in-plane mass terms are nearly perfect, and the out-of-plane mass terms are within one percent.

93 Table 3. Calculated mass matrix with respect to origin. Entries with absolute values less than 1E-5 set equal to zero.

0 0 0 0 − 29.27   0.585  0 0.585 0 0 0 29.27    0 0 0.589 29.43 − 29.50 0    0 29.45 1960 − 1474 0   0  0 0 − 29.50 − 1474 1966 0    0 0 0 3903  − 29.27 29.27

Table 4. Ratio between calculated rigid body mass matrix with respect to origin and model rigid body mass matrix with respect to the origin.

− − − 1.00  1.00 −  − 1.00 − − − 1.00    − − 1.01 1.01 1.01 −    − 1.01 1.01 1.01 −   −  − − 1.01 1.01 1.01 −    − − 1.00  1.00 1.00 − SUMMARY

A mathematical framework to calculate rigid body mass properties of flexible structures – structures in which rigid body modes are coupled with flexible body modes – has been presented and demonstrated on an analysis model of a plate structure. The key element is to account for the flexible motion of the structure by including flexible shapes in the coordinate transformation from physical (measured) coordinates to assumed shapes. The assumed shapes in this paper were based on the singular value decomposition of the FRF matrix, but there are certainly other estimates that could be employed Calculation of cross-orthogonality between rigid body deformation shapes and finite element mode shapes to identify target modes for sensor selection has been validated. Also, the use of pseudo-orthogonality to determine if a sensor set is sufficient to calculate rigid body mass properties has also been demonstrated. The benefit of using of constraint equations to force the rigid body mass matrix to take a particular form is still open to debate. Enforcement of these equations may not decrease overall error but may simply spread the error over the entire rigid body mass matrix.

REFERENCES

[1]

Toivola, J., and O. Nuutila, “Comparison of Three Methods for Determining Rigid Body Inertia Properties th from Frequency Response Functions,” Proceedings of the 11 International Modal Analysis Conference, Kissimmee, Florida, 1993.

[2]

Mangus, J., C. Passerello, C. VanKarsen, “Estimating Rigid Body Properties from Force Reaction th Measurements,” Proceedings of the 11 International Modal Analysis Conference, Kissimmee, Florida, 1993.

[3]

Stebbins, M. D. Brown, “Rigid Body Inertia Property Estimation Using a Six-Axis Load Cell,” Proceedings of the 16th international Modal Analysis Conference, Santa Barbara, California, 1998.

94 [4]

Schedlinski, Carsten, and Michael Link, “On the Identification of Rigid Body Properties of An Elastic th System,” Proceedings of the 15 International Modal Analysis Conference, Orlando, Florida, 1997.

[5]

Whitter, M., “Rigid Body Inertia Property Estimation Using the Dynamic Inertia Method,” Masters Thesis, Department of Mechanical Engineering of the College of Engineering, University of Cincinnati, 2000.

[6]

Lazor, D., “Considerations For Using The Dynamic Inertia Method In Estimating Rigid Body Inertia Property,” Masters Thesis, Department of Mechanical Engineering of the College of Engineering, University of Cincinnati, 2004.

[7]

Tuttle, R., T. Cole, and J. Lollock, “An Automated Method for Identification of Efficient Measurement th Degrees-of-Freedom For Mode Survey Testing,” 46 AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference, Austin, Texas, 2005.

[8]

Stabb, M., and P. Blelloch, “A Genetic Algorithm For Optimally Selecting Accelerometer Locations,” 13th International Modal Analysis Conference, Nashville, Tennessee, 1995.

Simulating Base-Shake Environmental Testing Jim Steedman NAVCON Engineering Network Fullerton, CA

ABSTRACT In most military aircraft and spacecraft applications, each payload structure must be pre-tested on a shake table to insure that it can withstand the vibration environment that it will experience during flight. Shaker testing is done using a control PSD which is designed to realistically represent the floor motion of the aircraft during takeoff, in flight, or during landing. Qualification testing is typically done by mounting the test article on one or more shakers, and exciting it with a closed loop shaker testing system so that the base of the payload responds with the pre-specified control PSD. When a test vehicle is too massive to be tested by mounting it on shakers, it is impossible to perform a base-shake test on a shake table. So the question arises; “Are there other more convenient driving points from which to excite the structure which will simulate a base-shake test?” In this approach, we derive a frequency domain Transmissibility model which is used to calculate PSDs for convenient driving points as functions of the base-shake PSDs. These calculated PSDs would then used to control a shaker test that simulates the base-shake test.

Brian Schwarz & Mark Richardson Vibrant Technology, Inc. Scotts Valley, CA

for the fixed base DOFs), a frequency domain model is derived that relates the responses of the moving DOFs to the fixed base responses. This Transmissibility matrix model is then used to calculate control PSDs for new driving points as functions of the base-shake control PSDs. Modal Test The base-shake simulation was done using a modal model of the vehicle sitting on its wheels in its tied down configuration, as shown in Figure 1. The modal model was used to synthesize elements of a Transmissibility model that relate base-shake responses to responses at other points on the structure. To develop a valid modal model, three different model tests were performed on the vehicle. From these three tests, a modal model was constructed that represented the dynamics of the vehicle vibrating on its wheels. The dominant modes of the model are its rigid body modes, i.e. the vehicle bouncing on its wheels. In addition, several of the lowest frequency elastic modes were also excited, and were including in the modal model.

The Transmissibility model is validated by using an inverse calculation to calculate base-shake PSDs as functions of the new driving point PSDs. Suitable driving points can then be chosen by comparing the calculated base-shake PSDs with the original pre-specified base-shake PSDs. INTRODUCTION When a test vehicle is too massive to be tested by mounting it on shakers, our assumption is that it can be tested by shaking it at other driving points which will have the same dynamic effects as shaking it from its base. To test at different driving points, new control PSDs must be calculated which will cause the test article to respond in a manner which is similar to its response during a base-shake test. In order to accomplish this, the dynamic properties between the base and the other driving points must be correctly modeled. To calculate new control PSDs, we start with a standard set of time domain equations of motion that model the dynamics of the structure. After partitioning the equations into two sets, (one for moving DOFs (degrees of freedom) and one

Figure 1 Test Vehicle in Base-Shake Configuration. Modal testing was done using an electro-dynamic linear stroke shaker, driven by a pure random signal. Twenty triaxial accelerometers were used to measure responses, and a load cell was used to measure the force input. FRFs (Frequency Response Functions) were calculated in the frequency span (0 to 50Hz), using 25 spectrum averages. A total of 60 FRFs were calculated from each of the three modal tests.

T. Proulx (ed.), Advanced Aerospace Applications, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 4, DOI 10.1007/978-1-4419-9302-1_8, © The Society for Experimental Mechanics, Inc. 2011

95

96

The modal model was then used to synthesize FRFs between each of the wheel hubs and other moving DOFs on the vehicle. In addition, the modal model was used to calculate driving point FRFs at the wheel hubs, which were then used to calculate the stiffness & damping of the tires. Tire stiffness & damping directly influence the response of the vehicle to the base-shake PSDs. BACKGROUND THEORY Time Domain Equations In all FEA (Finite Element Analysis) and modal testing work, it is assumed that Newton’s Second Law adequately describes the dynamic behavior of the mechanical structure. Hence, the linear, time-invariant dynamics of the vehicle can be represented with the following set of differential equations;

M  x( t ) C x ( t ) K  x ( t )  f ( t )

(1)

where:

M  = mass matrix (n by n) C = damping matrix (n by n) K  = stiffness matrix (n by n) x( t ) = acceleration (n vector) x ( t ) = velocity (n vector) x ( t ) = displacement (n vector) f ( t )= force (n vector)

(3)

where: u  {x}   F  u M 

{u F } = displacement of the fixed DOFs {u M } = displacement of moving DOFs {f F } = forces applied to fixed DOFs {f M } = forces applied to moving DOFs NOTE: The aircraft floor DOFs are referred to as the fixed base DOFs because all mode shape components are zero (or fixed) at the bottom on the wheels. This is unique to any base-shake problem.

 M FF M FM  s 2    C  FF C FM  s  K FF  M MF M MM  s 2    C  MF C MM  s  K MF

  K FM 

 uF     f F  u M 

  uF      f M  K MM  u M 

(4)

(5)

Assumption: No forces are applied at the moving DOFs.

Frequency Domain Equations Using Laplace transforms, the time domain equations of motion can be transformed into the frequency domain, and written as; 2

   u F   f F       K FM   u M  f M  K MM  

Separating the equations into two sets, one for the fixed base DOFs and the other for the moving DOFs;

n = DOFs (degrees of freedom) t = time variable

M  s  C s  K x(s)  f (s)

  M FF M FM  2  s    M MF M MM     C FF C FM  s   K FF  K  C  MF   MF C MM 

During a base-shake, forces are only applied at the fixed base DOFs. Using the moving DOFs equation and assuming that ( f M =0) gives:

M  s  C  s  K u   M  s  C  s  K  u   0 2

MF

(2)

MF

MF

F

2

MM

MM

MM

M

where:

Rearranging terms gives;

s = Laplace variable s    j = complex frequency x (s) = Laplace transform of the displacement (n vector) f (s) = Laplace transform of the force (n vector)

M  s  C  s  K  u   M  s  C  s  K  u 

NOTE: For convenience, the s-variable will be dropped from the displacement and force vectors in the following notation. Partitioning the matrices into the fixed base DOFs (subscript F) and the moving DOFs (subscript M), the equations become;

(6)

2

MM

MM

MM

M

2

MF

MF

MF

(7)

F

Solving for the moving DOFs gives;

u M   M MM  s 2  C MM  s  K MM 1 M MF  s 2  C MF  s  K MF  u F 

(8)

Assumption: There is no inertial coupling between the fixed base DOFs and moving DOFs. All of the base-shake forces are transmitted through the tires to the wheel hubs, and then to the rest of the structure.

97

Assuming that ( M MF   0 ), equation (7) becomes;

u M   M MM  s 2  C MM  s  K MM 1 C MF  s  K MF u F 

(9)

This equation expresses the moving DOFs as functions of the fixed base DOFs. FRFs for the Moving DOFs Using the moving DOFs equation (5) and setting ( {u F } = 0) gives;

M  s  C  s  K  0  M  s  C  s  K  u   f  2

MF

MF

MF

2

MM

MM

MM

M

(10)

M

or;

NOTE: The negative sign in front of the Transmissibility matrix is canceled by the negative signs in the off-diagonal terms of the damping C MF  and stiffness K MF  matrices. The diagonal terms, containing stiffness & damping relative to ground, are zero. Power Spectral Density The control PSD for the aircraft floor is specified as a Power Spectral Density (PSD). Multiplying the Transmissibility equation by the transposed conjugate of itself, gives a new equation in Power spectrum (or PSD) units;

u M u M t  Ts u F u F t Tst

u M   M MM  s 2  C MM  s  K MM 1 f M 

The term MMM  s2  CMM  s  K MM 1 in the above equation is simply the FRF matrix between the moving DOFs. Rewriting equation (9) in terms of the FRF matrix H MM s  for the moving DOFs;

u M   H MM s  C MF  s  K MF u F 

DOFs of the vehicle. The Stiffness & Damping matrix C MF  s  K MF  contains the stiffness & damping of the tires, between the fixed base (aircraft floor) and the wheel hubs.

(11)

Equation (11) expresses motions of the moving DOFs as functions of the motions of the fixed base DOFs. Stiffness & Damping Matrix The term C MF  s  K MF  is the stiffness & damping matrix between the fixed base DOFs (aircraft floor) and the moving DOFs. Meaningful (non-zero) stiffness & damping values only exist between the fixed base and the wheel hub DOFs. Therefore, the motion of all of the moving DOFs other than the wheel hubs depends on two dynamic properties;

1) FRFs between the wheel hubs and other moving DOFs. 2) Stiffness & damping of the tires, between the fixed base DOFs (aircraft floor) and the wheel hub DOFs.

(13)

where:

u F u F t = base-shake PSD matrix u M u M t = new control PSD matrix t – denotes the transposed conjugate This equation expresses the new control PSD matrix as a function of the base-shake PSD matrix. NOTE: The diagonal elements of these matrices contain Auto PSDs, and the off-diagonal elements contain Cross PSDs.

It is assumed that each wheel is subjected to independent random vibration. Therefore, the same base-shake PSD will be used for all diagonal elements and all off-diagonal elements are set to zero. Inverse Calculation The accuracy of the new control PSDs can be checked by performing an inverse calculation, i.e. calculating the baseshake PSD matrix as a function of the new control PSD t matrix. Pre-multiplying the previous equation by Ts 

and post-multiplying it by T s  gives;

Transmissibility Matrix The product of these two matrices (FRFs and Stiffness & Damping) is a unit-less Transmissibility matrix. The FRF matrix has units of (displacement/force) and the Stiffness & Damping matrix has units of (force/displacement), so their product is unit-less. Equation (11) can be re-written as;

Solving for the base-shake PSD matrix u F u F t gives;

u M   T s  u F 

where:

(12)

where:

Tst u M u M t Ts  Tst Ts u F u F t Tst Ts

u F u F t  As u M u M t Ast

(14)

A(s)  Tst Ts Tst 1

T s   H MM s C MF  s  K MF 

This equation expresses the base-shake PSD matrix as a function of the new control PSD matrix.

In summary, the FRF matrix H MM s  contains the dynamic properties between the wheel hubs and other moving

Clearly, there are cases where testing a vehicle using certain new control PSDs will not simulate the intended base-shake excitation. It depends on the dynamic properties represented

98

by the modal model at the chosen moving DOFs of the structure. For example, shaking at any moving DOF where one or more mode shapes are at or near a nodal point (zero motion) will not excite all of the modes, and therefore may not accurately simulate a base-shake test. The inverse calculation should help locate suitable driving points which will closely simulate a base-shake test. BASE-SHAKE PSDs For testing payloads in military aircraft, a base-shake PSD is pre-specified for the aircraft floor. It is assumed that the PSD is applied at the base of each wheel in three directions (X, Y, & Z). Therefore, the base-shake PSD matrix is a (12 by 12) diagonal matrix. Base-shake PSDs for the C-5 and C-17 aircraft are defined in Figure 2. TIRE STIFFNESS & DAMPNG The Stiffness & Damping matrix C MF  s  K MF  contains stiffness and damping values between the floor (bottom of each tire) and each wheel hub. This (12 by 12) off-diagonal matrix contains different values for each wheel and each direction. Each stiffness and damping element is obtained from the driving point FRF at each wheel hub, which is synthesized from the modal model.

Figure 2B C-17 Aircraft PSD

 Each element of the Stiffness matrix K MF  is obtained as the inverse of the flexibility line near DC of the driving point FRF. Examples are shown in Figure 3.  The elements of the Damping matrix C MF  are cal-

culated with the following formula; C = Mass x (damping decay constant from the IRF)

Figure 3 Flexibility Line at Low Frequency (in/lbf = 1/Stiffness)

Figure 2A C-5 Aircraft PSD

The damping decay constant is the slope of the envelope of the log magnitude of the driving point IRF (Impulse Response Function), the inverse FFT of each driving point FRF. The damping decay constant is obtained by curve fitting a straight line to the driving point IRF. Examples of logarithmic decay envelopes are shown in Figure 4. Mass is obtained from the mass line (at high frequency) of the driving point accelerance (acceleration/force) FRF in each direction at each wheel hub. The driving point accelerance is obtained by differentiating the synthesized (dis-

99

placement/force) FRF twice. Examples of mass lines are shown in Figure 5.

Figure 6 Stiffness & Damping matrix. TRANSMISSIBILITY MATRIX The Transmissibility matrix is the product of the moving DOFs FRF matrix and the Stiffness & Damping matrix. It is clear from this product that the Stiffness & Damping matrix scales each element of the moving DOFs FRF matrix to reflect the stiffness and damping of each tire in a direction.

Figure 4 Logarithmic Decrement of Impulse Responses.

If the tires had no stiffness or damping in them (a hypothetical case), then base-shaking the vehicle would cause no motion of the moving DOFs. On the other hand, if the tires were very stiff or had lots of damping, then base-shaking the vehicle would result in large motions of the moving DOFs. The moving DOFs FRF matrix need only be synthesized between the moving DOFs which might be used as the new driving points, and the DOFs of the wheel hubs. Rather than use all 60 moving DOFs (the number of components in the experimental mode shapes), a smaller set of 9 DOFs was considered. Figure 7 contains red arrows indicating the 9 candidate driving point DOFs. Figure 8 shows the Transmissibility’s for one moving DOF (-10Y) due to the 4 base-shake PSDs at each wheel in the Zdirection. The entire Transmissibility matrix contains 108 terms, nine rows for the moving DOFs and twelve columns for the fixed base DOFs (three directions at each wheel). NEW CONTROL PSD MATRIX A new control PSD matrix is calculated using the baseshake PSD matrix and the Transmissibility matrix in equation (13). The base-shake PSD matrix u F u F t is a (12 by

Figure 5 Accelerance Mass Line (in/sec^2-lbf = 1/mass).

Figure 6 shows all 12 elements of the Stiffness and Damping matrix in overlaid format. These 12 off diagonal terms (indicated by the DOFs in the spreadsheet), contain both a Stiffness (real part) and the Damping multiplied by frequency (imaginary part). This matrix is multiplied by the moving DOFs FRF matrix to obtain the Transmissibility matrix.

12) diagonal matrix. The Transmissibility matrix T s  is a (9 by 12) full matrix. The resulting matrix of new control PSDs is a (9 by 9) matrix. Elements of this matrix for only four moving DOFs due to base-shake PSDs in the Z direction are shown in Figure 9. The new driving Point DOFs are -17Z, 41Z, -45Z, and 55Z. The base-shake PSDs for DOFs 150Z, 190Z, 350Z, and 390Z were used to calculate the new control PSDs.

100

CONCLUSIONS The question that was addressed in this paper is; “Is it possible to shake an aircraft payload from other more convenient driving points using control PSDs that simulate a base shake test?” To simulate a base-shake, a unique Transmissibility model was derived to relate the base-shake PSDs to the new control PSDs required to simulate the baseshake. An experimentally derived modal model of the vehicle on its fixed base was used to synthesize the Transmissibility’s. The modal model was developed from three different shaker tests of the vehicle, using pure random excitation signal and 60 tri-axial accelerometers. The resulting FRFs were curve fit to obtain experimental mode shapes.

Figure 7 Convenient Driving Points.

The modal model was then used to synthesize FRFs between the wheel hubs and all other moving DOFs on the vehicle. By measuring motions at each wheel hub, all of the vehicle suspension dynamics were included in the modal model. The modal model was also used to synthesize driving point FRFs at the wheel hubs, from which the stiffness & damping of the vehicle tires were obtained. Then, Transmissibility’s were calculated as the product of the tire stiffness & damping times the FRFs between the wheel hubs and the other moving DOFs, Finally, the new control PSDs were calculated using the Transmissibility’s and the base-shake PSDs. The new control PSDs were validated using a Round Trip calculation as follows; Base-Shake PSDs > Transmissibility Model > Control PSDs Control PSDs > Inverse Model > Simulated Base-Shake PSDs

The round trip was then used to show that four convenient driving Points on the vehicle simulated the base-shake quite accurately, for the Z direction. If a real vehicle behaves in a linear manner, so that its modal model adequately represents its dynamics, then the Round Trip calculation can be used to prove that a base-shake simulation is indeed a valid test. Of course, conclusive proof can only be obtained by performing a true base-shake on a vehicle and comparing the base-shake and simulated baseshake results. Figure 8 Transmissibility’s for moving DOF (-10Y) SIMULATED BASE SHAKE The validity of the new control PSDs is confirmed by using the Inverse equation (14) to calculate base-shake PSDs from the new control PSDs. Figure 10 shows the base-shake PSDs which were calculated from the (4 by 4) control PSD matrix for DOFs -17Z, 41Z, 45Z, and 55 Z. For this case, it is clear that the (4 by 4) matrix of base-shake PSDs is reproduced with over three decades of accuracy.

REFERENCES 1. Zhuge, J., Formenti, D., Richardson, M. “A Brief History of Modern Digital Shaker Controllers” Sound & Vibration magazine, September, 2010.

101

Figure 9 C-17 Control PSDs for 4 Z-direction Driving Points.

Figure 10 C-17 Simulated Base-Shake PSDs.

Geometry-Based Updating of 3D Solid Finite Element Models T. Lauwagie, E. Dascotte Dynamic Design Solutions, Interleuvenlaan 64, B-3001, Leuven Belgium

Abstract Structural responses obtained with finite element (FE) simulations normally differ from those measured on physical prototypes. In the case of monolithic structures, the differences between the simulated and measured responses are mainly caused by inaccuracies in the geometry and material modeling. Such inaccuracies may result from the manufacturing process. The presented work illustrates how the geometry of CAD-based FE-models can be updated using a high-fidelity representation of the actual manufactured geometry, to improve the correlation between measured and computed resonant frequencies and mode shapes. The study presented in this paper was performed on a cast iron lantern housing of a gear box. In a first step, the resonant frequencies and modes shapes of the test structure were measured using impact testing. Next, a set of digital pictures were taken from a number of different angles. By means of photogrammetry, these pictures were converted into a surface model that represented the actual geometry of the lantern housing. This surface model was then compared with an FE-model derived from a CAD-model of the lantern housing. In this way, the regions where there was a substantial difference between the actual geometry and CAD-model could be identified. Finally, the geometry of the FE-model was corrected based on the measured geometry using a mesh morphing technique. For the considered test case, the correction of the geometry provided a significant improvement of the quality of FEM-test correlation of the modal parameters. 1. Introduction Structural responses obtained with finite element simulations normally differ from those measured on physical prototypes. The observed differences are mainly caused by inaccuracies in the geometry, material behavior and joint properties of the simulation model. Finite element model updating [1] is a commonly accepted technique to improve the validity of simulation models. By tuning physical element properties, model updating aims at reducing the differences between the measured and simulated responses as much as possible. However, in the case of 3D solid elements, the geometry of the model is not controlled by element properties, like a shell thickness in the case of 2D elements, but by the positions of the nodes of the elements. Direct updating of the individual nodal positions would lead to an excessive amount of independent updating variables and is therefore practically unfeasible. With 3D elements, the geometrical uncertainties are usually compensated in an indirect way. For example, an overestimation of a thickness has to be compensated by a reduction of the stiffness and mass density of the material in the considered area. Although such compensations can eventually provide models that correlate well with the test data set, the improvement in reliability of the model is limited as the modifications are physically not correct, which restricts the application range of the updated finite element model. The goal of the present work was to investigate the impact of the geometrical inaccuracies on the correlation between numerical and experimental resonant frequencies and mode shapes, and to verify if the correlation can be improved by updating the geometry of the FE-model using mesh morphing techniques [2-3]. To simplify matters, a monolithic cast-iron lantern housing was used. In this way the impact of any joint uncertainties was eliminated. A high-fidelity representation of the geometry was obtained by a combination of optical scanning and photogammetry. The point cloud that resulted from these optical measurements served as a starting point to generate a 3D solid finite element model representing the as-built geometry of the housing. To evaluate the impact of using the actual geometry on the correlation and model updating results, two FE-models were used: an FE-model derived from the measured geometry and an FE-model derived from the CAD model of the lantern housing. Both models had a similar mesh density and mesh quality. These two models were first correlated with the measured modal data and then updated by tuning the material properties. Finally, the potential benefits of geometrybased updating were evaluated by correcting the geometry of the CAD-based FE-model using the measured geometry in combination with mesh morphing techniques.

T. Proulx (ed.), Advanced Aerospace Applications, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 4, DOI 10.1007/978-1-4419-9302-1_9, © The Society for Experimental Mechanics, Inc. 2011

103

104 2. Measuring the Geometry To obtain an accurate geometrical description of the lantern housing, optical scanning and photogrammetric techniques were used to acquire a point cloud of the part as-built. Optical scanning was performed using a GOM ATOS I scanner, shown in Fig. 1.

Fig. 1 GOM ATOS I scanner which was used to digitize the lantern housing. The digitizing principle is based on a white light fringe pattern from a projector (center part on Fig. 1) onto the scanned object. Two cameras (left and right on Fig. 1) capture images of the object and reference geometries. Since the lantern housing is too large to fit the scanning volume of the scanner, photogrammetric techniques were applied to combine the partial scans. Photogrammetry uses photographical images of multiple marker systems to merge optical scans of different regions of the lantern housing. Fig. 2 shows pictures of the lantern housing during the scanning process. A white spray is applied to reduce reflections. Small marker stickers are used for local optical scanning. Large stickers and two reference bars on the ground are used for the merging software.

Fig. 2 Images showing the lantern housing during the digitizing process. Triangulation techniques are applied to reconstruct a point cloud from these images based on the distance and angle between the cameras, the projected grid information and the photogrammetry pictures. Using the GOM ATOS Professional software [4] the point cloud data is converted into the Standard Triangulation Language (STL) model shown in Fig. 3. An STL model is a triangle facet surface mesh representation based on the scanned point cloud. In case of the lantern housing, the minimal geometrical accuracy of this STL model, taking into account scanning and point cloud post-processing, is approximately 0.2 mm. However, the STL model resulting from the digitization process is not suitable to generate an FE mesh because it is not watertight. Therefore, the model was corrected with the STL fixing, design & meshing software package 3-matic [5].

Fig. 3 Detail views of typical scan surface mesh imperfections: non-watertight edges (left) and incomplete hole or slot information (left and right).

105 The defects along the edges were fixed using automated hole filling and stitching algorithms. For the virtual fixing of slots and holes CAD information was locally copied into the scan to complete the missing geometry. Once the STL model was watertight, the enclosed volume was meshed using 54040 10-noded tetrahedron elements. A section of the final volume mesh is shown in Fig. 4.

Fig. 4 Resulting 10-noded tetrahedron mesh 3. Evaluation of the Impact of the Measured Geometry The impact of using high-fidelity geometries was evaluated by comparing the reliability of the responses of the CAD-based model with those of the geometry-based model. Both FE-models had a similar number of elements ( 55000) to exclude the influence of the mesh density. 3.1. Modal Testing A standard experimental modal analysis was performed to measure the resonant frequencies and mode shapes up to 1.5 kHz. The test structure was suspended with a number of elastic bands to simulate free-free boundary conditions. The frequency response functions were measured using a roving hammer test and using 6 tri-axial accelerometers. The input and response signal were measured up to 2 kHz using 2048 spectral lines. The average of 5 individual FRFs measurements was used as measured FRF. The first 18 modes of the lantern housing could be extracted from the measured FRFs. Fig. 5 shows the AutoMAC of the measured mode shapes.

Fig. 5 The AutoMAC of the measured mode shapes 3.2 Initial Correlation 3.2.1 The CAD-Based FE-Model The correlation analysis between the results of the CAD-based FE-model and the test provided 18 mode shape pairs. The mode shape order in the two data sets was identical. The correlation results showed that the FE-model underestimated all the frequency about 6.9 %. The MAC values ranged between 40.1 and 97.7, with an average value of 82.0. 4.2.2 The Geometry-Based FE-Model The correlation analysis between the results of the geometry-based FE-model and the test also provided 18 mode shape pairs. As for the CAD-based model, the order of the FE-modes corresponded with the order of the test modes. The correlation

106 results showed that the geometry based FE-model underestimated all the resonance frequency about 8.6 %. The MAC values ranged between 88.8 and 97.8, with an average value of 83.8. 3.3 Updating of the Material Properties Both FE-models were updated using the FEMtools Error! Reference source not found. model updating module. The updating procedure that was used consisted of two separate steps. In the first step the overall mass of the FE-model was set to the mass value of the test structure by modifying the mass density of the material in the model. In the second step the stiffness of the model was updated by modifying the Young’s modulus of material. The Young’s modulus was defined as a global parameter, i.e. the Young’s modulus remained uniform over the whole FE-model during updating. The 18 measured resonant frequencies were used as targets for the updating procedure. The mode shape data was only used for mode tracking purposes, not as target values for the updating. 3.3.1 The CAD-Based FE-Model The CAD-based FE-model underestimated the mass by 3 kg, i.e. 108.0 kg instead of 111.0 kg. To increase the mass by 3 kg, the mass density of the material had to be raised from 7100 to 7295 kg/m 3. The mass correction resulted in a drop of the overall frequency correlation with 1.2 %, i.e. -8.1 % instead of -6.9%. The stiffness updating of the FE-model increased the Young’s modulus from 110.5 GPa to 130.7 GPa. This resulted in frequency residuals ranging between -1.72 % and 3.64. Note that a global updating of the mass and stiffness does not have a significant effect on the mode shapes. Hence, the updating did not provide any improvement in the mode shape correlation. 4.3.2 The Geometry-Based FE-Model The geometry-based FE-model had an overall mass of 105.0 kg instead of the 111.0 kg of the test model. This required an increase of the mass density of the material from 7100 kg/m3 to 7503 kg/m3. This increase resulted in an overall frequency drop of 2.5 %; increasing the underestimation of the resonant frequencies by the FE-model. The second step of the updating procedure increased the Young’s modulus from 110.5 GPa to 139.9 GPa resulting in frequency residuals ranging between -0.29 % and 0.24 %. 4.3.3 Comparison Table 1 provides an overview between the correlation results of the updated CAD-based FE-model and the updated geometry-based FE-model with the measured shapes and frequencies. The frequency match of the updated geometry-based FE-model is significantly better than the frequency match of the CAD-based FE-model. However, the most remarkable difference between the two models is their correlation with the test modes. While the CAD-based FE-model is missing a few mode shape pairs, the geometry based FE-model has an excellent correlation for all 18 considered modes. More details and background information on this correlation analysis can be found in [7]. CAD-based FE-model

Geometry-based FE-model

Average |freq. residual|

1.04%

0.14%

Min. |freq. residual|

0.01%

0.00%

Max. |freq. residual|

3.64%

0.29%

Frequency residuals

…

107 MAC FEM-EMA

Average MAC

82.0

94.3

Minimum MAC

40.1

88.9

Maximum MAC

97.7

98.2

Updated mass density

3

7295 kg/m

7503 kg/m3

Updating Young’s modulus

130.7 GPa

139.9 GPa

Table 1 Comparison of the CAD-based and geometry-based FE-model 4. Geometry-Based Updating The correlation analysis revealed that the errors in the CAD geometry have a significant impact on the accuracy of the FEmodel. A reliable FE-model thus requires a better approximation of the geometry. However, instead of using the scan data to generate a geometry-based FE-model, one could also use the measured geometry to update the geometry of the CAD-based FE-model. In the case of 1D of 2D elements, this could be done by simply correcting the geometrical properties of the elements. However, in the case of 3D elements this will require the repositioning the nodes of the FE-model. 4.1 Comparison of the CAD and Measured Geometry Fig. 6 presents the differences between the measured and CAD geometry; red indicates that the surface geometry-based model is below the CAD model, blue indicates that the geometry-based model is above the CAD model and green indicates there is no difference. The color maps reveal that the difference between the two models is of a systematic nature; the major difference between the two models seems to be a tilt of the top and bottom surfaces and an incorrect diameter for the cylindrical bulging in the center of the bottom surface. The tilt of the planes is most likely a result of warping and/or shrinkage during the casting process.

Fig. 6 The difference between the measured and CAD geometry. 4.1 Lattice-Based Mesh Morphing The observed inaccuracies in the CAD geometry could, for example, be eliminated by tilting the surfaces of the FE-model using mesh morphing techniques. The latticed-based mesh morphing technique [2-3] starts by defining a number of hexahedral lattice cells that envelop a part of the FE-mesh, i.e. the mesh associated to the lattice cell. When one of the vertices of the lattice cell is moved, all the nodes of the associated mesh will move as well. The new position of the nodes is determined using a Bezier interpolation between the lattice vertices. In this way an FE-mesh can be deformed using a limited number of control points.

108 4.2.2 Application to the Lantern Housing Six mesh deformations are considered to morph the CAD-based model and bring it closer to the actual geometry of the structure. Fig. 7 gives on overview of the considered mesh deformations: the first three tilt the surfaces at the bottom, the next two tilt the surfaces at the top, and the last one changes the diameter of the bulging at the bottom.

Fig. 7 The six considered mesh deformations The mesh morphing was executed in FEMtools [6] as follows. In a first step three points were selected on the bottom surface in order to simulate the case were the geometry is only measured in a discrete number of points, instead of a full scan. The geometry measurements in those three points were used to compute the required tilt to align the CAD surface with the measured surface. Next the mesh was morphed by tilting the lattice box using the estimated tilt. Fig. 8 shows the result of the first mesh deformation. It confirms that the difference between the CAD and measured geometry on the bottom surface has been removed. The other mesh deformations were applied in a similar way.

Fig. 8 The differences before (left) and after (right) the first mesh deformation 4.3.2 Effect on the Modal Correlation The effect of the various mesh deformations on the modal correlation is presented in Fig. 9. On this plot, the frequency and mode shape differences between the original CAD and geometry-based model are taken as a reference, i.e. 100%. Both models use the material properties of the updated geometry-based FE-model, i.e. E = 139.9 GPa,  = 7503 kg/m3. The first mesh deformation removed about 20% of the frequency differences and about 35% of the mode shape difference. The impact of the first mesh deformation on the MAC matrix, which now has all the diagonal terms, is significant. Combined, the six considered mesh deformations remove about 80% of the frequency and mode shape differences. This implies that the difference between these two models is due to a limited number of systematic errors. Differences resulting from approximations/simplifications made during the model generation only seem to play a secondary role. These results also confirm that updating the geometry of a CAD-based FE-model using a limited number of discrete geometry measurements is a feasible approach to increase the reliability of the FE-model.

109

Fig. 9 The impact of the geometry modifications on the modal correlation 5. Conclusions The presented work evaluates the impact of using a high-fidelity representation of the geometry of a monolithic cast-iron structure on the correlation between measured and simulated responses. The use of the as-built geometry of the structure, obtained by optical scanning of a prototype, provides a significant improvement in the correlation between the measured and simulated mode shapes in a wide frequency range. It was shown that only a limited number of geometry measurements are needed to update the CAD-based geometry using mesh morphing techniques. With geometry updating it is possible to eliminate most of the uncertainty on the geometry. As such, geometry updating eliminates, or at least reduces, the need for equivalent parameter changes to compensate the effects of geometrical inaccuracies. As the updating process provides parameter changes that are physically more relevant, the application range in which the updated FE-model can be used as a reliable predictive tool for design optimization can be increased. Acknowledgements We would like to express our gratitude to Hansen Transmissions N.V. for providing the test data and FE-models of the lantern housing, and to the KULeuven for providing the modal test data. References [1] E. Dascotte, Model Updating for Structural Dynamics: Past, Present and Future Outlook. In proceedings of the International Conference on Engineering Dynamics (ICED), April 16-18, 2007, Carvoeiro, Algarve, Portugal.

[2] T.W. Sederberg, S.R. Parry, Free-Form Deformation of Solid Geometric Models, SIGGRAPH 1986, Vol. 20 No.4. [3] S. Coquillart, Extended Free-Form Deformation: A Sculpturing Tools for 3D Geometric Modeling, SIGGRAPH 1990, [4] [5] [6] [7]

Vol. 24, No.4. GOM ATOS Professional Software Manual, 2010 Materialise, 3-matic 5.0 manual, 2010 FEMtools, www.femtools.com

T. Lauwagie, F. Van Hollebeke, B. Pluymers, R. Zegels, P. Verschueren, E. Dascotte, The Impact of High-Fidelity Model Geometry on Test-Analysis Correlation and FE Model Updating Results. In proceedings of the International Seminar on Modal Analysis 2010 (ISMA), September 20-22, 2010, Leuven, Belgium.

A PZT-Based Technique for SHM Using the Coherence Function

Jozue Vieira Filho¹, Fabricio Guimarães Baptista¹ and Daniel J. Inman² ¹UNESP – Sao Paulo State University Department of Electrical Engineering Ilha Solteira, Sao Paulo, Brazil ²Virginia Tech Center for Intelligent Material Systems and Structures Department of Mechanical Engineering Blacksburg, Virginia, USA ABSTRACT

This paper presents a new approach for damage detection in structural health monitoring systems exploiting the coherence function between the signals from PZT (Lead Zirconate Titanate) transducers bonded to a host structure. The physical configuration of this new approach is similar to the configuration used in Lamb wave based methods, but the analysis and operation are different. A PZT excited by a signal with a wide frequency range acts as an actuator and others PZTs are used as sensors to receive the signal. The coherences between the signals from the PZT sensors are obtained and the standard deviation for each coherence function is computed. It is demonstrated through experimental results that the standard deviation of the coherence between the signals from the PZTs in healthy and damaged conditions is a very sensitive metric index to detect damage. Tests were carried out on an aluminum plate and the results show that the proposed methodology could be an excellent approach for structural health monitoring (SHM) applications. Keywords: Structural Health Monitoring, PWAS, Piezoelectric Transducers, PZT.

Introduction Structural health monitoring (SHM) systems have become an important way to increase the safety and reduce maintenance costs in a variety of structures, such as civil infrastructure [1] and aircraft [2]. These systems are able to identify structural damage at an early stage using, for example, acoustic emission, comparative vacuum, Lamb wave and electromechanical impedance (EMI) methods. Among the various techniques, the EMI [3, 4] and Lamb wave [5, 6] are notable by using small, lightweight and inexpensive piezoelectric wafer active sensors (PWAS) bonded to the structure to be monitored, such as PZT (P-Lead Zirconate Titanate) piezoceramics. These devices exhibit the piezoelectric effect, which is the property that the material has to convert mechanical energy into electrical energy (direct effect) and electrical energy into mechanical energy (reverse effect). Therefore, it is possible to identify structural damage by analyzing the signals acquired from PZT transducers attached to the structure of interest and excited through an appropriate frequency range. As in both EMI and Lamb wave techniques, this new approach requires that data be previously collected when the structure is in a condition considered healthy and this data are then used as a reference, commonly known as the baseline. The baseline must be continuously compared with updated data provide by transducers in order to determine if damage has occurred. Some drawbacks for both EMI and Lamb wave based methods are due to excitation and analyzed frequency range, which have to be defined according to the monitored structure. Also, when applied in some structures such as plates, the index obtained in healthy condition is not far from the one obtained in damage condition [7], which becomes difficult to determine if damage has occurred. In the proposed methodology using the magnitude square of the complex coherence (MSC) function, a wide frequency range can be used and the index obtained in damaged condition is much higher than the index obtained in healthy condition.

T. Proulx (ed.), Advanced Aerospace Applications, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 4, DOI 10.1007/978-1-4419-9302-1_10, © The Society for Experimental Mechanics, Inc. 2011

111

112

Proposed Methodology The basic concept of the proposed methodology is illustrated in Figure 1. The three piezoelectric transducers are bonded to a plate, which is the structure to be monitored. One of the transducers operates as an actuator, which excites the structure at an appropriate frequency range. The other transducers operate passively as sensors. At first glance the proposed method seems like a lamb wave based system. However, it does not use the same principle. In fact, this configuration is similar to the active vibration based method, but the vibration is generated by PZT actuator in a wide frequency band.

Fig.1 Proposed methodology for damage detection based on coherence-function The excitation signal x used to excite the actuator and the acquisition of the response signals y1 and y2 from the sensors are performed using an adapted version of the measurement system developed in [8]. The system uses a multifunctional data acquisition (DAQ) device model USB-6259 which is controlled by LabVIEW® software, both from National Instruments. The number of sensors depends on the DAQ capacity and generically it can be k sensors. A PC microcomputer run LabVIEW® software that controls the DAQ device and provides the discrete form x[n] and yk [n] of the excitation and response signals, respectively, where k is the selected sensor and n is the sample. For the correct application of this methodology, the D/A (digital to analog) and A/D (analog to digital) converters, which are integrated to the DAQ device, must be synchronized. Therefore, each sample n is synchronously generated and sampled in the excitation and response signals. This is an important difference between the proposed method and lamb wave based method because the sensor response is obtained synchronously while the PZT actuator is been excited. The coherence function is computed between pairs of two different sensors k, but it can be computed as well between the excitation signal x[ n] and the response signal y[n] from each sensor k. The magnitude square of the complex coherence (MSC) between a generic signal x(t) and y(t) is based on the power spectra of both signals [9] and it is given by: Υ𝑥𝑥𝑥𝑥 [𝑓𝑓] =

�Φ𝑥𝑥𝑥𝑥 [𝑓𝑓]�

2

Φ𝑥𝑥𝑥𝑥 [𝑓𝑓].Φí€µí±¦í€µí±¦í€µí±¦í€µí±¦ [𝑓𝑓]

(1)

In equation (1), Υ𝑥𝑥𝑥𝑥 [𝑓𝑓] is the MSC of the complex coherence function, Φ𝑥𝑥𝑥𝑥 [𝑓𝑓] and Φí€µí±¦í€µí±¦í€µí±¦í€µí±¦ [𝑓𝑓] are the auto power spectrum of x(t) and y(t), respectively, Φ𝑥𝑥𝑥𝑥 [𝑓𝑓] is the crossed power spectrum and f represents the frequency. Common methods to compute the power spectra are based on periodograms [10], but considering x(n) and y(n) as sampled version of x(t) and y(t), respectively, they can be represented through fast Fourier transform (FFT) as follow: Φ𝑥𝑥𝑥𝑥 [𝑓𝑓] = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐(𝐹𝐹𝐹𝐹𝐹𝐹[𝑥𝑥(í€µí±łí€µí±ł)]) . 𝐹𝐹𝐹𝐹𝐹𝐹[í€µí±¦í€µí±¦(í€µí±łí€µí±ł)]

(2)

Φí€µí±¦í€µí±¦í€µí±¦í€µí±¦ [𝑓𝑓] = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐(𝐹𝐹𝐹𝐹𝐹𝐹[í€µí±¦í€µí±¦(í€µí±łí€µí±ł)]) . 𝐹𝐹𝐹𝐹𝐹𝐹[í€µí±¦í€µí±¦(í€µí±łí€µí±ł)]

(4)

Φ𝑥𝑥𝑥𝑥 [𝑓𝑓] = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐(𝐹𝐹𝐹𝐹𝐹𝐹[𝑥𝑥(í€µí±łí€µí±ł)]) . 𝐹𝐹𝐹𝐹𝐹𝐹[𝑥𝑥(í€µí±łí€µí±ł)]

(3)

113

In fact, the discrete signals x(n) and y(n) are fragmented in N segments to compute the power spectra. Then, the equations (2), (3) and (4) are computed for each segment to compose an average. Also, it is possible to use a large quantity of measurements to compute equations (2), (3) and (4) and average them. A well know method used to compute the power spectra based on the FFT was proposed by Welch [11] and it is used in this work. Equation (1) provides values between 0 and 1. Values close to 1 indicate that the signals are coherent, i.e., the signals have powers distributed in the same frequency range. However, in this work the coherence is used as a signature of the structure in a healthy condition. According to Figure 1, for example, the MSC between sensor 1 and sensor 2 in healthy condition will be different of the MSC in a damage condition and the goal of this work is to detect this variation with accuracy. As in the electromechanical impedance based methods, damage changes the power spectra of the signals received in each sensor and by consequence the MSC. This variation can be detected by using common metrics proposed in the literature such as the root mean square deviation (RMSD) and the correlation coefficient deviation mean (CCDM) indexes. Nevertheless, the MSC is itself related to the correlation function [10] and it is important to take this into account. Therefore, it is expected that the difference between the same MSC in health and damaged condition should bring to good results. As mentioned before, the coherence based method can be applied using a wide frequency band to excite the PZT actuator. So, in this work it was used a chirp signal to excite the actuator. The chirp signal makes a sweep from an initial low frequency to a final high frequency, or vice-versa. The sequence x[n] (n = 0, 1,..., N-1) of a chirp signal is given by:

 2π  FS 

x[ n] = A ⋅ sin 

 ( f2 − f1 )  n + f1    2N   

n

(6)

In equation (6) f1 and f2 represent the initial and final frequency values, respectively, A is the amplitude, n is the sample, N is the number of samples and FS is the sampling rate. The sample rate is determined by the frequency band of the excitation signal and the DAQ device as well. In this work, f1 = 0 kHz, f2 = 125 kHz, A = 5 V, FS = 250 kS/s, and N = 65536. The time step of the analog signal is given by 1/ FS and the frequency resolution of the spectrum and coherence function is given by FS/N. The values adopted in this work are enough to analyze changes in the coherence function and thus detect structural damage. Experimental Setup To verify the proposed methodology, tests were carried out on an aluminum plate with 500 x 230 x 2 mm. Four piezoceramic wafers PSI-5H4E of 20 x 20 x 0.267 mm from Piezo Systems were bonded to the specimen using “super glue”. One of the transducers was used as actuator (PZT X) and the three other ones were used as sensors (PZT 1, 2 and 3). Four removable damages (A, B, C, and D) were simulated bonding a steel nut of 4 x 2 mm and about 1 g to the structure at different distances from the actuator and the sensors. This procedure is illustrated in Figure 2.

Fig.2 PZT’s configuration and damages position A, B, C, and D in the aluminum plate

114 For each healthy and damaged condition, the coherence functions between the signals from the actuator and each sensor and between the sensors were calculated making an average of 20 measures. All measurements were taken with the specimen in free-free configuration, i.e. with the two sides suspended by elastic and at a room temperature. Results The coherence functions between the excitation signal (from PZT X in Figure 2) and the response signals (from PZT 1, 2 and 3) and between all response signals (from PZT 1,2; PZT 1,3; PZT 2,3) can be computed using the equation (1) for the host structure in the healthy condition and with damages A, B, C and D, respectively. A first step was dedicated to evaluate some MSC in healthy and damage conditions in different frequency bands. As an example of result, Figure 3 shows MSC between PZT X (actuator) and PZT 2 (sensor) and Figure 4 shows the MSC between sensor 1 and sensor 2, both in healthy and damaged (damage A) conditions in a frequency band from 50 kHz to 75 kHz. As it can be verified, both MSC changes significantly when damage occurs.

Fig.3 MSC in healthy and damage condition (between actuator X and sensor 2)

Fig.4 MSC in healthy and damage condition (between sensors 1 and 2)

115 Based on the results shown in Figure 3 and Figure 4, it was verified that the only difference between the MSC in healthy and damaged conditions is its spectral distribution. The mean of MSC in healthy and damage condition are similar. So, in order to get more information, a histogram of the difference between the MSC in healthy and damaged conditions was computed. Figure (5) shows the histogram using data shown in Figure (3) e (4) in the whole band. The histogram for healthy condition was obtained considering two measurements in healthy conditions at different times.

Fig.5 Histogram for the difference between MSC According to Figure 5, if the MSC in healthy condition is considered as a baseline, the healthy and damaged conditions can be easily indentified. This test was repeated for other MSC and similar results were obtained. However, it is necessary to choose a metric to detect damage. The classical RMSD and CCDM were tested and they don’t work correctly. The histograms presented in Figures (5) shows a different statistical between the results on healthy and damage conditions. After some tests, it was verified that the standard deviation can be useful in detecting damage and the following metric is proposed: Mstd= std(MSCm - MSCb)

(7)

In equation (7), Mstd is the metric to quantify damage, MSCb represents the baseline and MSCm is the magnitude of the coherence function measured after the baseline is defined. To present the results for all MSC, the standard deviation (Mstd) was computed for the different sensors. Figure 6 shows the results for the coherence between sensors 1, 2 and 3 and Figure 7 shows the results for the coherence between the actuator (PZT X) and sensors 1, 2 and 3.

116

Fig.6 Mstd (Std Index) between different pairs of sensors

Fig.7 Mstd (Std Index) between the actuator (Sensor X) and sensors 1, 2 and 3 According to Figure 3 and Figure 4, the MSC have a higher amount of peaks and higher amplitudes when the host structure is in a damaged condition. Also, from Figure 6 and Figure 7, it is observed that the sensors have different sensitivities according to the damage position. Despite the difference in sensitivity, all sensors can efficiently detect structural damage. As other SHM techniques, the proposed method has to be implemented using a threshold to separate healthy from damage condition. In order to show how sensitive is the proposed method, in Figure 8 it is presented a normalized result based on the results presented in Figure 7. The first result in healthy condition was used as a reference. Figure 8 shows that the proposed method can easily identify damage conditions.

117

Std Index Ratio 60 50 40 30 20 10 0 healthy 1

damage A 2

Série1

3damage B Série2

4 damage C 5 damage D

Série3

Fig.8 Normalized results from Figure 7

Conclusions In this paper an alternative methodology for structural health monitoring is proposed. A PZT patch is excited as an actuator and other PZT patches are used as sensors to get information on the health of the monitored structure. The new methodology is based on the magnitude of the complex coherence function which is computed between the signals from the pairs of PZT sensors and also between the actuator and the sensors. Tests were carried out on an aluminum plate and the results show the effectiveness of the proposed method to detect damage. Also, it was observed that the coherence between sensors could be a good approach in future works related to damage localization, which is a highlighted issue in SHM applications. Acknowledgements The authors gratefully acknowledge the support of the Capes Foundation, Ministry of Education of Brazil (BEX 3634/09-4 and BEX 0125/10-5). This work was also supported in part by AFOSR MURI Grant Number FA9559-060-0309 under the direction of Dr. David Stargel.

References [1] [2] [3] [4] [5] [6]

BROWNJOHN, J. M. W. Structural health monitoring of civil infrastructure. Phil. Trans. R. Soc., A365, p. 589–622, 2007. ROACH, D.; RACKOW, K. Health monitoring of aircraft structures using distributed sensor systems. Sandia National Laboratories, FAA Airworthiness Assurance Center, Albuquerque, NM 87185, 2006. SUN, F. Piezoelectric active sensor and electric impedance approach for structural dynamic measurement. Master Thesis. Center for Intelligent Material Systems and Structures, Virginia Polytechnic Institute and State University, Blacksburg, 1996. PARK, G.; SOHN, H.; FARRAR, C.; INMAN, D. J. Overview of piezoelectric impedance-based health monitoring and path forward. Shock and Vibration Digest, v. 35, p. 451-463, 2003. KESSLER, S.; SPEARING, S.; SOUTIS, C. “Damage detection in composite materials using Lamb wave methods,” Smart Materials and Structures, v. 11, n. 2, pp. 269-278, 2002. GIURGIUTIU, V. “Lamb Wave Generation with Piezoelectric Wafer Active Sensors for Structural Health Monitoring,” SPIE's 10th Annual International Symposium on Smart Structures and Materials and 8th Annual

118 International Symposium on NDE for Health Monitoring and Diagnostics, 2-6 March 2002, San Diego, CA. paper # 5056-17. [7] PARK, G; KABEYA, K; CUDNEY, H.; INMAN, D. “Impedance-based structural health monitoring for temperature varying applications,” JSME International Journal. Series A, Solid Mechanics and Material Engineering, vol.42, n.2, pp. 249-258, 1999. [8] BAPTISTA, F.G.; VIEIRA FILHO, J. “A New Impedance Measurement System for PZT-Based Structural Health Monitoring,” IEEE Transactions on Instrumentation and Measurement, v. 58, n. 10, pp. 3602-3608. [9] G. CLIFFORD CARTER, CHARLES H. KNAPP, ALBERT H. NUTTALL, Estimation of the Magnitude-Squared Coherence Function Via Ovelapped Fast Fourier Transform Processing, IEEE Transactions on Audio and Electroacoustics, Vol. AU-21, No. 4, August 1973, pp.337-349. [10] L.R. Rabiner and B. Gold, Theory and Application of Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1975. [11] PETER D. WELCH, The Use of Fast Fourier Transform for the Estimation of Power Spectra: A Method Based on Time Averaging Over Short, Modified Periodogram”, IEEE Transactions on Audio Electroacoustics, Vol. AU-15 (June 1967), pages 70–73.

The Best Force Design of Pure Modal Test Based Upon a Singular Value Decomposition Approach

J. M. Liu (1,2) Q. H. Lu (1) H. Q. Ying (2) (1) Dept. of Engineering Mechanics, Tsinghua University, Beijing 100084, China (2) China orient institute of noise and vibration, Beijing 100085, China

NOMENCLATURE FRFs frequency response functions H frequency response function matrix FEM finite element model

ωk

natural frequency (radians/second)

ψ

mode shape (vector)

ξ

damping coefficient

F

force vector

ABSTRACT Pure normal mode test, a routine ground vibration test (GVT) for large aircrafts, needs many shakers. When conducting such test for a pure mode, two major problems remain unsolved: One is how to determine the best shaker number; another is how to determine the best distribution of the shakers after shaker number is determined. This paper will answer these two important questions. When one mode is to be excited, the number of the shakers and the corresponding locations of shakers can be determined by a singular value decomposition approach, an algorithm proposed in this paper. The force appropriation can also be solved by SVD method with FRF matrix. The force appropriation effect is quantified by two indexes, the pure index and effective index. And the complete FRF matrix can be constructed by modal parameters of initial modal test or FEM. At the first section of this paper, the pure normal mode test theory development is reviewed. At the end, a simulation example is given to help understanding the new theory. 1

INTRODUCTION

In 1958, Asher proposed a method for selecting force amplitudes for single-mode isolation via multi-point sine testing [1]. This method is referred as pure normal mode test. He defined the steady state response as {U} = [B]{F}

(1)

Where {U}, [B], and {F} are complex matrices denoting the system response, the admittance matrix, and the force. The admittance is comprised of real [B’], and imaginary [B’’] components, which denote in-phase and out-of-phase (with respect to force contributions, respectively). In the system resonance, the real part of the system response is zero, which is denoted by the following equation [B’]{F}={0}

T. Proulx (ed.), Advanced Aerospace Applications, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 4, DOI 10.1007/978-1-4419-9302-1_11, © The Society for Experimental Mechanics, Inc. 2011

(2)

119

120

This set of homogeneous equations has a nontrivial solution (i.e., {F} ≠ 0) if and only if the determinant of [B’] vanishes. Asher defined [B] to be of size M-by-M, with M shakers and M measurement locations. A candidate resonant frequency occurred at every frequency that the determinant of [B’] vanished; at each of these frequencies, the force appropriation required to excite pure modal response is found by solving Equation (2) for {F}. Asher later explored some of the mathematical properties of the admittance matrix [2], with particular emphasis on the conditioning of the matrix and the effective degrees of freedom of the structure. In that paper, he also relaxed the restriction that [B] had to be square matrix; therefore, the number of shakers and measurement locations could be different. Craig and Su performed force appropriation simulation based on Asher’s method [3]. They showed that candidate resonant frequencies (for which all measurement locations have a real component of response near zero) could be separated from actual system resonant frequencies (for which all structural degree of freedom—greater than or equal to the number of measurement locations —have a real component of response identically equal to zero). Smith and Strond described the Lockheed MODLAB (Mobile Dynamics Analysis Laboratory) system, in which Asher’s method for force appropriation is implemented for selective tuning of modes [4, 5]. Ibanez extended the basic Asher technique to account for more measurement locations than shakers [6]. Vanishing of the determinant of Equation (2) for a square matrix implies that at least one of the eigenvalues is zero. Though rectangular matrices do not have classical eigenvalues, they do, nevertheless, possess singular values. The vanishing of singular value is, therefore, equivalent to the vanishing of the determinant of a square matrix. Ibanez gave the conceptual procedure for force appropriation with the extended method, and an example result was showed [7]. In this example, the extended Asher algorithm had been applied to a hypothetical structure. Jer-Nan Juang and Jan R. Wright presented a new multi-point appropriation method based upon a Singular Value Decomposition method [8]. The appropriated force vector which minimizes the in-phase response components while maximizing the required quadrature components is found, subject to a fixed force norm. The method can accommodate more transducers than shakers and also more shakers than effective degree of freedom. The method is demonstrated upon a six degree of freedom theoretical model and it was shown that an estimate for the effective degrees of freedom can be obtained. Kenneth S. Smith and Marc Trubert introduced the 1988 modal survey of the Galileo spacecraft core structure relied on the traditional tuned sine dwell method, with significant enhancements [9]. For each mode, response data was acquired at 32 frequencies in a narrow band enclosing the resonance. Modal frequencies and damping were estimated using all response, the narrow band FRF could be looked as one degree of freedom. Robert J. Dieckelman etc. introduced Modern Sine Excitation GVT Techniques [10]. The application of recent commercial technologies such as bar coding, VXI, ICP transducers and the Internet to the GVT had been successfully adapted by the Boeing Company Aircraft and Missiles Dynamics Laboratories to revitalize sine testing. Kelvin L. etc. introduced method of calculating modal mass from multiple-shaker sine tests [11]. Michael W. etc. stressed the importance of multiple-shaker sine tests in GVT tests [12]. G. Gloth and M. Sinapius introduced the detection of non-linearity in swept-sine measurements [13]. It explained in detail how the excitation level can be determined in the case of swept-sine excitation. It gave examples of impedance plots from numerical simulations and complex, large aerospace structures. The impedance plots were compared with curves that stem from phase resonances techniques. William R. and Nash M. found that by Multivariate Mode Indicator Function (MvMIF), the resonance frequency can be

121

determined in the pure normal modal test [14, 15]. To complete pure normal modal test, best force design is needed, which includes 3 parts, the number of shakers, location of shakers and force appropriation. At present, Jer-Nan Juang’s multi-point appropriation method based upon a Singular Value Decomposition can complete force appropriation, in the premise of all FRFs with input points relating to shakers position are measured. If there is some index to value the force appropriation effect and all the FRFs relating to any inputs are known, the remaining parts of the best force design, the number of shakers and the location of shakers can be determined. In this paper, based on the measured FRF matrices, the modal analysis is completed with SIMO, MISO or MIMO method. By ERA or PolyMAX method, closely-modes can be identified[16, 17]. The synthesized FRF between any output and any input can be calculated with the modes identified or the results of Finite Element analysis. To determine the effect of excitation, the pure index and effect index are put forward. 2 BEST FORCE DESIGN THEORY When modal parameters are identified, the FRF of any output point j to any input point i can be synthesized as

ψ irψ jr 2 r =1 mr (ω − ω + j 2ξ rωrω ) n

H ij (ω ) = ∑

2 r

Where n is the total mode number. If the mode shape is normalized by unit modal mass, the equation can be written as

ψ irψ jr 2 r =1 ω − ω + j 2ξ rωrω n

H ij (ω ) = ∑ If harmonic force with amplitude

2 r

Fi and frequency ω excite at i point, the harmonic response at point j will be n

( Fiψ ir )ψ jr

r =1

ω r2 − ω 2 + j 2ξ rωr ω

yij (ω ) = ∑ If there are total e harmonic forces of frequency

ω k with adjustable amplitudes, the harmonic response at point j

will be

e

ψ jr ∑ Fiψ ir

n

y j (ω k ) = ∑

i =1

2 2 r =1 ω r − ω k + j 2ξ r ω r ω k

(3)

When the location of forces is fixed, the k-th pure mode can be obtained by force vector F appropriation. The influence of other modes can be neglected with only k-th mode left, that is e

y j (ω k ) =

ψ jk ∑ Fiψ ik i =1

j 2ξ k ω k2

.

122 e

∑ Fψ Where

i =1

i

ik

2ξ k ω k2

= c is constant, thus y j (ω k ) = − jcψ

(4)

jk

This is a pure mode which is quadrature with in-phase. 2.1 THE FORCE APPROPRIATION If the location of e forces is fixed, equation (3) can be changed as e

y j (ω k ) = ∑ ( i =1

n

(ω r2 − ω k2 )ψ jrψ ir

r =1( r ≠ k )

(ω r2 − ω k2 ) 2 + 4ξ r2ω r2ω k2

∑

n

2ξ r ω r ω kψ jrψ ir

r =1( r ≠ k )

(ω r2 − ω k2 ) 2 + 4ξ r2ω r2ω k2

−j

∑

−j

ψ jkψ ik ) Fi 2ξ k ω k2

in matrix form,

Y = ( B − Cj − Aj ) F Y ∈ C m×1 ， A ∈ R m × e ， B ∈ R m × e ， C ∈ R m × e ， F ∈ R e×1 Where m (>e) is the total response number. Elements relating to A, B and C are

ψ ikψ jk aij = 2ξ k ω k2 Let J1

,

bij =

n

∑

r =1( r ≠ k )

(ω r2 − ω k2 )ψ irψ jr (ω r2 − ω k2 ) 2 + 4ξ r2ω r2ω k2

,

cij =

n

2ξ r ω r ω kψ irψ jr

r =1( r ≠ k )

(ω r2 − ω k2 ) 2 + 4ξ r2ω r2ω k2

∑

= F T AT AF , J 2 = F T ( BT B + C T C ) F

If the modes are not closely distributed with light damping, matrix C can be neglected. One of the two indexes below can be used for force appropriation. 1.

The pure index J 1 / J 2 . The larger the better.

2.

The efficient index J 1 / F . When J 1 / J 2 is satisfied to some extent, such as ( J 1 / J 2 >1000.0), the larger the efficient index, the better.

When the FRFs are based on measured data, the matrix C is not able to be separated from FRF imaginary part, so the pure index really used is defined as F

T

( AT A + C T C ) F / F T ( B T B) F . Only when the modes are not closely distributed with

light damping, matrix C can be neglected, the pure index almost equals to J 1 / J 2 . So the force appropriation method based on measured FRFs is suitable for the condition of sparsely modes with light damping. The pure index and efficient index of this paper is suitable for all conditions with the following 2 algorithms.

Algorithm 1: Consider J 2 first: The singular value decomposition of

BT B + C T C is written as

123

BT B + C T C = USU T S = diag ( s1 , s 2 , L se ) , s1 ≥ s 2 L ≥ se , U ∈ R e×e ， U TU = I To minimize J 2 , Assuming

F = U pd , Where U p

(5)

∈ R e× p , is a matrix constructed by the last p columns of matrix U , d ∈ R p×1 is the vector need to be solved.

This time p

J 2 = ∑ d i2 se− p +i

(6)

i =1

J1 = d TU Tp AT AU p d The singular value decomposition of

U Tp AT AU p is

U Tp AT AU p = Vdiag(σ1 , σ 2 ,Lσ p )V T ， σ 1 ≥ σ 2 L ≥ σ p , V ∈ R p× p ， V TV = I Let d be q-th column of matrix V, then

J1 = σ q By equation (6) J 2 can be computed, by equation (5) F and

(7)

F can be computed.

The value of p varies from 1 to e. For each p, there are p columns of matrix V, so there are p kinds of choice for the vector d. There are total (1+e)e/2 cases. In each case, J1 , J 2 , F and

F can be computed. According to pure index, the case in which

J 1 / J 2 is maximized can be found. According to efficient index, the case in which J1 / F is maximize and at the same time ( J 1 / J 2 >1000.0) can be found. The F vector in these cases is the force appropriation. Algorithm 2: Consider J 1 first: T

The singular value decomposition of A A is written as

AT A = USU T S = diag ( s1 , s 2 , L se ) , s1 ≥ s 2 L ≥ se , U ∈ R e×e ， U TU = I To maximize J 1 ,

124

Assuming

F = U pd , Where U p

(8)

∈ R e× p ，is a matrix constructed by the first p columns of matrix U , d ∈ R p×1 is the vector need to be solved.

This time p

J 1 = ∑ d i2 si

(9)

i =1

J 2 = d T U Tp ( B T B + C T C )U p d T

The singular value decomposition of U p ( B

T

B + C T C )U p is

U Tp ( B T B + C T C )U p = Vdiag(σ 1 ,σ 2 ,Lσ p )V T ， σ 1 ≥ σ 2 L ≥ σ p , V ∈ R p× p ， V TV = I Let d be q-th column of matrix V, then

J2 = σ q By equation (9) J 1 can be computed, by equation (8) F and

(10)

F can be computed.

The value of p varies from 1 to e. For each p, there are p columns of matrix V, so there are p kinds of choice for the vector d. There are total (1+e)e/2 cases. In each case, J1 , J 2 , F and

F can be computed. According to pure index, the case in which

J 1 / J 2 is maximized can be found. According to efficient index, the case in which J1 / F is maximize and at the same time ( J 1 / J 2 >1000.00) can be found. The F vector in these cases is the force appropriation. The force appropriation with better index can be obtained after comparing the two methods mentioned above. 2.2 THE BEST LOCATION OF SHAKERS When the shaker number is fixed, the shaker location can be arranged at different points. The number of the candidate arranged plans is very large. For example, if there are 8 measuring points with 2 shakers, the number of plans for arranging shakers is 8*7/2=28. In all the plans, by force appropriation their pure index or their efficient index can be got. The plan with the best index will be the best location of shakers. Two kinds of indexes are helpful because for the plans with same pure index, the best plan is one with maximum efficient index, and for the plans with same efficient index, the best plan is one with maximum pure index. 2.3 THE BEST NUMBER OF SHAKERS By the method of finding the best location of shakers, a curve of index varying with the number of shakers can be obtained. We can find a point on the curve after which the index will not improve conspicuously with number increasing. The number responding to that point is the best number of shakers.

125

Theoretically, the pure index will always improve with the number of shakers increasing until the number of shakers arrives the number of freedom degrees minus 1. So the efficient index is very important to decide the best number of shakers. In real tests, the max number of shakers is limited by laboratory conditions. 3

NUMERICAL EXAMPLES

The example is a simulation plate. Only vertical vibration is measured. The 24 measured points are arranged as figure 1. The analysis frequency range is from 0 to 500Hz. There are total 8 order modes in this range. The modal frequencies and modal damping ratios are listed in table 1.The mode shapes are illustrated as figure 2.

Figure 1: Measured points setup Table 1: Frequencies and damping ratios modes

Frequency [Hz]

Damping ratio [%]

1

96.370

0.100

2

123.860

0.090

3

202.120

0.300

4

221.780

0.300

5

261.340

0.100

6

334.010

0.260

7

407.550

0.890

8

415.780

0.150

126

Figure 2: Mode Shapes To excite the first order “pure” mode, computed the best pure index with the different number of shakers as table 2 and the best efficient index with the different number of shakers as table 3. Figure 3 corresponding to table 2 and Figure 4 corresponding to table 3. Table 2: The best pure index with different number of shakers (efficient index for reference) Number Of

Location of Shakers

J1 / J 2

( J1 / F )

2

4.1312e5

5.0895e2

2,7 or 18,23

3

7.8639e5

6.9591e2

2,7,18

4

6.3706e8

1.0179e3

2,7,18,23

5

6.9106e8

1.0311e3

1,2,7,18,23

2,7,8,18,23

2,7,17,18,23

2,7,18,23,24

2,4,5,7,18,23

2,7,18,20,21,23

Shakers

6

7.3445e8

1.0048e3

2,7,23

2,18,23

7,18,23

Table 3: The best efficient index with different number of shakers (pure index for reference) Number Of

Location of Shakers

( J1 / J 2 )

J1 / F

2

2.3272e5

9.9755e2

1,8 or 17,24

3

2.9571e5

1.4963e3

1,8,17

4

6.3990e6

1.9951e3

1,8,17,24

5

2.5459e6

2.2496e3

1,2,8,17,24

1,7,8,17,24

1,8,17,18,24

1,8,17,23,24

1,2,7,8,18,14

1,8,17,18,23,24

Shakers

6

5.1507e6

2.5040e3

1,8,24

1,17,24

8,17,24

127

Figure 3: The best pure index with the different number of shakers

Figure 4: The best efficient index with the number of shakers According to Figure 3 and Figure 4, the number of shakers should be 4. In some location plan, sometimes the force appropriation will be different based on the pure index or the efficient index. In this numerical example, if the number of shakers is 4, the Location of shakers is in 1,8,17,23. To excite the 1st order mode, the force vector should be [1,-1,-2.07,2.07 ] with best pure index, J 1 / J 2 =3.59956e6, should be [1,-1,-0.72,0.72 ] with best efficient index,

J 1 / F =1.1556e3. The force vector

J 1 / F =1.5065e3, J 1 / J 2 =3.59956e6.

If the number of shakers is fixed to 4 for all modes, the best location of shakers with pure index is as table 4. The best location of shakers with efficient index is as table 5.

128

Table 4: The best location with pure index (efficient index for reference) Modes

J1 / J 2

Location of 4

( J1 / F )

Shakers

Force Appropriation

1

6.3706e8

1.0179e3

2,7,18,23

[1.0,-1.0,-1.0,1.0]

2

7.3471e5

9.3331e1

9,12,13,16

[1.0,-1.07,-1.07,1.0]

3

8.3567e20

1.7033e1

9,12,13,16

[1.0,1.78,1.78,1.0]

4

1.3939e14

4.9505e2

1,8,17,24

[1.0,1.0,-1.0,-1.0]

5

8.0826e6

2.5427e2

1,9,16,17

[1.0,-1.0,1.0,-1.0]

6

4.4860e7

1.5036e1

6,11,14,22

[1.0,-0.02,3.25,1.0]

7

7.2499e4

3.7275e-1

3,6,19,22

[1.0,-1.0,-1.0,1.0]

8

1.4411e5

2.1261e1

4,8,17,21

[1.0,-0.37,-0.37,1.0]

Table 5: The best location with efficient index (pure index for reference) modes

4

( J1 / J 2 )

Location of 4

J1 / F

Shakers

Force Appropriation

1

6.3990e6

1.9951e3

1,8,17,24

[1.0,-1.0,-1.0,1.0]

2

2.9761e4

1.5598e2

1,9,16,24

[1.0,1.0,1.0,1.0]

3

3.2169e6

2.3930e2

3,6,19,22

[1,0,1.0,1.0,1.0]

4

5.6070e12

4.9505e2

1,8,17,24

[1.0,1.0,-1.0,-1.0]

5

7.9539e4

9.2036e2

1,8,17,24

[1.0,-1.0,1.0,-1.0]

6

7.0159e3

6.4368e1

1,8,17,24

[1.0,-1.0,1.0,-1.0]

7

3.1343e4

8.7738e-1

1,8,17,24

[1.0,-1.0,-1.0,1.0]

8

6.6106e4

5.5835e1

1,8,17,24

[1.0,1.0,1.0,1.0]

CONLUSIONS

The pure index is often used in force appropriation, but the efficient index is first put forward in this paper. Sometimes based on pure index or efficient index, different force appropriation will be produced. The efficient index will be more practical when large pure index responding to a small efficient index. Best force design includes 3 parts, the number of shakers, location of shakers and force appropriation. At present, only force appropriation can be completed because all the algorithms are based on the measured FRF matrices. In this paper, the best force design is based on the modal analysis outcome that can be completed with the measured FRF matrices or based on the FEM outcome, the modal damping is given by experience. All 3 parts of best force design can be completed with the new theory put forward by this paper. The pure index based on measured data will only suit for the condition that the modes are not closely distributed and modal damping is small. But the pure index and the efficient index based on modal analysis outcome or FEM outcome will suit for all conditions. The modal parameters from pure modal test can be used to replace part of initial modal analysis outcome. The test and best force design can proceed with iteration.

129

REFERENCES [1] G.W.Asher, A method of Normal Mode Excitation Utilizing Admittance Measurements, Proceeding, National Specialist’s Meeting, IAS; Dynamics and Aeroelasticity (Institute of the Aerospace Science, New York, November 1958). [2] G.W. Asher, A Note on the Effective Degrees of Freedom of a Vibrating Structure, AIAA Journal, Vol. 5 , No. 4, April 1967, pp. 822-824 [3] P R.Craig,et al, On Multiple Shaker Response Testing. AIAA Journal,Volume 7, No.12,1974,pp. 924-931 [4] S.Smith, et al, “MODLAB –A Computerized Data Acquisition and Analysis system

for Structural Dynamic Testing.

Proceedings, 21st International Instrumentation Symposium, Vol. 12(Instrument Society of America, Philadephia 1975), pp.183-189 [5] R C.Strond MODLAB —— A New System for Structural Dynamic Testing. Shock and Vibration Bulletin,1975,46,(5) [6] P. Ibnanez, Force Approppriation by Extended Asher’s Method, SAE Paper 760873, Aerospace Engineering and Manufacturing Meeting, San Diego, 1976. [7] Ibanez P. and Blakely K D, Automatic Force Appropriation – A Review and Suggested Improvements. Proc, 2nd IMAC 1984. [8] Juang J N and Wright J R, A Multi-point Force Appropriation Method based on a Singular Value Decomposition Approach. Proc. 7th IMAC,1989 [9] Kenneth S. Smith and Marc Trubert, An Enhanced Sine Dwell Method as Applied to the Galileo Core Structure Modal Survey. Proc. 8th IMAC,1990 [10] Robert J. Dieckelman, etc. Modern Sine Excitation GVT Techniques. Proc. 20th IMAC 2002. [11] Kelvin L. Napolitano, Ralph D. Brillhart, Calculating Modal Mass From Multiple-Shaker Sine Tests.Proc. 21 IMAC 2003. [12] Michael W. Kehoe and Lawrence C. Freudinger, Aircraft Ground Vibration Testing at the NASA Dryden Flight, Research Facility – 1993, NASA Technical Memorandum 104275 [13] G. Gloth, M. Sinapius, Detection of Non-Linearities in Swept-Sine Measurements. Proc. 21 IMAC 2003. [14] William R. the Multivariate Mode Indicator Function in Modal Analysis Proc, 3th

IMAC 1985

[15] Nash M. Use of the Multivariate Mode Indicator Function for Normal Mode Determination. Proc, 6th IMAC 1988 [16] Jr-Nan Juang and Richard S. Pappa. Effects of Noise on Modal Parameters Identified by the Eigensystem Realization Algorithm. Journal of Guidance, Control, and Dynamics, Vol. 9, No. 3, May-June, 1986, pp. 294-303 [17] Van H der Auweraer,Leuridan J. Multiple Input Orthogonal Polynomial Parameters Estimation. Proceedings of ISMA 11: workshop on parameter estimation, Leuven, Belgium, September, 1986, pp.16.

IMAC-XXIX Conference and Exposition on Structural Dynamics

MODAL IDENTIFICATION AND MODEL UPDATING OF PLEIADES Fabrice BUFFE (1) , Nicolas ROY (2) , Scott COGAN (3) (1) CNES, 18, av. Edouard Belin, 31401Toulouse Cedex 9, FRANCE Tel. +33 (0) 5 61 28 34 65 - e-mail [email protected] (2) TOP MODAL, Ecoparc II, av. José Cabanis, 31130 Quint-Fonsegrives, FRANCE Tel. +33 (0) 5 61 83 59 72 - e-mail [email protected] (3) FEMTO-ST Institute – Applied Mechanics Department, 24 Rue de l’Epitaphe 25000 Besançon, FRANCE Tel. +33 (0) 3 81 66 60 22 - e-mail [email protected]

THEME: Advanced Aerospace Applications. Satellite Modeling and testing

Abstract This paper presents all the updating activities performed on the finite element model of PLEIADES. The model updating is usually limited to a correction of modal data, by changing the most sensitive physical design parameters. In this paper, the modelization errors are localized and corrected thanks to a residual energy criteria: the Constitutive Relation Error (CRE). This method was originally developed by the LMT Cachan, and then implemented by the FEMTO Institute (Besançon, FRANCE) for application in an industrial context. The updating of PLEIADES is based on a modal approach: The experimental modes are identified using the Real Time Modal Vibration Identification (RTMVI) method. First, the model of the payload is updated with respect to a subsystem test performed on the instrument. Next, the model is condensed and included in the satellite model. The final step is to update the entire model using tests at satellite level. Primodal, a structural analysis tool developed by TOPMODAL (Toulouse, FRANCE) is used for correlation and updating.

T. Proulx (ed.), Advanced Aerospace Applications, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 4, DOI 10.1007/978-1-4419-9302-1_12, © The Society for Experimental Mechanics, Inc. 2011

131

132

Notation CRE FEM PFM FRF MAC RFR RTMVI SVD

Constitutive Relation Error Finite Element Model ProtoFlight Model Frequency Response Function Modal Assurance Criterion Resonance Frequency Research Real Time Modal Vibration Identification Singular Value Decomposition

Presentation of the context Pleiades Picard is a constellation of 2 satellites for Earth observation. It is both a civil and military mission. The structure includes a high resolution instrument under Thales Alenia Space responsibility, and a platform which was under ASTRIUM responsibility. The mass of the entire satellite is around 900 kg.

Fig. 1. Artistic View of Pleiades Methodology and goals of model updating The goal of model updating is to deliver a correlated model to the launcher authority. This model shall be representative in terms of modal behavior, to predict coupling between satellite and launcher modes.

133 This updating methodology is only based on a modal approach. The goal is to improve the MAC between prediction and test results and the difference between analytical and experimental eigenfrequencies. The final model may not be representative in term of stress if some properties have been changed significantly (as the Young's modulus). 2 updating methods have been used in parallel: - the first one is based on the MAC and eigen-frequencies. The updating is directly based on the optimization of the difference between the model and the test results. The choice of updating parameters depends on their sensitivity with respect to frequencies or modal shapes. It is essential to have a good initial matching between analytical and experimental modes. - the second one is based on the CRE. This method allows locating errors in the model, which gives a accurate indicator for the choice of updating parameters. This method depends on the instrumentation, and may not be accurate if the location and number of the sensors is not sufficient to have a good visibility of the model. Tests used for the correlation The tests used to update the instrument model have been performed in the beginning of 2008 on a PFM to qualify the payload. 132 sensors were used to measure the acceleration at different points of the structure. RFR are used to update the model. For the entire satellite, tests on a PFM have been performed at the end of 2008. A total of 141 sensors located on the Bus were used to update the model.

Context of the study This study is the following step of a collaboration between TOPMODAL, the FEMTO-ST institute and the CNES. The CRE (Contitutive Equation Error) has been implemented in the structural analysis tool, PRIMODAL, developed by TOPMODAL. This methodology was used for several years by the FEMTO institute through another tool, AESOP. Some previous study proved that the methodology is appropriate to large structure. The finite element model of PICARD has recently been updated successfully with AESOP.

134 This work on Pleiades was carried out by Pierre-Alain REBOUL during his internship in CNES. The goal was to validate this new implementation in PRIMODAL.

Modal identification The CRE needs the experimental eigenvalues and eigenvectors. As no modal survey test has been performed, these modal parameters have to be identified from sine-sweep base excitation vibration tests. FRF are obtained by normalizing the responses by the acceleration measured on pilot sensors. Identification has been performed with the RTMVI method. It assimilates each mode to a single dof system. This method is particularly efficient when modes are uncoupled, which is the case on almost all spacecraft structures. An improvement has been added to Primodal to take into account coupling between modes.

Fig. 2. Modal Identification with RTMVI in presence of 3 coupled modes

Fig. 3. SVD representation in presence of 3 coupled modes The SVD tool has been added to PRIMODAL. It allows separating quickly independent components from a set of sensors responses, and assessing if a peak corresponds to a mode.

135 From this representation in Fig. 3, we can see that there are 3 independent components, and therefore 3 modes to be identified in this frequency range.

Theoretical approach of the CRE [M ] [K ] [K elm ] [K red ]

Mass matrix (n*n) Stiffness matrix (n*n) Stiffness matrix of one element (i*i) Stiffness matrix reduced on the sensor nodes (c*c)

U exp

Modal shape from identification (1*c)

λexp

Eigenvalue from identification (1*1)

Π

Projection of the sensors nodes (n*c)

U ,V ,W Virtual fields of displacement (1*n)

U elm ,Velm Displacement of dof link to the considered element (1*i)

Supposing that only stiffness errors are taking into account, the fields U and V are calculated by minimizing the following term:

(U − V )t [K ](U − V ) + (ΠU − U exp )t [K red ](ΠU − U exp ) (2) Respecting the condition:

[K ] V

= λexp [M ] U

(3)

We can combine the 2 terms by adding a factor comprised between 0 and 1. If we are confident of tests results, we can weight the second term, which represent the difference between the extended field U and the experimental acceleration measured on the dof sensors. The localization criterion is defined for each element. It has the dimension of deformation energy, and can be calculated for a zone of the model by summing the error of elements belonging to this zone.

eelm = (U elm − Velm ) [K elm ](U elm − Velm ) t

(4)

The error used for updating is calculated on the entire model: t t e = (U − V ) K (U − V ) + (ΠU − U exp ) K red (ΠU − U exp ) (5)

136 To update the model, the residual energy e has to be minimized. This formula of CRE does not permit to localize mass error. But it is often sufficient, because the mass of the structure is generally well known. If we wish to localize mass error, we have to modify the formula (2). The field U, V and W are calculated by minimizing the following term:

λ U −V

2 K

+ (1 − λ ) U − W

2 M

+

α 1−α

ΠU − U exp

2

(6)

Kred

Respecting the condition:

[K ] V

= λexp [M ] W

(7)

With λ and α comprised between 0 and 1. The equivalent of the formula (4) for mass error is calculated with the following equation:

eelm = (U elm − Welm ) [M elm ](U elm − Welm ) t

(8)

The error used for updating is calculated on the entire model with the formula (6). One advantage of this method is that analytical and experimental modes do not need to be paired. The minimization of the CRE does not guarantee that the eigenfrequencies converge. It is generally the case when the model has a good visibility i.e. when the sensors allow locating a zone of the model when it is erroneous. If a zone has a low visibility, we may locate it even if no error is present. A modification of this zone could decrease the CRE without improving the eigenfrequencies. In this case, another updating method shall be used. The quality of a model updating is generally judged by comparing analytical and experimental eigen- frequencies. Usual updating methods are based on parameter sensibility. One advantage of the CRE is that only erroneous parameters are modified

137

Updating of the Pleiades instrument

Fig. 4 : FEM of the Pleiades instrument Comparison before model updating The first step is to calculate the MAC and the differences between eigenfrequencies.

Fig. 5. MAC and eigenfrequency errors before model updating The MAC is presented Fig. 5. Vertically (resp. horizontally), analytical (resp. experimental) modes are presented. The diagonal is clearly identified, which confirms that analytical and experimental modes can be paired. The first method of updating, based on MAC and eigenfrequencies, will be favoured. The eigenfrequencies which indicate that the model is too stiff can be improved. A computation of the CRE confirms that the errors in the model come from stiffness.

138

Fig. 6 In red, stiffness errors. In green, mass errors, in blue, errors from tests measurements. Updating procedure For the instrument, the choice of updating parameters is based on sensitivity. As we want to improve eigenfrequencies, parameters must have an influence on the modes. The first method of updating is favoured. However, to be sure that the modification of the model is robust, the MAC and the CRE are observed at each step of the updating procedure. The improvement of the MAC is essential to keep an accurate representativity of the modal shapes. The improvement of the CRE provides confidence in the modification of the model, and ensures that parameters which have been modified were really erroneous. The parameters modified in the instrument model are: - thickness of structural plate - Young's modulus of materials which constitute junction items For large properties, the coefficient applied on these parameters is never higher than 1.3. But for junction items, this coefficient has been sometimes increased until 3. Final comparison The 2 following figures shows the improvements on the model. -

blue: before updating purple: after updating

139

Fig. 7: Eigenfrequency errors before and after updating

Fig. 8: MAC before and after updating Frequencies have improved without degrading the MAC (the 3 last modes have a low effective mass). The global value of CRE is decreased from 1.98 to 1.76. As a conclusion, we can say that the initial model was already well correlated. The improvement concerns only the eigenfrequencies. A model is generally stiffer in reality, thus providing margins during the preliminary design phases. As tests have been performed on the Flight model, the model to be delivered to the launcher authority shall be as predictive as possible, and there is no need to keep these margins.

Updating of the satellite Pleiades Once the model of the instrument was updated, it has been condensed and included in the satellite model. This allows reducing the size of the model. Since the instrument has already been updated, only the model of the Bus structure will be modified.

140

Fig. 9 : FEM of the satellite Pleiades The instrument is located inside the platform. Comparison before model updating

Fig. 10 : MAC before model updating

Fig. 11: Eigenfrequency errors before model updating

The diagonal is less identifiable as before. It is more difficult to pair analytical and experimental modes. In consequence, the CRE method is favoured. All 15 first modes are taking into account, even if the pairing is difficult. From the MAC, we can observe that the distinguishability between some modes is weak. It is due to the fact that no sensors are located on the instrument during sat-

141 ellite test. The 4 first modes concern both the platform and the instrument, in phase or in opposition. The model is stiffer than the real structure, except for the 2 first modes, because they mainly concern the instrument, which has already been updated.

Fig. 12 : In red, stiffness errors. In green, mass errors, in blue, errors from tests measurements. We can observe here that the mass and test error has a non negligible contribution. Only the 10 modes where the error is visible (for which the effective mass is significant) are taken into account in the following updating procedure. Updating procedure Updating parameters have been chosen through the CRE localization. It concerns mainly junction element (bar properties, thickness of small shell…). The CRE has been minimized, by keeping an eye on the evolution of eigenfrequencies. Final comparison The 2 following figures shows the improvements on the model. - blue: before updating - purple: after updating

142

Fig. 13: MAC before and after updating Globally, the MAC has improved (except for some modes). But for the third and fourth modes, the MAC is not satisfactory enough. It is certainly due to the lack of sensors on the instrument during satellite test. The CRE is highly dependant to the instrumentation, and is limited if the visibility of the model is not good enough.

Fig. 14: Eigenfrequency errors before and after updating The eigenfrequencies have improved, without degrading the 2 first modes.

Conclusion The initial model was already well correlated in terms of MAC, but a difference between analytical and experimental eigenfrequencies remained. The purpose of this updating is to deliver to the launcher authority a model as predictive as possible. There is non need to keep a margin on frequencies. The main goal of this study was to reduce the eigenfrequency errors. The results are satisfactory, the improvement in terms of eigenfrequencies is significant. The difficulty is to satisfy simultaneously different criteria: eigenfrequencies, MAC and CRE. Primodal is an efficient tool which allows evaluating clearly the influence of each parameter on all these criteria. It is far from an automatic procedure, Primodal gives numerous indicators which help the user to choose the correct parameters, and the appropriate methodology.

143 However some limitations can be mentioned: -

In the instrument updating step, the error localization was deficient, because localized parameters have a low influence on eigenfrequencies. Other more sensitive parameters have been chosen to change the modes

-

The CRE was limited during the second step of the model updating, due to the lack of sensors on the instrument.

Bibliography 1.

Cogan S. AESOP. Analytical-Experimental Structural Optimization Platform Version 5.0 User Guide

2.

Roy N. PRIMODAL An Innovative Software tool for Structural Dynamics in Industry User’s Manual Version 1.7

3.

Ladevèze P., Leguillon D. Error estimate procedure in the finite element method and application. SIAM Journal of Numerical Analysis, Vol. 20 (3), pp. 485-509, 1983

4.

Cogan S. Model Updating based on the extended Constitutive Relation Error. Study between FEMTO-ST, TOPMODAL and CNES, 2009

5.

P. Ladevèze, M. Reynier, N. Maia, Error on the constitutive relation in dynamics, in H.D. Bui, M. Tanaka, et al. (Eds.), Inverse Problems in Engineering, 1st ed., Balkema, Rotterdam, 1994, pp. 251-256.

Aircraft GVT Advances and Application – Gulfstream G650 Ralph Brillhart, Kevin Napolitano ATA Engineering, Inc., 11995 El Camino Real, San Diego, California 92130 Lloyd Morgan, Robert LeBlanc Gulfstream Aerospace Corporation, 500 Gulfstream Road, Savannah, Georgia 31402-2206 ABSTRACT Ground vibration testing (GVT), one of the critical tests which occur during aircraft development, is typically one of the last tests to take place prior to embarking on the flight test program, providing valuable information for the validation of the aeroelastic stability of the aircraft. Historically, the GVT is required by the aviation regulators in the certification process. This highly visible and time-constrained test has evolved over the years as new data collection tools, both hardware and software, have become available. The Gulfstream G650 aircraft serves as an example of how modern approaches have allowed this required test to provide highly evolved information much more efficiently and with improved confidence, dispelling the myth that testing has to be time-consuming, costly, and complicated in order to be considered a success. INTRODUCTION Gulfstream Aerospace Corporation has recently added a new flagship aircraft to its fleet of long-range business aircraft. The G650 is a technologically advanced twin-engine business aircraft with an advanced fly-by-wire flight control system as well as many other significant improvements. It is designed to cruise at speeds of Mach 0.85 to 0.90, with a maximum speed of Mach 0.925 and will have a range of up to 7,000 nautical miles (13,000 km). The aircraft has an overall length of 99 feet 9 inches (30.40 m), an overall span (wingspan plus winglets) of 99 feet 7 inches (30.35 m), and a height of 25 feet 8 inches (7.82 m). Figure 1 shows the G650 aircraft; Figure 2 provides a three-view display with dimensions. A critical test in preparation for the aircraft flight test and certification program was the ground vibration test (GVT), which was used to characterize the dynamic properties of the aircraft in multiple configurations prior to flight test. The objective of the GVT was to measure the aircraft responses on the ground to confirm and validate the finite element model predictions. Two primary aircraft configurations were tested: zero wing fuel with stabilizer nose neutral and full wing fuel with stabilizer nose neutral. Overall aircraft dynamic behavior was characterized for these configurations, with further testing completed for aircraft control surfaces and other subsystems. The GVT is a critical path test since it occurs shortly before the flight test program begins and helps confirm the finite element models used to predict the aircraft dynamic loads and aerodynamic responses. These models in turn are used to guide the flight test program by predicting the flight stability of the aircraft. Historically, the GVT of an aircraft can be a time-consuming process [1, 2]. Preparing the aircraft, installing instrumentation, and conducting a large number of tests for different aircraft configurations can take a considerable amount of time. Often, new aircraft designs have required weeks to complete all of the testing required to satisfactorily achieve the GVT results; a single change in configuration could require days of testing. Advances in GVT methods over the past forty years take advantage of improvements in instrumentation and computing hardware as well as modal methods enhancements. These advances allow a GVT program such as performed on the G650 to be completed very efficiently in order to meet the very compressed time schedule. Gulfstream teamed with ATA to prepare for and conduct the G650 GVT. With proper test planning, implementation of new modal testing tools, and the ability to work around the clock, the complete GVT program for the G650 was completed in less than four days. Completing the test within this short time provided significant cost savings by freeing the aircraft for other processes in preparation for the flight test program.

T. Proulx (ed.), Advanced Aerospace Applications, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 4, DOI 10.1007/978-1-4419-9302-1_13, © The Society for Experimental Mechanics, Inc. 2011

145

146

Figure 1. Gulfstream G650 aircraft.

Figure 2. Three-view layout of G650 with dimensions. TEST PREPARATION ACTIVITY Appropriate test preparation can yield significant rewards in testing efficiency and accuracy of results. These activities are those which can be completed well in advance of the aircraft availability for the GVT. The preparation activities can guide the test program and minimize the number of test-time actions that are required after the aircraft has been committed to the GVT program, thereby reducing the total time when the aircraft is not available for other activities. The pretest analysis model preparation and evaluation are typically a significant test preparation activity. In collaboration between test and analysis engineers, the finite element model can be used to effectively select the appropriate accelerometer measurement locations as well as locations where excitation of the aircraft should be applied in the test. Further, the finite element model can be used to develop a simplified test display model which can be used to evaluate test results as soon as they become available. The test display model allows both test and analysis information to be displayed and compared to assure that all appropriate test data has been acquired

147

during the test program. This approach was used for the G650 program where Gulfstream provided the aircraft finite element model (FEM) and ATA developed the corresponding test display model shown in Figure 3. This test display model shows all of the measurement locations which were selected for the GVT (392 sensors). It was prepared so that back expansion to the full display model could be performed using all of the measurement results. Evaluation of the FEM results using the test display model allowed all of the measurement local coordinate systems to be checked well in advance of the test and verified that the model results could be directly compared to the test data. Using the finite element model to prepare for the test allows the quantity of test measurements to be selected with high confidence and verifies that the locations and number required can properly define the modes of interest. Proper selection of the number of measurements allows the data acquisition system to be configured appropriately. Given that, today, many data acquisition systems can acquire hundreds, if not a thousand, channels of data, this does not generally impose a significant constraint on the test preparation. Nonetheless, it can assist in ensuring that the data acquisition system is properly selected for the job at hand. At the same time as the measurement locations are defined, preparation for the installation of the sensors can be made before the aircraft is available. Geometry definition is prepared for each sensor to clearly define where the sensors are to be installed. Measurement tools and templates are also developed which will speed the test setup process. Once the locations are known, some of the layout for sensor installation can be performed prior to the test, while the aircraft is undergoing other final preparation Predefining sensor location tags or bar codes which will later be used in automating the sensor hookup can also be accomplished at this early stage. ATA has implemented a bar code system in which all sensors and measurement locations are identified with a bar code tag as shown in Figure 4. The system also takes advantage of recent sensor advances which use TEDS identification [3] of sensors through the data acquisition system. This allows accurate, automated sensor installation to be verified all the way through to the data collection hardware. A significant amount of measurement setup entry can be performed before the test setup, so that the final channel definition is quickly completed as the sensors are installed. A complete set of test configurations was defined for the GVT in order to validate the dynamic behavior of the aircraft. The first series of tests focused on identifying the overall airframe modes in the empty fuel and full fuel configurations. This was followed by a significant number of control surface tests in the second series. These included multiple flight control and hydraulic system configurations. Also included were structural mechanical interaction (SMI) tests in which the control surfaces were used to excite the aircraft while evaluating the aircraft behavior. Collecting this test data helped develop a better characterization of the flight control system in preparation for subsequent test programs, showing added value in the GVT. A third series of testing was conducted to focus on specific components of the aircraft including the nose boom and the ram air turbine (RAT). Table 1 provides a summary of the tests conducted during the course of the GVT. It should be noted that most of these tests were conducted using multiple excitation types and levels to characterize the linearity of the aircraft as well. Gulfstream and ATA selected the sequence of tests based on priority of the information to be obtained as well as the most efficient order to complete the tests. With the pretest analysis, test sequence definition, and instrumentation layout completed, the GVT was started once the aircraft was available.

148

Figure 3. Test display model of the G650. Arrows indicate where measurements were made.

Figure 4. Barcodes placed on sensors and at measurement locations enhance setup.

149

Table 1. Many aircraft test configurations were defined at the planning stage. Test Series 1

Targeted component overall aircraft

Aircraft configuration

Hydraulics configuration

Fuel empty

all on

1

overall aircraft

Fuel full

all on

2

SMI rudder

Fuel empty

all on

2

elevator rotation

Fuel empty

dual actuation

2

elevator rotation

Fuel empty

EBHA Inactive EHSA Active

2

elevator rotation

Fuel empty

EBHA Active EHSA Inactive

2

elevator rotation

Fuel empty

EBHA Electric EHSA Inactive

2

aileron rotation

Fuel empty

dual actuation

2

aileron rotation

Fuel empty

Right Hydraulic off

2

aileron rotation

Fuel empty

Left Hydraulic off

2

aileron rotation

Fuel empty

Electric Mode

2

elevator hinge line - SMI support

Fuel empty

all on

2

aileron hinge line - SMI support

Fuel empty

all on

2

rudder rotation

Fuel empty

dual actuation

2

rudder rotation

Fuel empty

Right Hydraulic off

2

rudder rotation

Fuel empty

Left Hydraulic off

2

rudder rotation

Fuel empty

EBHA active in Electric mode

2

flap -starboard side

Fuel empty

all on

2

rudder hinge line - SMI support

Fuel empty

all on - yaw damper off

2

rudder hinge line - SMI support

Fuel empty

all on - yaw damper on

2

rudder hinge line - SMI support

Fuel empty

all on - alternate mode yaw damper on

3

wing flutter exciters

Fuel empty

all on

3

nose boom - impact test

Fuel empty

n.a.

3

RAT - impact test

Fuel empty

n.a.

IMPLEMENTING NEW METHODS After thorough pretest preparation for the GVT, further steps were implemented to allow the GVT to be completed as efficiently as possible. The aircraft was prepared for the GVT and installed on a soft suspension while the aircraft instrumentation was being installed. Figure 5 shows the aircraft in the hangar test area. The suspension system, designed and built by Gulfstream, used a series of bungee loops installed with special support hardware at the landing gear. The nose gear suspension is shown in Figure 6. This bungee suspension provides a simple, effective isolation system that can be easily adjusted for varying load conditions. ATA provided the data acquisition system used for the GVT such that all data channels could be acquired simultaneously. This system incorporated VXI hardware with B&K Test for I-DEAS software. This allowed multiple tests to be conducted quickly, without requiring any data collection system channel configuration changes. In addition to the 392 sensor locations defined, six shaker force signals, the shaker command signals, and input location accelerations were also acquired for over 400 channels. Separate data collection and data analysis computing systems were provided to allow data to be acquired at the same time that data was being analyzed and compared to finite element predictions. ATA provided IMAT (Matlab based) software for data processing tasks and for comparison to analysis predictions. In many cases, time domain data was acquired and stored for postprocessing into frequency response functions on another computer system. This parallel processing of data allowed the testing to be completed without major interruptions. Further, since Matlab and IMAT were heavily used, this provided flexibility in how the data could be evaluated. Specially developed tools which might not be available in standard software products allowed quick on-site decisions to be made. The six shakers for the overall airframe studies were installed at the aircraft wingtips, the horizontal stabilizer tips, and the engine nacelles. This allowed the entire aircraft to be excited using both random and sine excitation while collecting all of the acceleration data. Multiple force levels were applied to study how the behavior of the aircraft changed with varying excitation force. Subsequent sine testing also made use of the shakers installed at these locations.

150

All of these steps have become a common part of an efficient GVT process being used today. In order to increase the efficiency of the test program, ATA implemented two new techniques. The first involved an excitation approach called Multi-Sine [4]. The second involved using a new data analysis tool called AFPoly [5]. Incorporating these methods into the GVT process helped assure that there were not delays either in completing the excitation and data collection process or in the modal parameter extraction process. Multi-Sine is an excitation technique in which multiple shakers are used to apply multiple sinusoidal frequencies to the aircraft at the same time. This is somewhat analogous to the typical multiple shaker random excitation approach which is widely used for modal testing today. By using Multi-Sine, multiple sine sweeps are effectively conducted simultaneously. Since all shakers are used together, as in a multishaker random test, there is no need to change the shaker configurations between sweeps. This saves time in the overall process. Figure 7 shows an example of the excitation forces applied to the G650 (time and frequency response) for a combination of symmetric sine sweeps. The three distinct frequencies shown in the spectra indicate the three different sinusoidal components that were applied and swept together. After the completion of the sweep, the time domain data was processed to yield the frequency response functions which were used in data analysis. The use of sinusoidal excitation allows higher amplitude excitation to be applied at a given frequency, which can result in higher quality frequency response function data which has better coherence than a corresponding set of multishaker random excitation. Further, with proper configuration, the sine sweep can be conducted in less time than required for the random excitation, which needs more time in order to allow for sufficient data averaging. In this test program, the sine sweep excitation was performed in about seventy-five percent of the time required for the random excitation. Improved efficiency in the test approach allowed all of the various test excitation techniques to be used for a thorough characterization of the aircraft. Due to the significant number of tests to be completed, data processing efficiency was also important. ATA implemented AFPoly as the key modal parameter estimation tool in addition to other methods which are in widespread use. AFPoly allowed the data sets to be processed quickly once frequency response data was available for a given configuration. Similar to time domain polyreference, a stability plot is generated using AFPoly which guides the modal selection process. The advantage of AFPoly, is there are fewer indicated computational poles to clutter the stability plot, thereby making the parameter selection process more straightforward. Figure 8 shows a stability plot generated during an AFPoly parameter estimation process. This method also allows a broader frequency band to be analyzed in a single process, which means that the test results for a given test can be generated more quickly. Linearity assessments were made as part of the GVT program by conducting data acquisition at a variety of excitation force levels. This testing could be completed quickly and easily since all data was acquired in a single set. With the data being available for immediate assessment, linearity could be studied without any substantial delay in the testing. Figure 9 shows a mode indicator plot demonstrating the effect of excitation level on frequency content for the overall aircraft. Specific study of the control surfaces showed more characteristic nonlinear behavior as seen in Figure 10 where the elevator response frequencies decrease with increasing force level. These trends were documented for all control surfaces during the second series of tests. Having all of the analysis pretest predictions available during the GVT allowed direct frequency comparisons to be generated and mode shape comparisons to be made. Figure 11 shows a comparison of test and analysis shapes which assisted during the test program. ATA developed a number of specialized tools which allowed the test and analysis shapes to be compared quickly as results became available. These documented which analysis modes had been identified and helped clarify whether further test data was needed.

151

Figure 5. G650 aircraft undergoing GVT.

Figure 6. Bungee suspension system installed at the nose landing gear.

152

Figure 7. Multi-Sine excitation allows for multiple sine sweeps to be conducted simultaneously. Excitation forces time domain (top) and spectra (bottom) are shown.

Figure 8. AFPoly stability plot for modal parameter extraction.

153

10

10

10

10

10

10

10

Mode Indicator Function Sh1 Starboard Wing - Accel

0

-1

-2

-3

-4

-5

-6

5

10 15 20 (Hz) - low burst random 1001X+ 1PMIFFrequency G650_16b_ff.afu 1001X+ 1PMIF G650_17b_ff.afu - mid burst random 1001X+ 1PMIF G650_18b_ff.afu - high burst random

25

30

Figure 9. Power Mode Indicator Function used for linearity assessment using multiple excitation levels.

154

Frequency Response Function Sh3 Starboard Tail - Accel 0

-180

-360 0

Acceleration/Excitation Force (G/lbf)

10

-1

10

-2

10

-3

10

10

15

20 1003X+ 1003X+ 1003X+ 1003X+ 1003X+ 1003X+ 1003X+

25

30 Frequency (Hz)

35

40

1003X- G650_36sym_fee.afu - Elevator Testing - Inboard Actuators - Dual Actuation 1003X- G650_37_fee.afu - Elevator Testing - Inboard Actuators - Dual Actuation x2 1003X- G650_38_fee.afu - Elevator Testing - Inboard Actuators - Dual Actuation x4 1003X- G650_39sym_fee.afu - Elevator Testing - Inboard Actuators - Dual Actuation x5 1003X- G650_40sym_fee.afu - Elevator Testing - Inboard Actuators - Dual Actuation x6 1003X- G650_41sym_fee.afu - Elevator Testing - Inboard Actuators - Dual Actuation x7 1003X- G650_42sym_fee.afu - Elevator Testing - Inboard Actuators - Dual Actuation x8

Figure 10. Linearity study of the elevator control surface.

45

50

155

Figure 11. Comparison of test and analysis mode shape results helped test completion decisions.

156

RECOMMENDED PRACTICES A number of steps were taken to allow the G650 GVT to be conducted as efficiently as possible. As many tasks as possible were completed prior to the aircraft availability in order to prevent delays once the aircraft was committed to the GVT program. These pretest efforts made sure that all of the instrumentation was properly identified and configured so that final installation could be completed quickly. Substantial channel table configuration was also completed ahead of time to minimize test delays. Use of barcodes and other instrumentation automation features such as TEDS were essential to the setup process. The test display model which was completed as part of the pretest activity, made it possible to check the channel layout and data processing tools prior to the test. Once the test was started, having a test configuration sequence which was efficiently ordered allowed the test program to be conducted with minimal delays between each test set. Test data acquisition tools were also critical to the process. Use of a data acquisition system which could acquire all data channels simultaneously minimized the total acquisition time required for each test configuration. This allowed multiple data sets to be acquired to study structural variability with excitation level, which was particularly important for control surface studies. The use of Multi-Sine excitation helped in this regard, since the total number of sine sweeps required was minimized. The amount of time required for data processing was also important since it influenced the assessment of the data acquired in any given configuration. Having the modal parameters extracted in a short time after the data was acquired allowed decisions to be made about whether a given test configuration had been completed. AFPoly was an important tool in this regard. Other useful data analysis and processing tools allowed for comparison between test and analysis results while the testing was being conducted. SUMMARY In completing the Gulfstream G650 GVT, a total of 120 unique test runs were performed to complete the full study of the aircraft. This included testing for the overall airframe modal behavior in two aircraft fuel states and a large number of control surface tests. The control surface testing also included SMI evaluation providing valuable assessment of the control system behavior. In order to complete such an extensive GVT program in less than four days, an efficient process had to be employed which included the implementation of new modal testing tools. The evolution of modal testing has reached the point where a large number of sensors can be installed, verified, and measured while applying multiple shaker input to the aircraft. Multishaker techniques allow a complete characterization of the aircraft to be developed without the large number of shaker moves which had been required in the past. Implementation of the Multi-Sine excitation technique allows further improvement by making it possible to conduct a combination of sine sweep tests in significantly less time. Improved data quality may result since larger excitation forces can be applied. Modal parameter estimation can be performed using a wide variety of software tools. Any number of these can be used in the data analysis process. Those tools which allow a clear assessment of the modal parameters with fewer data set iterations will permit the modal testing to be completed more efficiently. The use of the AFPoly software for the G650 GVT demonstrated that it could be effective in enhancing the data analysis process, clearing the way for test site decisions. There are always ways to improve the process of testing so that it does not have to be considered a time consuming and expensive endeavor. Since the GVT is such a critical test process required in the development of a new aircraft, new tools such as those used in this G650 GVT are essential to the further improvement and efficiency of the modal testing process. ACKNOWLEDGEMENT The authors of this article thank Gulfstream Aerospace for the opportunity to discuss the ground vibration test program on the G650 aircraft.

157

REFERENCES 1. Hunt, D., A Comparison of Methods for Aircraft Ground Vibration Testing, Third International Modal Analysis Conference, Union College, Schenectady, NY, Jan., 1985. 2. Hunt, D., A Comparison of Methods for Ground Resonance Testing, IMAC, 1997. 3. Brillhart, R., and M. Dillion, Automated Test Setup in Modal Testing, 10th International Modal Analysis Conference, Los Angeles, CA, Feb, 2002. 4. Napolitano, K., D. Linehan, Multiple Sine Sweep Excitation for Ground Vibration Tests, 27th International Modal Analysis Conference, Orlando, FL USA, Feb, 2009. 5. Vold, H., M. Richardson, K. Napolitano, D. Hensley, Aliasing in Modal Parameter Estimation - A Historical Look and New Innovations, IMAC, Feb, 2007.

Aircraft Dynamics And Payload Interaction – SOFIA Telescope Ralph Brillhart, Kevin Napolitano ATA Engineering, Inc., 11995 El Camino Real, San Diego, California 92130 Tracy Duvall L-3 Communications, Integrated Systems, 7500 Maehr Road, Waco, Texas 76715 ABSTRACT The Stratospheric Observatory for Infrared Astronomy (SOFIA) is now in operation out of NASA Dryden Flight Research Center and is providing astronomical science observations not possible from other Earth- and spaceborne observatories. A 2.7 meter telescope which weighs 34,000 pounds has been installed in the aft fuselage of a Boeing 747SP aircraft. This required significant structural changes to the airframe. Further, the telescope is installed on an isolation system with a dynamic control system for properly positioning and controlling the pointing of the telescope. The major structural modifications of the aircraft and the dynamic interaction of the telescope with the aircraft are important issues related to the aircraft flight stability. Ground vibration testing (GVT) was performed to recharacterize the dynamic properties of the aircraft with the telescope installed. This testing was performed prior to delivery to NASA DFRC, where the flight test program was conducted. In addition to standard GVT measurement parameters, the testing involved evaluation of the acoustic cavity where the telescope is installed. Dynamic testing was also performed to investigate the structural coupling interaction that occurs when the telescope control system is activated. INTRODUCTION The Stratospheric Observatory for Infrared Astronomy (SOFIA) is a joint project of the National Aeronautics and Space Administration (NASA) and the Deutsches Zentrum für Luft-und Raumfahrt (DLR, the German aerospace center) which places an advanced, airborne, infrared, telescope observation platform in service. Following an extensive twenty-four-year development and flight test process, this flying telescope has now started its service phase with its first science missions. DLR is responsible for the entire telescope assembly and design while NASA is responsible for the aircraft. SOFIA offers several key advantages over either land-based or space-based telescopes [1]. Since it can be positioned at cruising altitudes of over 40,000 feet, it places the telescope above 99.8 percent of the atmosphere’s water vapor, which is far better than even the highest land-based observatories that can be above 90 percent of the atmosphere’s water vapor. Water vapor effectively blocks most infrared wavelengths, rendering telescopes blind to infrared radiation. Space-based telescopes can of course be placed outside the earth’s atmosphere, where water vapor is not a concern, but telescope mobility and telescope cooling requirements limit the ability to place the telescope where desired as well as its useful life. Access to the telescope for maintenance is also a limiting challenge for space-based systems. SOFIA overcomes these limitations by installing the telescope in a flying test bed 747SP aircraft. This approach has been applied previously in the Kuiper Airborne Observatory (KAO) [2], although with a considerably smaller telescope. SOFIA uses a reflecting telescope whose primary mirror is 2.7 meters across; larger than the Hubble telescope and almost three times the size of that in the KAO. While this telescope is the largest ever placed in an aircraft, it is modest in size relative to ground-based telescopes. Its advantages lie in the ability to get above the infraredblocking water vapor and its high maneuverability; it can be flown to virtually any viewing location around the world. Furthermore, this telescope system will provide access to wavelengths from 0.3 μm to 1000 μm, which means it has a wider spectrum coverage than any other ground- or space-based system. Since no window material is transparent over this whole range, one requirement for the observatory platform was that the telescope be directly exposed to the atmosphere, in this case the stratosphere. In order to accommodate the large telescope, a large aircraft was required. In 1984, when the SOFIA program was first proposed, the Boeing 747 was the largest aircraft available to fulfill the flight requirements (altitude and

T. Proulx (ed.), Advanced Aerospace Applications, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 4, DOI 10.1007/978-1-4419-9302-1_14, © The Society for Experimental Mechanics, Inc. 2011

159

160

range) and would accommodate the three-meter class telescope. The 747SP was the longest-range airliner available at the time, so it was selected as the airborne platform for SOFIA. The actual aircraft was not made available to the SOFIA program until 1997, when a series of baseline flight studies was performed. The size of the SOFIA telescope provided a number of challenges for the aircraft platform. Further, the requirement that the telescope be exposed directly to the atmosphere meant that substantial structural modifications would need to be made to the aircraft. L-3 Communications, Integrated Systems in Waco, Texas, was tasked with making the aircraft modifications necessary to accommodate the telescope. Those modifications were started following baseline ground and flight testing studies. Most of the aft fuselage of the aircraft was completely reworked to accommodate the telescope. Figure 1 shows an illustration of the SOFIA aircraft and position of the telescope.

Figure 1. SOFIA aircraft illustration showing open door to telescope cavity (top) and closed door (bottom). [Illustration courtesy of NASA] Aircraft modifications In order to provide the opening for the telescope aperture, almost one fourth of the aircraft circumference was cut away in the fuselage just forward of the empennage section on the port side of the aircraft. Figure 2 shows the telescope cavity in the aircraft fuselage with the fuselage door open as it would be for telescope observations. With this change in the fuselage, bulkheads were installed forward and aft of the telescope cavity. Since the telescope would have no window installed in the aperture, exposing the telescope to the atmosphere, the bulkheads were designed to accommodate the significant difference in pressure between the cavity and the scientific instrument compartment where the scientists operating the telescope would be during observation flights. Temperature differences between the two sections of the aircraft were also key in defining the design loads. The aft bulkhead was a closed section which would carry structural loads as well as the pressurization loads. The forward bulkhead was designed to accommodate the attachment of the telescope flange which also included the variable vibration isolation system installed in the forward bulkhead as part of the telescope system. Figure 3 shows a cutaway display of the telescope system indicating the interface between the aircraft bulkhead and the telescope isolation system. In addition to the static pressure and temperature differences which would be applied to the bulkheads, these needed to accommodate the aerodynamic fluctuating pressures inside the telescope cavity resulting from the opening in the side of the aircraft. During takeoff and landing, the telescope cavity is closed to the atmosphere using a set of doors which cover the telescope opening. These can be seen in Figure 2. This allows the telescope to be thermally conditioned before and after the flight, preventing condensation on the telescope. Once the aircraft has reached the observation cruising altitude, the doors are opened to expose the telescope. Two doors are used to cover the opening: an upper rigid door (URD) and a lower flexible door (LFD). The segmented LFD can be adjusted, moving with the telescope aperture, to control the amount of light which enters the telescope. When the URD is closed, a door seal is also inflated to ensure proper conditioning of the telescope. The forward fuselage was also modified to provide the mission control and science operations section to accommodate the telescope instrumentation and control equipment, the scientists, and an educational section. This allowed for telescope observations to be controlled and adjusted as needed during the flight. This configuration puts the operators in closer proximity to the telescope than any other land- or space-based system, meaning that telescope adjustments can be made in virtual real time. Figure 4 shows the forward bulkhead and the forward end of the telescope system as seen from the operations section of the aircraft.

161

Figure 2. SOFIA telescope cavity viewed from outside the 747SP aircraft.

Figure 3. Cutaway view of SOFIA telescope system.

162

Figure 4. View of telescope instrument end and forward bulkhead from inside the scientific instrument compartment. Measurement of aircraft parameters The substantial changes to the airframe structure required verification testing to be completed before flight testing, such as a new aircraft requires. As part of the aircraft testing performed after acquisition in 1997, baseline characterization of the aircraft was developed which was used as a starting point for dynamic analysis as the structural changes were implemented. This included an initial ground test as well as flight testing to establish the aircraft characteristics before structural modifications were made. Once all of the aircraft modifications were completed and the telescope assembly was installed in the aircraft, a ground vibration test (GVT) was conducted to establish dynamic characteristics of the airplane/telescope assembly prior to undertaking flight testing. This GVT was conducted in June of 2006. The purpose of the GVT was to observe the modal behavior of the aircraft to verify the finite element models used to study the stability of the aircraft in flight. The significant structural changes to the aft fuselage needed to be verified to confirm that they did not negatively impact the structural dynamics. In addition to the evaluation of the aircraft structure, the GVT was needed to evaluate how the telescope would affect the dynamics of the aircraft and also how the telescope itself would behave dynamically. The telescope assembly (TA) uses a vibration isolation system (VIS) to minimize the aircraft vibration from the telescope while in flight. The VIS is locked out or caged when the telescope is not in use, but then uncaged once observations are to be made. One study of interest was to see how the overall aircraft dynamics changed as a result of the VIS configuration. In order to study the telescope effects on the aircraft, a large matrix of tests was performed on multiple test configurations to study how the telescope orientation (elevation angle) and aperture elevation angle would change the aircraft dynamics. Further configurations took into account the telescope isolation system as well as drive control system settings. Additionally, the outer door position and door seal positions were studied. All of these

163

changes in mass and stiffness distribution in the payload area of the telescope could have an influence on the overall aircraft dynamic behavior and stability. Table 1 provides a summary of test configurations which were evaluated as part of the GVT. During all of the variations of the telescope parameters, the general configuration of the aircraft remained constant. It was tested in an unfueled condition with the aircraft supported on the landing gear and partially deflated tires. The landing gear used for supporting the aircraft were the nose gear and the fuselage main gear; the wing landing gear were retracted. Table 1. List of test configurations showing variety of telescope parameters. Aircraft Configuration #

Aircraft Hydraulic Systems

Telescope Bearing Hydraulic System

Telescope Elevation Angle (Deg)

Aperture Elevation Angle (Deg)

1

1-4 On

On

40

20

Caged

Braked

Braked

Closed

2

1-4 On

On

40

20

Uncaged

Braked

Closed

3

1-4 On

On

40

20

Uncaged

Braked Unbraked; Open Loop

Braked

Closed

4

1-4 On

On

40

20

Caged

5

1-4 On

On

40

20

Uncaged

Telescope VIS

6

1-4 On

On

40

20

Caged

7

1-4 On

On

20

20

Uncaged

8

1-4 On

On

60

20

Uncaged

9

2-4 On

On

60

20

Uncaged

10

1-3 On

On

60

20

Uncaged

11*

1-4 On

On

60

20

Uncaged

12

off

On

60

20

Uncaged

13

off

On

40

20

Caged

14

off

On

20

20

15

off

On

60

60

16

off

On

40

17

off

On

40

18

all on

On

40

Telescope RIS

Braked Braked Unbraked; Open Loop Unbraked; Open Loop Unbraked; Open Loop Unbraked; Open Loop Unbraked; Open Loop Unbraked; Open Loop Unbraked; Open Loop

Coarse Drive

Unbraked; Open Loop Unbraked; Open Loop

Cavity Door System Position

Closed Closed

Braked

Closed

Braked

Closed

Braked

Closed

Braked

Closed

Braked

Closed

Braked

Closed

Braked

Closed

Braked

Braked

Closed

Caged

Braked

Braked

Open 20

Caged

Braked

Braked

Open 60

40

Caged

Braked

Braked

Open 40

40

Caged

Braked

Braked

Open 40

20

Caged

Braked

Braked

Open 40

* Note that configuration 11 is the same as configuration 8 except that the URD door seal was deflated in configuration 11 to simulate failure.

Overall aircraft responses were measured in the GVT as is typical for tests of this type. In addition, a substantial amount of instrumentation was installed on the telescope assembly, suspension components, both forward and aft bulkheads, and the aperture and door assemblies. Figure 5 shows a test display model indicating where accelerometers were installed on SOFIA. Over 350 sensors were used for the GVT program. In general, sensors stayed at the same locations throughout the test program and were all taken simultaneously using a multichannel VXI data acquisition system with I-DEAS software. One exception to this was a detailed survey of the telescope cavity section of the fuselage, described later. The overall aircraft was studied for the frequency range of 0.2 to 25 Hz. Over 120 different test runs were performed during the GVT to study all parameters of interest. These runs included various excitation types comprised of impact, burst random, and sinusoidal inputs. Initial testing of the aircraft was performed with the VIS caged so that the telescope was held rigidly to the aircraft. Subsequent tests were performed with the VIS uncaged, making the isolation system active and allowing the telescope to move relative to the aircraft. When this configuration change occurred, the dynamic behavior of the aircraft was noticeably different, as indicated in the mode indicator function (MIF) [3] shown in Figure 6. While most of the fundamental modes of the aircraft did not

164

change, several of the intermediate frequency modes were affected. The information derived in this part of the test program was critical in validating the finite element predictions of how the TA interacted with the aircraft in the variety of flight configurations.

Figure 5. SOFIA test display model showing measurement locations for GVT. Arrows indicate sensor locations and orientation. Note the detailed measurements around the telescope cavity.

165

Figure 6. Vibration isolation system (VIS) connection to aircraft affected overall aircraft behavior. Telescope cavity investigations A detailed study of the fuselage structure around the telescope cavity was performed to characterize the shell modes of the aircraft body in this area. A matrix of accelerometers was installed on the starboard side of the fuselage, opposite the telescope opening. Figure 7 shows the fuselage section with accelerometers installed on the skin opposite the telescope door opening. The higher density of accelerometers allowed short wavelength modes of the skin to be evaluated. Various excitation types (impact, random, sine) were then applied to the fuselage and the aft bulkhead to characterize the local behavior to about 200 Hz. Excitation on the bulkhead was applied to evaluate what type of coupling would occur between the fuselage skin and the bulkhead. Since this study was concentrated on higher frequency, localized behavior of the fuselage and bulkheads, the sensors on the fuselage exterior were removed once this test was finished, and the balance of GVT testing was completed with the remaining sensors on the aircraft and telescope. In addition to the fuselage shell mode behavior, the acoustic behavior of the cavity was of interest. Since the telescope cavity would be exposed to the atmosphere and fluctuating pressures, the acoustic cavity modes could result in dynamic pressure oscillations which might affect the telescope performance. Piezoelectric microphones were installed in the cavity while the fuselage was excited in order to identify fundamental acoustic modes. These were validated with simple wavelength calculations and helped establish the internal acoustic modes which were present. Excitation of the aft bulkhead was also used in investigating the acoustic behavior of the cavity. Increased frequency limits were used for all of the acoustic investigations, with data collected to about 140 Hz. Modal behavior of the doors was studied with localized excitation and response measurements on both the URD and the LFD. In addition to characterizing the door dynamics, a door seal failure was simulated to evaluate if this resulted in any significant changes in dynamic behavior. It was found that if the door seal failed, the door

166

appeared to decouple from the aircraft above about 13 Hz, giving rise to a dominant door mode around 30 Hz, Figure 8 shows the door behavior with and without the door seal active. These studies helped establish the characteristics of the local structural components around the telescope cavity and how they could influence the overall aircraft behavior as well as the telescope performance.

Figure 7. Fuselage skin opposite the telescope door was instrumented with a fine grid of sensors.

167

Figure 8. Study of door seal showed effect of simulated failure on door dynamics. Telescope control interaction The test configurations that were defined were developed to characterize how the varying telescope orientations, suspension conditions, as well as telescope door positions influenced overall dynamic behavior. Numerous tests were performed to establish the overall aircraft dynamics. With considerable interest in the telescope, bulkhead, and isolation system behavior, a series of tests was performed by exciting on the telescope and bulkhead structures. Figure 9 shows an example where a shaker system was attached to the telescope structure inside the scientific instrument compartment. Excitation of the telescope allowed the effectiveness of the isolation system to be studied as the system was activated. This can be seen in Figure 10. Further testing was performed to study the telescope interaction with the aircraft. Rather than using one of the modal exciters, the telescope itself was used as the excitation system. This type of control system interaction test is typical of that used for aircraft control systems. In this case, the interaction between the telescope control system and the aircraft response was evaluated. The measurements in this case allowed evaluation of the type of responses that might be expected during the activation of the telescope while in flight. This testing was conducted using sinusoidal input signals to the telescope servo control drive system. Variations in drive command signal levels allowed the study of the excitation levels on the expected responses. This study was performed to investigate the frequency range between about 0.5 and 15 Hz during telescope movements in the elevation, crosselevation, and line-of-sight axes. They were conducted with the VIS in both caged and uncaged conditions. These tests helped evaluate a telescope drive system which might be “out-of-control,” where both the aircraft response and the TA active control system safeguards could be studied. As seen in Figure 11, the telescope responses were isolated from the aircraft forward bulkhead for low frequency excitation. Performing these tests allowed critical telescope behavior and aircraft interaction to be evaluated while conducting the overall GVT, taking advantage of the GVT instrumentation.

168

Figure 9. A shaker was used to directly excite the telescope balance weights.

Figure 10. Excitation of the telescope allowed evaluation of the vibration isolation system.

169

Figure 11. Transfer functions between the telescope servo drive and structural components showed isolation contribution. SUMMARY Significant structural changes to the carrier aircraft for the SOFIA telescope required a detailed GVT to be performed to validate the structural dynamics of the completed aircraft prior to the flight test program. The dynamics of the telescope as installed in the aircraft were also important to establishing the behavior of the system once in service. Combining a number of telescope characterization tests with the GVT program assured a more fully understood dynamic system. This ground test program for the SOFIA aircraft encompassed a wide range of tests which captured important dynamic characterization. Testing was completed in about a two-week period and provided a thorough documentation of the dynamics of the SOFIA platform. Over eighteen different test configurations were evaluated in the test program using over 350 accelerometers and other measurement instrumentation. The modal parameters obtained in the test program were critical in defining the overall aircraft responses to be used in airworthiness certification, flight test safety, and validation of the analytical modeling. Interaction of the TA with the aircraft was a critical component in this process. It further served to establish the dynamic behavior of the TA vibration isolation system and the servo-elastic properties of the TA. Use of the engineering information obtained in this ground test program provided necessary information leading up to successful subsequent testing of the SOFIA platform. A successful GVT program paved the way for subsequent flight testing and confirmed the structural modifications of the aircraft as well as helped demonstrate performance characteristics of the telescope. With the successful completion of this GVT program and the aircraft flight test program, SOFIA is now entering its operational phase, promising to provide new images of space for an extended period of time.

170

ACKNOWLEDGEMENT The authors of this article thank L-3 for the opportunity to discuss the ground vibration test program on the SOFIA aircraft. REFERENCES 1. Bell TE: SOFIA Flies to the Stars, Exploring the universe from the atmosphere. The Bent, 18-23 (Fall 2010, Vol. CI/No. 4) 2. Dolci WW (1997) Milestones in Airborne Astronomy: From the 1920’s to the Present. American Institute of Aeronautics and Astronautics. 975609 3. Williams R, Crowley J, Vold H, The Multivariate Mode Indicator Function in Modal Analysis, Proceedings of the 3rd International Modal Analysis Conference, 1985, pp. 66-70.

Application of Modal Analysis for Evaluation of the Impact Resistance of Aerospace Sandwich Materials Amir Shahdin*, Joseph Morlier1,*, Guilhem Michon*, Laurent Mezeix†, Christophe Bouvet* and Yves Gourinat* * Université de Toulouse, ICA, ISAE, 10 av. Edouard Belin BP54032, 31055 Toulouse, France †

Université de Toulouse, INPT-ENSIACET/CIRIMAT, 118 route de Narbonne, 31077 Toulouse, France

1

Corresponding author, Email: [email protected], Phone no: + (33) 5 61 33 81 31, Fax no: + (33) 5 61 33 83 30

ABSTRACT Impact resistance of different types of composite sandwich beams is evaluated by studying vibration response changes (natural frequency and damping ratio). This experimental works will help aerospace structural engineer in assess structural integrity using classification of impact resistance of various composite sandwich beams (entangled carbon and glass fibers, honeycomb and foam cores). Low velocity impacts are done below the BVID limit in order to detect damage by vibration testing that is hardly visible on the surface. Experimental tests are done using both burst random and sine dwell testing in order to have a better confidence level on the extracted modal parameters. Results show that the entangled sandwich beams have a better resistance against impact as compared to classical core materials.

Keywords Structural Integrity Assessment, Composite Sandwich beams, Vibration Testing, Impact Resistance

Introduction The aim of composite sandwich structures is to increase the stiffness and specific strength and to reduce the weight so it is advantageous to employ them in aerospace applications where the challenge is to produce light structures. However damage due to impact in these structures may negate many of the benefits of sandwich construction. The facesheets can be damaged through delamination and fibre breakage; the facesheet and core interface region can be debonded and the core can be damaged through crushing and shear failure mechanisms. Safe and functional effectiveness of stressed sandwich structures can often depend on the retention of integrity of each of the different materials used in its manufacture. For aeronautical structures, a field where this problem has been quite studied, the components have to undergo low energy impacts caused by dropped tools, mishandling during assembly and maintenance, and in-service impacts by foreign objects such as stones or birds. In these low energy impacts normally, a small indentation is seen on the impact surface. This level of damage is often referred to as Barely Visible Impact Damage (BVID). Although not visually apparent, low energy impact damage is found to be quite detrimental to the load bearing capacities of sandwich structures, underscoring the need for reliable damage detection techniques for composite sandwich structures. In recent years, vibration based damage detection has been rapidly expanding and has shown to be a feasible approach for detecting and locating damage [1-3]. Any structure can be considered as a dynamic system with stiffness, mass and damping. Once some damages emerge in the structures, the structural parameters will change, and the frequency response functions and modal parameters of the structural system will also change. This change of modal parameters can be taken as the signal of early damage occurrence in the structural system. Shift in natural frequency is the most common parameter used in the identification of damage [4-6]. However, in structures made of composite materials there seems to be a tendency to use

T. Proulx (ed.), Advanced Aerospace Applications, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 4, DOI 10.1007/978-1-4419-9302-1_15, © The Society for Experimental Mechanics, Inc. 2011

171

172 damping as a damage indicator tool, as it tends to be more sensitive to damage than the stiffness variations, mainly when delamination is concerned [7-10]. A comprehensive review of low-velocity impact responses of composite materials is presented by Richardson and Wisheart [11]. Dear et al. [12] studied the impact toughness of different lightweight sandwich panels and composite sheet materials. They emphasized on the degree of damage inflicted on the contact surface, throughthickness and rear surface of the materials when subjected to different impacts. Vaidya et al [13] studied composite sandwich structures composed of aluminum foam core and found it optimal for resisting low-velocity impacts. Similarly several novel sandwiches have been developed with a view to enhance the impact toughness [14-16]. In this paper we have made Low Velocity Impact (LVI) and post impact vibration tests to measure the impact resistance. More precisely we evaluated the resistance against impact of entangled sandwich materials in comparison with standard sandwiches with honeycomb and foam cores, uniquely based on vibration test results (i.e., decrease in natural frequency which signifies loss of rigidity, and increase in damping which corresponds to friction in damaged zones). Vibration tests verify the presence of high damping in the entangled sandwich specimens making them suitable for specific applications like the inner paneling of a helicopter cabin, even if the structural strength of this material is on the lower side.

Material and Specimen Six sandwich beam specimens are studied in this article, Fig. 1. The main emphasis of this article is on the two entangled sandwich beams with carbon and glass fibers as core materials. The honeycomb and foam sandwich beams are only presented for comparison purposes. Therefore comparison with standard sandwich beams is essential in order to evaluate the performance of entangled sandwich materials.

Fig. 1 The six sandwich beams tested in this article with (a) carbon prepregs and (b) glass woven fabric as skin materials Sandwich beams have entangled carbon fiber, honeycomb and foam as core materials. For the skin, three of them have unidirectional carbon-fiber/epoxy prepregs of T700/M21 (four plies each with a stacking sequence of [0/90/90/0]); the other three are made of glass woven fabric, impregnated with epoxy resin (two plies each with a total thickness of 0.5 mm containing 50 % of resin). The thickness of the skins in case of glass woven fabric is kept similar to carbon fiber skins. The honeycomb sandwich beams are made of Nomex-aramid honeycomb core (nominal cell size of 6.5 mm and a core thickness of 10 mm). In case of the foam sandwich beams, the foam core has also a thickness of 10 mm. The core, in case of the carbon entangled sandwich beam, consists of carbon fibers that are made of a yarn of standard carbon filaments having a diameter of 7 μm. The length of the carbon fibers is 10 mm and their elastic modulus is 240 GPa. In case of the glass entangled sandwich beam, the core consists of glass fibers made of glass filaments having a diameter of 14 μm. The length of the glass fibers is 10 mm with an elastic modulus of 73 GPa. For the cross-linking of carbon and glass fibers, epoxy resin and injection hardener are used. A better vaporization is achieved if the resin is heated up to 35°C before being sprayed on the carbon and glass fibers. This allows the mixture of resin and hardener to become less viscous. The fabrication of these entangled sandwich beams is a relatively complex process. The carbon and glass fibers are cut with the help of a fiber-cutting machine. A blow of compressed air then separates the fibers. The fibers vaporized by the resin are then placed in the mold between the two skins. A fiber core density of approximately 200 kg/m3 is chosen for the entangled sandwich core in case of both carbon and glass fibers.

173

Experimental Methods High quality vibration tests The experimental equipment used for vibration testing is shown in Fig. 2. The experimental set-up is that of a free-free beam excited at its center, based on Oberst method [17]. The test specimen is placed at its center on a force sensor which is then assembled on a 100N electrodynamics shaker. The response displacements are measured with a non-contact and high precision laser vibrometer. The shaker, force sensor and the laser vibrometer are manipulated with a data acquisition system.

Fig. 2. Diagram of the experimental set-up Each specimen is tested with two types of excitations: burst random and sine dwell. Response is measured at 27 points that are symmetrically spaced the length of the beam to have reliable identifiable mode shapes. Burst random excitation permits to have rapidly the overall dynamic response of the structure (natural frequencies and mode shapes). It is a broadband type excitation signal. 50% burst percentage is used for burst random excitation in the frequency range of 0-2650Hz. However if we need precise damping measurements then sine-dwell testing becomes inevitable. Its advantage is its capability of detecting non-linear structural dynamic behavior. As this method requires larger acquisition times, acquisition is carried out only around the first four bending modes previously identified by burst random testing. The modal parameters are extracted for burst random and sine-dwell testing respectively. The estimation method is a new non-iterative frequency domain parameter estimation method based on weighted least squares approach. One of the specific advantage of this technique lies in the very stable identification of the system poles and participation factors as a function of the specified system order, leading to easy-to-interpret stabilization diagrams. Low Velocity Impact tests The sandwich beams tested in this article are damaged by drop weight impacts below the Barely Visible Impact Damage limit (BVID), in order to simulate damage by foreign impact objects. The impact tests are carried out by a drop weight system as shown in Fig. 3-a, and a detailed cut away of the drop assembly is shown in Fig. 3-b. The impactor tip has a hemispherical head with a diameter of 12.7 mm. A force sensor is placed between the impactor tip and the free falling mass of 2 kg. The velocity before the impact is measured with an optical velocity sensor from which the energy of impact can be verified.

174

Fig. 3. Arrangement of the test equipment for the impact test (a) and detailed cutaway of the drop assembly (b) The impact energy is chosen in such a way that each sandwich beam has approximately the same level of damage (i.e., below the BVID limit which is nearly invisible on the surface). The impact parameters and the indentation depths measured for the six sandwich beams are listed in Table 1. The data obtained during the drop weigh impact tests carried out on the beams are presented in Fig. 4. In Fig. 4-a and 4-b, the impact forces are drawn as a function of time for the six sandwich beams. These curves are globally smooth and almost sinusoidal at low impact energy. It can also be seen, from the force-displacement plots (Fig. 4-c and 4-d), that after the first damage in the classical sandwiches with honeycomb and foam cores, there is a decrease in the force signal followed by oscillations which signifies damage and loss of rigidity in the material. But in case of both glass and carbon fiber entangled sandwich beams (Fig. 4-c and 4-d), after the appearance of first damage the material continues to rigidify which is shown by a progressive increase of force signal. It can be observed from the force displacement curves of the honeycomb and foam core sandwiches (Fig. 4-c and 4-d) that the energy dissipation seems mostly due to the rupture mechanism. However in case of entangled sandwiches the behavior is different, it is possible that the energy dissipation might be predominantly due to damping as no oscillations or force signal loss is observed (i.e., no apparent damage signs).

Fig. 4. Impact test data (a,b) force-time (c,d) force-displacement for the carbon and glass sandwich beams

175 Table 1. Impact test parameters of the sandwich beams Type of Specimen

Energy of Impact

Indentation just after impact (mm)

Velocity of impact (m/s)

(J)

Point 1

Point 2

Entangled Carbon

4.9

0.1

0.15

2.21

Honeycomb Carbon

3.9

0.1

0.5

1.98

Foam Carbon

3.8

0.1

0.2

1.98

Entangled Glass

6.2

0.2

0.15

2.49

Honeycomb Glass

3.8

0.2

0.4

1.98

Foam Glass

3.8

0.15

0.25

1.98

Shift in modal parameters due to damage The tested sandwich beams have two states. First one is the undamaged state (UD) and the second is the damage state due to two impacts (D1). Vibration tests are carried out on the six sandwich beams after each of these two states. The effect of impact damage on the modal parameters of the three types of sandwich beams is studied in the following sections with the help of frequency and damping changes between the undamaged (UD) and the damaged case (D1) defined by Eq. 1 and 2: Change in frequency between UD and D1, (Δf) =

f UD (k) − f D1 (k) f UD (k)

Change in damping between UD and D1, (Δ ζ ) =

ζ D1 (k) − ζ UD (k) (2) ζ UD (k)

(1)

where fUD(k) is the damped natural frequency for the undamaged specimen for the kth mode and fD1(k) is the damped natural frequency for the specimen damaged at two impact points (D1) for the kth mode.

Fig.5. Sandwich test beams with location of the two impact points, 27 measurement points and one excitation point (Point 14) The effect of physical properties on poles in the complex s-plane is illustrated in Fig. 6. It can be observed that a change in stiffness affects only the frequency, while changes in mass and structural damping affect both modal damped frequency (ωd) and modal damping ( σ ). For this study, the primary interest is to study the decrease in the modal damped frequency (ω d) and the increase in modal damping ( σ ) due to damage in the sandwich specimens [17].

176

Fig. 6. Movement of pole due to mass stiffness and damping effect

Results and discussions Effect of impact damage on modal parameters Frequency and damping ratios are the global parameters of the specimen, and are extracted from measurements carried out on the 27 measurement points. The modal parameters (natural frequency and damping) help in monitoring globally the health of a specimen. As a result of impact damage, there is a decrease in the natural frequencies for the six sandwich beams as discussed before in section 3.3. It can be noticed that this decrease is less prominent in case of both the carbon and glass entangled sandwich beams as compared to the honeycomb and foam sandwich beams. But the interesting fact is that for all the sandwich beams, even as the impact damage does not produce a visible damage on the surface, the change in frequency between the undamaged and the damaged cases is quite noticeable. This proves that there is a notable loss of rigidity without any signs of damage on the surface. The fact that the change in natural frequency between the damaged and the undamaged case is small in the entangled beams as compared to the foam and honeycomb beams can be seen in Fig. 7, which presents a comparison between the frequency response functions of the undamaged and the damaged cases for the six sandwich beams for the 1st bending mode. The frequency response functions presented in Fig. 10 are obtained with the sine-dwell testing. For the six sandwich beams, the shift in natural frequencies is slight in case of entangled sandwich beams proving that they have a loss of rigidity that is less pronounced as compared to the honeycomb and foam sandwich beams. Furthermore in case of the three carbon sandwich beams (Fig. 7-a), the frequency response functions of the carbon entangled sandwich beam are more acute (smaller in width) as compared to the honeycomb and foam sandwich beams. This phenomenon is less evident in case of the glass entangled sandwich beams (Fig. 7-b).

Fig. 7 Comparison of the frequency response functions estimated by sine-dwell testing for the undamaged case (UD) and damaged at 2 points (D1) for the 1st bending mode for (a) Point 11 for the three carbon sandwich beams and for (b) Point 21 for the three glass sandwich beams

177 Therefore it can be concluded that the entangled sandwich beams show a better resistance to impact as compared to the honeycomb and foam sandwich beams, whereas all the beams have more or less the same level damage. The shift in modal parameters is less in case of the entangled sandwich beams, which signifies that they possess better impact toughness as compared to the standard sandwich beams with honeycomb and foam as core materials. Furthermore, the results underline the fact that the damping change ratios are more prominent than the frequency change ratios. The maximum damping change ratio is 310 % whereas the maximum frequency change ratio is 25 %. It can be concluded from the above results that damping seems more sensitive to damage than the natural frequency variations in case of honeycomb sandwich beams. So it is reasonable to assume that damping may be used instead of natural frequency as a damage indicator tool for structural health monitoring purposes. However, the fact that damping is a parameter that is relatively difficult to estimate as compared to natural frequency has to be taken into account.

Conclusion The aim of the article is to evaluate the resistance against impact of entangled sandwich beams by the pole shift which signifies loss of rigidity (decrease in natural frequency) and increase in friction in damaged zone (increase in damping ratio). A simple case of symmetrical impacts is studied and Low Velocity Impacts are done below the BVID limit in order to detect damage by vibration testing that is hardly visible on the surface. From the results it can be concluded that both carbon and glass fiber entangled sandwich beams show a better resistance to impact (based only on decrease in natural frequency) as compared to the honeycomb and foam sandwich beams, whereas all the beams have more or less the same level damage.

References [1] Doebling SW, Farrar CR, Prime MB. A summary review of vibration-based damage identification methods. Shock and Vibration Digest, 30, 91-105, 1998. [2] Yan YJ, Cheng L, Wu ZY, Yam LH. Development in vibration-based structural damage detection technique, Mechanical Systems and Signal Processing, 21, 2198-2211, 2007. [3] E.P. Carden EP, P. Fanning P. Vibration based condition monitoring: A review, Structural Health Monitoring, 3(4), 355–377, 2004. [4] Khoo LM, Mantena PR, Jadhav P. Structural damage assessment using vibration modal analysis. Structural Health Monitoring, 3(2), 177-194, 2004. [5] Gadelrab RM. The effect of delamination on the natural frequencies of a laminated composite beam. Journal of Sound and Vibration, 197(3), 283-292, 1996. [6] Yam LH, Cheng L. Damage detection of composite structures using dynamic analysis. Key Engineering Materials, 295-296, 33-39, 2005. [7] Shahdin A, Morlier J, Gourinat Y. Correlating low energy impact damage with changes in modal parameters: A preliminary study on composite beams. Structural Health Monitoring 8(6), 523-536, 2009. [8] Adams RD. Damping in composites. Material Science Forum, 119-121, 3-16, 1993. [9] Gibson RF. Modal vibration response measurements for characterization of composite materials and structures, Composites Science and Technology, 60, 2769-2780, 2000. [10] Saravanos DA, Hopkins DA. Effects of delaminations on the damped dynamic characteristics of composites. Journal of Sound and Vibration, 192, 977-993, 1995. [11] Richardson MOW, Wisheart MJ. Review of low-velocity impact properties of composite materials. Composites Part A, 27, 1123-1131, 1996. [12] Dear JP, Lee H, Brown SA. Impact damage processes in composite sheet and sandwich honeycomb materials. International Journal of Impact Engineering, 32, 130–154, 2005. [13] Vaidya UK, Pillay S, Bartus S, Ulven C, Grow DT, Mathew B. Impact and post-impact vibration response of protective metal foam composite sandwich plates. Materials Science and Engineering A, 428, 59–66, 2006. [14] Hosur MV, Abdullah M, Jeelani S. Manufacturing and low-velocity impact characterization of foam filled 3-D integrated core sandwich composites with hybrid face sheets. Composite Structures, 69(2), 167-181, 2005. [15] Sutherland LS, Guedes Soares C. Impact characterisation of low fiber-volume glass reinforced polyester circular laminated plates. International Journal of Impact Engineering,, 31(1), 1-23, 2005. [16] Abdullah MR, Cantwell WJ. The impact resistance of polyporopylene based fiber-metal laminates. Composites Science and Technology, 66(11-12), 1682-1693, 2006. [17] Wojtowicki JL, Jaouen L. New approach for the measurements of damping properties of materials using oberst beam. Review of Scientific Instruments, 75(8), 2569-2574, 2004.

AN INTEGRATED PROCEDURE FOR ESTIMATING MODAL PARAMETERS DURING FLIGHT TESTING William Fladung Gordon Hoople ATA Engineering, Inc. 11995 El Camino Real, Suite 200 San Diego, California 92130

ABSTRACT This paper presents the methodology used for the flight test conducted as part of the gross weight increase program for NASA’s WB-57F aircraft. The primary focus is on the synergy of data processing procedures to track the trends of the frequency and damping of the modes of interest in near-real time during the flight test. These procedures include spatial filtering with FEM mode shapes, time domain parameter estimation, and a pole density diagram. This method can be used to estimate modal parameters from free-decay response-only measurements using only a few acceleration response measurements on the aircraft. The authors will show that with this method, a robust estimate for frequency and damping can be extracted and statistics of the estimated values can be evaluated at each flight test point. The paper will also discuss the preliminary testing and analyses as well as the overall logistics of the flight test program, including instrumentation selection and installation, data acquisition and transfer, and operational considerations.

INTRODUCTION ATA Engineering, Inc., (ATA) recently completed work on a flight test program for NASA Johnson Space Center (JSC) Aircraft Operations Division. ATA provided engineering support for a series of flight tests on the WB-57F aircraft for several increased gross weight configurations. The WB-57F began its life in the 1940s as an English Electric Canberra B.2, a post-World War II fighter/bomber. This design was licensed to Martin for U.S. manufacturing and renamed the B-57A. The next iteration, the B-57B, added guns and a new cockpit for close air support. This aircraft was then repurposed for reconnaissance as the RB-57. General Dynamics then developed the RB-57F for high altitude reconnaissance, a design which almost doubled the original wing span. Finally, the aircraft was repurposed for weather reconnaissance as the WB-57F. Today there are only two WB-57Fs in operation, both operated by NASA out of Ellington Airfield in Houston, Texas. The WB-57F now serves as a high altitude research aircraft that can fly in excess of 60,000 feet for approximately 6.5 hours. The aircraft can carry payloads at multiple locations throughout the fuselage as well as at four specially designed pods that mount on the wings, as shown in Figure 1.

T. Proulx (ed.), Advanced Aerospace Applications, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 4, DOI 10.1007/978-1-4419-9302-1_16, © The Society for Experimental Mechanics, Inc. 2011

179

180

Spearpod Superpod

Pallet Payloads Figure 1 – WB-57F aircraft. ATA’s involvement in the WB-57F program began in 2005 with a feasibility study to determine whether the aircraft could support a gross weight increase. The need for a gross weight increase was driven by the desire for increased flight range, endurance, and additional payload. Over the next five years, ATA conducted a three-phase program culminating in a flight test. The first phase of the program was information gathering. In this phase, the program requirements were developed, the load cases defined, and a ground vibration test conducted. The second phase of the program was analysis. Detailed finite element and aerodynamic models were developed and analyzed for maneuver, gust, and turbulence loads. The current flight envelope was evaluated and modifications were explored in conjunction with a flight stability analysis. The third phase of the program consisted of final test verification and substantiation documentation. A second ground vibration test was conducted with updated hardware and, finally, flight testing was performed. The flight test, conducted in March of 2010, consisted of multiple configurations, which are presented in Table 1. Flight testing was required to verify ground testing and analysis and to establish a safe operational flight envelope for the various aircraft configurations and weights above the original maximum gross weight. The primary objective of the flight test program was to identify and track the frequency and damping of the critical flutter mode of the structure across multiple configurations and flight points (speed and altitude). An instrumentation checkout flight and four configurations were completed in five days of testing and are described in Table 1. This aggressive schedule of one flight per day was met by performing all configuration changes during a night shift. Flights lasted three to five hours, and two altitudes, 35,000 and 50,000 ft, were tested per flight. Nominally, there were ten flight points per altitude; however, this number was reduced because of known errors in the flight instruments near Mach number and velocity limits. Table 1 – Configurations for the WB-57F flight test. Configuration Checkout C1 C2 C3 C4

Pallet Payload x x x x

Spearpod Superpod

x x

x x

The primary focus of this paper will be on the synergy of data processing procedures to track the trends of the frequency and damping of the modes of interest in near-real time during the flight testing. The first section of the paper will describe instrumentation, the next will discuss data collection, and the final section will present data processing.

181

TEST INSTRUMENTATION The flight test involved applying an excitation to the test article and measuring the acceleration responses at the instrumented locations. In order to make these measurements, additional hardware was installed on the aircraft. A ruggedized data acquisition system and ten DC accelerometers were installed in the aircraft; locations are shown in Figure 2. The WB-57F is configured as a platform for science experiments and is equipped with a standard Windows XP computer. This provided a simple method to allow an onboard instrumentation officer to monitor data acquisition in real time. In addition, a telemetry system was configured to allow engineers on the ground to review and process data during the flight.

eDAQ Power Switch

GPS Antenna

eDAQ Trigger Switch

Cockpit

SOMAT eDAQ

Bulkhead

On Board Computer

DC Accelerometers Throughout Aircraft

Telemetry System

Nose Accelerometer

Aircraft Nose Figure 2 – Test instrumentation overview.

A sixteen-channel SoMat eDAQ data acquisition system, shown in Figure 3, was used to record time response data from ten accelerometers as well as GPS data during flight testing. The eDAQ is a rugged system, which was able to meet the harsh temperature and vibration requirements seen during flight. The eDAQ was installed in the partially pressurized nose of the aircraft and was powered directly with DC power from the aircraft. Remote switches were wired from the cockpit to the eDAQ to allow the instrumentation officer to both power on the unit and trigger data acquisition. The eDAQ software was set up and controlled with a Web interface from the ground over a satellite connection.

182

Figure 3 – The data acquisition system installed in the nose of the plane. Ten PCB 3741D4HB30G DC accelerometers were installed on the aircraft for flight testing; the locations are shown in Figure 4. Accelerometer locations were chosen based on the ground vibration tests to optimally characterize modes relevant to flight stability. All of the accelerometers installed for the flight test were oriented to measure vertical accelerations perpendicular to the plane of the wing.

2

1

Accelerometers Data Acquisition System

3

4

10 9 8 7

5

6

Figure 4 – Test instrumentation locations. Accelerometer locations are marked with yellow circles, and the data acquisition system is marked with a green triangle.

DATA COLLECTION There were two main challenges associated with data collection for this flight test. The first challenge was to excite a modal response of the aircraft—something that had been difficult to accomplish in the original flight testing in the 1960s. The instrumentation checkout flight was used to evaluate multiple excitation methods. In the end, two methods proved successful: sine dwell and decay and stick pulse. For the sine

183

dwell and decay method, the pilot moved the stick in a sinusoidal fashion to induce a pitching motion in the aircraft that matched the frequency of the first wing bending mode. After a few cycles of exciting the aircraft, the stick was released and the decay monitored. The second excitation method used was to apply a stick pulse to impart an impulsive excitation that provides broadband excitation to the system. Details about how the data were processed are provided in the next section. The second challenge of data collection was to transfer the data from the aircraft to the ground and disperse them quickly to engineers at multiple locations. While data were collected and monitored in real time on the aircraft, the final engineering judgment about the results was made by a team on the ground scattered over several states. The data acquisition system was primarily controlled from the ground over the satellite telemetry link, but acquisition was triggered by a switch in the aircraft. After completing each data point, the aircraft radioed the ground and the data was downloaded through the satellite telemetry link to engineers in Houston. Once downloaded, time domain data were reviewed on the data acquisition computer. Simultaneously, a second engineer processed the data on another computer in order to extract frequency and damping information. This second computer screen was displayed on a large-screen television and also broadcast via WebEx to allow other engineers in Houston, San Diego, and Phoenix to all assist in interpreting the data. Finally, a third computer in Houston, which was connected to a projector, was used to track data trends as the flight testing progressed. In addition, a conference call was ongoing throughout the flight test for communication with the flight director, located at air traffic control, as well as with the participants who were not at the airfield. This flow of information is summarized in Figure 5. IDAN Computer

SoMat eDAQ

Network Switch

CISCO 876 Router (w/ ISDN S/T)

WB-57 INMARSAT Antenna

Aircraft INMARSAT (satellite)

Ground

ISDN Landline

CISCO 876 Router (w/ ISDN S/T)

Data Acq Computer

USB

Estimation Computer

Figure 5 – Data transmission and review process.

Trending Computer

184

DATA PROCESSING At each flight point, the measurements were first reviewed in the time domain and then processed to obtain frequency and damping values for the primary mode of interest—the first wing bending at approximately 1.25 Hz. Time domain data were downloaded immediately after each flight point using the telemetry system and imported into MATLAB using ATA’s IMAT© software. A special set of tools was created for the IMAT spFRF application to facilitate reviewing the flight data and extracting modal information. Time domain data were first reviewed without performing any signal processing to ensure good measurement quality. Figure 6 shows a wingtip accelerometer response that contains five distinct events, indicated by the orange rectangles: four stick pulses and a sine dwell. The power spectral density (PSD) of the time segment marked by the gray frame cursor is plotted below the time response. Sine dwell and free decay

1g offset of DC accelerometers

Stick pulses

200 time point frame

Auto spectra of time segment in gray frame

Figure 6 – Measured response from a wingtip accelerometer. A series of postprocessing operations was performed to facilitate the extraction of frequency and damping using a time domain modal parameter estimation algorithm. The first was to remove the 1-g offset from the DC accelerometers’ response by simply subtracting the mean of the signals, as shown in Figure 7. However, the low frequency force response signal due to aircraft control inputs remained, so a 0.5-Hz high-pass filter was instead used to remove these components as well as the DC offset. The filter was a four-pole, Butterworth, infinite impulse response implemented as a zero-phase forward and reverse digital filter. The resulting output from this filter is shown in Figure 8. Removing these components not only helped to make the signal more intelligible from a visual perspective, but also reduced the effects of the low frequency content in the parameter estimation.

185

Flight correction inputs

Removed DC frequency content

Figure 7 – 1-g DC offset removal for a wingtip response.

0.5 Hz high-pass filter

Figure 8 – High-pass filter applied to a wingtip response. The next step was to spatially filter the measurements to enhance the mode of interest, which was the wing first bending mode. The corresponding mode shape from the finite element model was used as the spatial filter. The objective of spatial filtering is to create a virtual channel that is a linear combination of the measured channels, the weighting of which are the coefficients of the mode shape used as the spatial filter. The effect of this weighted combination of measured channels is to enhance the mode of interest by diminishing the contributions of the other modes to the response. The spatially filtered channel was used to estimate the frequency and damping of the mode of interest. The resulting output from the spatial filter

186

is shown in Figure 9. As can be seen from the PSD, the effect of spatial filtering is to diminish the magnitude of the higher frequency modes. The effect in the time domain is a reduction of the high frequency content added to the response of the primary mode of interest, which can be considered as noise in the identification of the frequency and damping of this mode.

Spatial filter channel

Figure 9 – Spatially filtered response for wing first bending mode. For most of the flight points, the free decay from the sine dwell input was used to extract the modal parameters. The accelerometer located on the aft end of the fuselage most clearly identified where the sine dwell excitation stopped and the free decay began. The cursor in Figure 10 is positioned at the point in time when the structure is just beginning to enter free decay and indicates the start of the time range that was processed to estimate the modal parameters. The spatially filtered channel was used to determine when the free decay had damped out, and the cursor indicating the end of the time range was positioned accordingly, as shown in Figure 11.

Start time for parameter estimation

Figure 10 – Selection of parameter estimation start time from fuselage aft response free decay.

187

End time for parameter estimation

Figure 11 – Selection of parameter estimation end time from spatially filtered response free decay. For a stick pulse excitation, the start of the parameter estimation time range was selected at the end of the input pulse from the aft fuselage response, as shown in Figure 12, and the end of the time range was selected from the spatially filtered channel. End of stick pulse

Figure 12 – Selection of the parameter estimation time block for a stick pulse input. The poles (frequency and damping) were computed from the spatially filtered response in the selected time range using the least squares complex exponential (LSCE) algorithm [1], which is a high-order, scalar polynomial, time domain modal parameter estimation method. In the interest of programmatic simplicity and computational speed, this algorithm was implemented for a single response for the flight test processing. Though slightly longer than three lines in MATLAB, the code to implement the LCSE algorithm for this specific application, listed in Figure 13, is rather straightforward and quite fast.

188

function s = lsce_siso(y,dt,Np) % % % % % %

S = LSCE_SISO(Y,dt,Np) S Y dt Np

= = = =

poles free decay response time interval number of poles

% define sizes m = 2*Np; N = length(y); Ns = N-m;

% model order % number of time points % number of time shifts

% allocate data matrices YY = zeros(m,Ns); Yn = zeros(1,Ns); % fill data matrices for kk = 1:Ns YY(:,kk) = y(kk+m-1:-1:kk); Yn(:,kk) = y(kk+m); end % solve for coefficients a = -Yn/YY; % form companion matrix C = [-a;eye(m-1),zeros(m-1,1)]; % solve eigenvalues of companion matrix z = eig(C); % convert eigenvalues to s-domain s = log(z)/dt;

Figure 13 – MATLAB function to implement LSCE algorithm for a single response. The time range was processed in a number of overlapped frames. The frame size was 150–200 time points and, depending on the duration of time range, ten to 25 frames were processed with an overlap of over 90%. For each frame, the algorithm was run for model orders yielding a minimum of one pole to a maximum of twenty to thirty poles. This procedure generates a large number of pole estimates, which are plotted on a pole density diagram [2] as shown in Figure 14. This is essentially a plot of the poles on the s-plane, where the imaginary axis is plotted as the X axis in Hertz and the real axis is plotted as the Y axis in percent critical damping. The blue line plotted on the pole density diagram is an overlap-averaged PSD used as a mode indicator function. The idea behind the pole density diagram is that the poles corresponding to the underlying dynamics of the system will be repeatedly estimated for successive model orders and time frames. And while these multiple pole estimates will not have exactly the same frequency and damping values, the system poles should form a high-density cluster on the pole density diagram. The remaining poles generated by the overdetermined parameter estimation model are computational poles, which are attributable to noise in the measured data. These computational poles are not expected to be consistent for successive model orders and time frames and should have a low density. The density of each pole is determined by defining a frequency tolerance (e.g., 1%) and a damping tolerance (e.g., 10%). The density of each pole is equal to the number of poles that lie within the rectangle defined by these tolerances. The pole densities then determine the color of the round markers in the plot.

189

The color map for the pole density diagram, which is shown on the right of Figure 14, was chosen such that low-density poles fade into the white background, while high-density poles appear as a dark cluster. There were several reasons the pole density diagram was implemented instead of using a traditional stability diagram available in most modal parameter estimation software packages. First, the modal parameters were going to be estimated with response-only measurements from in-flight excitation and not from noise-minimizing frequency response functions, or the equivalent impulse response functions, measured with controlled inputs. Furthermore, the institutional knowledge base of the original flight testing indicated that exciting the modes of interest may be difficult, and measuring adequate system response was a concern from the onset. As such, it was presumed that extracting consistent estimates on a stability diagram could be difficult, if at all possible—the spread of the damping estimates appears to support this expectation. Secondly, the overlapped time frame scheme was adopted in order to generate as many estimates as possible from a relatively short duration of response. A stability diagram, however, is constructed for increasing model orders for a single data block and does not readily incorporate multiple data blocks. A pole density diagram can combine pole estimates from any number of algorithm solutions into one display. Lastly, with a stability diagram, only one pole for each mode is selected and all of the other estimates are discarded. This is an acceptable procedure when the frequency and damping of the poles fall within a relatively tight tolerance (e.g., 1% for frequency and 10% for damping are typical) and the selected pole is a valid representative of the cluster. For the flight data, the frequency estimates were relatively consistent, but as may be expected, the spread of the damping estimates was much larger—so selecting only one pole for the damping value becomes ambiguous and the uncertainty of the estimate is disregarded. Averaging the pole cluster, however, uses all of the available information and provides a standard deviation as an, admittedly qualitative, indicator of the validity of the estimates. It should be noted that while the flight test examples below are focused on extracting only one mode of interest using LSCE, the pole density diagram methodology is certainly applicable to indentifying more than one pole from the output of any parameter estimation algorithm, as demonstrated in [2].

Cluster of poles for mode of interest

Figure 14 – Pole density diagram for time range of spatially filtered response.

190

The pole for the mode of interested was selected by graphically zooming in on the cluster, as indicated in Figure 15, and then pushing a blue square button on the toolbar. This creates an averaged pole for the cluster, which has a frequency equal to the mean of the frequency values of the poles in the cluster and a damping equal to the mean of the damping values of the poles in the cluster. The blue square plotted in Figure 15 represents this averaged pole at the geometric centroid of the cluster. The standard deviation of the frequency and damping were also computed, plotted, and reported to provide a relative statistic pertaining to the spread of the estimates comprising the cluster, also shown in Figure 15. The primary focus during the flight tests was to monitor changes in damping of the first wing bending mode. As such, the parameter estimation results obtained from the pole density diagrams for each flight point were trended as damping vs. speed and frequency vs. speed plots during the flight tests. These plots also included error bars indicating the span of the standard deviations for the frequency and damping estimates. As might be expected, the variance on the damping estimates was typically larger than for the frequency estimates.

Standard deviation on frequency

Averaged pole at cluster centroid

Standard deviation on damping

Figure 15 – Zoomed region of plot cluster for primary mode of interest. The example used above to demonstrate the signal processing and parameter estimation methodology was naturally chosen because of its favorable aspects: well-excited first wing bending mode, enhanced by spatial filtering, and one tight cluster of poles for the mode of interest. As might be expected, however, not all flight points had as favorable a response. Below are a few other examples to illustrate some results for less than ideal conditions. First, shown in Figure 16 is a pole density diagram from the same flight point above, but for a measured response on the wingtip—not the spatially filtered response. The time range, frame size, frame overlap, and number of poles were the same as for the spatially filtered response in Figure 14. While there is a high-density pole cluster at the 1.25-Hz mode, it is not quite as tight as for the spatially filtered response. Also, a low-density cluster forms for the mode at 6.5 Hz. This example shows that while the modal

191

parameters could be extracted from a measured response, spatial filtering helps to concentrate the parameter estimation algorithm on the mode of interest.

Figure 16 – Pole density diagram for measured response on wingtip. Next, shown in Figure 17 is a pole density diagram for the spatially filtered response from the stick pulse shown in Figure 12. Since this type of excitation creates a broadband input, the higher frequency modes have more response than for the sine dwell input. In this case, even after spatially filtering the response, medium-density clusters form at the 4.5- and 6.5-Hz modes. However, there is a prominent high-density cluster at the 1.25-Hz mode of interest, but with a larger spread in the damping estimates than for the sine dwell input. This type of result was typical for stick pulse inputs and, consequently, the sine dwell input was used to extract the modal parameters for most of the flight points.

192

Figure 17 – Pole density diagram for a stick pulse spatially filtered response. Finally, shown in Figure 19 is a pole density diagram for the spatially filtered response of the sine dwell input from another flight point. As can be seen from the time response data shown in Figure 18, this sine dwell input is not a clean, single frequency excitation as it was for the previous examples. The pole density diagram generated from this response has a closely-grouped, high-density cluster at a higher frequency 4.5-Hz mode. Even so, meaningful frequency and damping estimates could be extracted from the cluster at the 1.25-Hz mode of interest.

Figure 18 – Spatially filtered response for sine dwell input showing time range processed to generate the pole density diagram in Figure 19.

193

Figure 19 – Pole density diagram for the sine dwell spatially filtered response shown in Figure 18.

SUMMARY This paper described the synergy between instrumentation, signal processing, and parameter estimation as implemented on the flight test of the NASA WB-57F. The data processing procedure combined temporal and spatial filtering of the response measurements to enhance the first wing bending mode, which was the primary mode of interest for flight stability of the aircraft. Overlapped time frames of free decay response were processed with the least squares complex exponential algorithm to compute estimates of the modal parameters. A pole density diagram was used to select and average the frequency and damping values. These parameters were tracked across the flight points in order to inform the decision of whether to proceed to the next flight point. The successful flight tests were the culmination of several years of work that included analysis, modeling, testing, and model correlation. During the flight tests, measured acceleration responses were telemetered via a satellite link from the aircraft to Ellington Field in Houston. Using a variety of multimedia resources, personnel in three states were involved in data processing, parameter trending, and decision making.

ACKNOWLEDGEMENTS The authors would like to express their gratitude to the staff at the NASA JSC Aircraft Operations Division for the opportunity to be involved in this project and for allowing us to share our methodology with the IMAC community.

194

REFERENCES [1] Brown, D.L., Allemang, R.J., Zimmerman, R.D., and Mergeay, M., “Parameter Estimation Techniques for Modal Analysis,” SAE Paper No. 790221, SAE Transactions, Vol. 88, 1979, pp. 828-846. [2] Phillips, A.W., Allemang, R.J., Pickerel, C.R., “Clustering of Modal Frequency Estimates from Different Solution Sets,” Proceedings, International Modal Analysis Conference, Society of Experimental Mechanics (SEM), 1994, pp. 501-514.

Multiple-site Damage Location Using Single-site Training Data

R.J. Barthorpe, K. Worden University of Sheffield, Mappin Street, Sheffield S1 3JD, United Kingdom

ABSTRACT The identification of multiple-site damage is a challenging problem in data-based structural health monitoring (SHM). It is generally accepted that higher level damage identification via statistical pattern recognition requires the adoption of a supervised learning approach, with the need for data to be gathered from the structure in all damaged states of interest. The number of states for which data would be required to cover all damage combinations grows exponentially with the number of locations at which damage may occur. Damage state data sets of this extent are unlikely to be available in practical applications. The objective of this paper is to explore an interesting approach to the problem of multiple-site damage location. It is postulated that if sufficient information can be gleaned from single-site damage data to allow identification of multiplesite damage, then the requirement to gather data for all combinations of damage location may be circumvented. In the present study this possibility is assessed using data from an experimental structure. The experimental structure used is a full-scale, laboratory-based aircraft wing section. Damage sensitive features identified using single-site data are shown to perform well when applied to the multiple-site location problem. 1.

INTRODUCTION

It is more or less accepted that damage identification at Level 2 (damage location) of Rytter’s damage identification hierarchy [1] via statistical pattern recognition requires the adoption of a supervised learning approach, thus requiring the gathering of data from the structure in its damaged state. In the particular case that damage is believed to occur at only one of a finite discrete set of k locations, this would require the data to be gathered for each of the corresponding k damage states. In the more general case of damage occurring concurrently at more than one location, the naïve approach might be to gather damage data for all combinations of damage location. The number of states for which data would be required in order to cover all combinations thus grows exponentially with the number of locations k. It is commonly accepted that damaged state data even for single site damage comes at a premium, and it would appear unwise to rely on the availability of damage data for all possible combinations of damage location in the general case. It is apparent that this lack-of-data problem, a major obstacle in the diagnosis of single-site damage, is a potentially critical issue in multiple-site damage location. Despite the multi-site damage identification task representing an important and challenging problem in SHM it is one that has received relatively little dedicated attention in the literature. Where multiple damage identification has been addressed, it has typically been done so with the aid of a law-based model. In [2-4] a variety of essentially model-based approaches are developed. It is noted that the structures investigated in these studies have typically exhibited a low level of complexity. The approach pursued in this paper is intended to allow diagnosis of multiple-site damage without resort to a law-based model or full damage state data. Only a greatly reduced subset of the full dataset corresponding to single-site damage is used. The method is demonstrated for a complex decommissioned structure. The aim of the developed approach is to dramatically reduce the number of states for which training data is required. This is pursued by examining the characteristics of features selected using observations of single-site damage, and evaluating the success of these features when presented with multiple-site damage data. The study is experimental in nature, with the structure used being the wing of a Piper Tomahawk trainer aircraft. The features used are discordancy measures, extracted from transmissibility data.

T. Proulx (ed.), Advanced Aerospace Applications, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 4, DOI 10.1007/978-1-4419-9302-1_17, © The Society for Experimental Mechanics, Inc. 2011

195

196 In §2 the wing structure and the procedure for data acquisition are described. The wing structure features five removable inspection panels and the removal of these panels is adopted as a proxy for the introduction of damage. Particular attention is paid to test sequencing in order that the effects of systematic and random test variability are represented in the datasets used for training and testing. The raw data recorded are FRF spectra from a network of 15 sensors. These spectra are used to form transmissibility functions between sensor pairs. Thirteen such transmissibility pairs are considered as a source from which features may be drawn. In §3 the extraction and selection of features from the gathered data is described. In §4 and §5 feature evaluation is pursued, first visually and latterly through the application of a quantitative summary statistic. The conclusions of the study are presented in §6. 2.

EXPERIMENTAL DATA ACQUISITION

The experimental structure considered in this study is an Aluminium aircraft wing, shown in situ in the laboratory in Figure 1. The wing is mounted in a cantilevered fashion on a substantial, sand-filled steel frame. The sensor network and data acquisition equipment may also be observed. The wing is in fact mounted upside-down in order to allow access to the inspection panels mounted on the underside of the wing. The reasons for this are explained later. The wing includes various complicating features, including stiffening elements (both accessible and inaccessible), inspection panels, riveted and welded connections, as well as auxiliary structures such as aileron mounting points.

Figure 1. Piper Tomahawk trainer aircraft wing Fifteen PCB 353B16 piezoelectric accelerometers were mounted on the upper (as mounted) surface of the wing using ceramic cement. The location of the sensors, inspection panels and sub-surface stiffening elements (dotted lines) are shown schematically in Figure 2. The sensors were placed in an ad hoc fashion on the basis of previous experience, with no formalised sensor placement optimisation undertaken. The sensors were placed so as to form transmissibility ‘paths’ between sensor pairs. These are indicated in Figure 2, with the transmissibility paths denoted T1 to T13. Placement directly above stiffening elements was avoided as it was believed that such locations would offer poor observation of localised changes in structural flexibility.

197

Figure 2. Schematic sensor placement diagram Experimental data acquisition was performed using a DIFA SCADAS III unit controlled by LMS software running on a desktop PC. All measurements were recorded within a frequency range of 0-2048 Hz with a resolution of 0.5 Hz. The structure was excited with a band-limited white Gaussian signal using a Gearing and Watson amplifier and shaker mounted beneath the wing. Both the real and imaginary parts of the accelerance FRFs were recorded at 15 response locations using single-axis accelerometers. Five-average samples were recorded in all cases as this was found to offer a good compromise between noise reduction and acquisition time. In order to introduce damage in a repeatable, and in some sense realistic, way advantage was taken of the presence of inspection panels on the underside of the wing, with the wing being mounted upside-down to enable access. Five such panels were employed in the present study. The advantages of using the removal of a panel as a proxy for damage are that it is nondestructive; it is repeatable; and the primary effect of panel removal (a localised reduction in structural stiffness) is expected to be similar to the effect of introducing gross damage at the same location. The disadvantage is that the repeatability is not perfect. During preliminary studies it was found that the removal and reattachment of the panel led to substantial variability in the FRF observations. Accordingly, care was taken in order to reduce this variability to as great an extent as possible by using a torque-controlled screwdriver and raking care over screw tightening, An attempt to accommodate the outstanding variability in the recorded data attributable to this and other sources of variability was made by introducing randomisation and repetition into the test sequence.

Figure 3. Inspection panel P2 in normal condition (left) and damaged condition (right) Two tests were conducted, resulting in two data sets being available with which to develop and test classifiers. A test sequence was developed for each test to reflect the testing objectives, and taking into consideration the sources of measurement variability identified. Randomisation and blocking were applied in the specification of the test sequences in order to account for the effects of measurement noise and panel boundary condition variability.

198 Dataset A comprises 1000 normal state observations and 1000 single-site damage state observations. The primary objective for the dataset was to allow features to be selected using observations of normal condition and single–site damage condition data only, but which are capable of generalising to the multi-site damage identification task using data from a separate test. This is a demanding objective, and particular attention was paid to full randomisation of the panel boundary condition. ‘Randomisation’, as used here, refers to the removal and replacement of panels to ensure that the latent boundary condition variation (that which is present despite the use of a torque controlled screwdriver and care over the order of screw tightening) be represented in the dataset. Dataset B comprised 2300 normal, 1000 single-site damage and 1300 multi-site damage state observations. The primary objective of this test was to provide data with which to evaluate the features selected using the single-site data of Dataset A. Randomisation was limited to the removal and replacement of individual panels, rather than full randomisation of the panel boundary conditions. For both tests, a large number of response measurements were taken with the aim of capturing random measurement variability. For Dataset A, repetition was applied to further enhance the dataset. In developing the test sequences, a balance is sought such that both random and systematic variabilities may be well represented in the datasets without exceeding a reasonable timescale for testing. 3.

FEATURE EXTRACTION AND SELECTION USING SINGLE-SITE TRAINING DATA

The objectives of the feature selection and evaluation task may be summarised in several steps. 

Extraction of a candidate feature set from raw experimental data.



Selection of a low-dimensional feature set containing features that are sensitive to single-site damage, using a training set drawn from Dataset A.



Evaluation of these features for the identification of single-site damage, using a testing set drawn from Dataset B.



Evaluation of these features for the identification of multi-site damage, using a testing set drawn from Dataset B.

Each of these steps is covered in turn. The hypothesis of this study is that it may be possible to find features that offer a good level of discrimination between multiple-site damage states, despite only single-site damage states being available for feature selection. Damage leads to the structural response of the structure deviating from that observed when it is in its initial, undamaged condition. The structural response features considered in this study are transmissibility spectra. By comparing examples of undamaged and damaged spectra, regions of the spectra that are sensitive to particular damage states may be identified. These regions form the basis of features that may subsequently be used to train a statistical damage classifier. In the interests of developing a statistical classifier, it is desirable that the feature set used is of low dimension. Achieving a suitably concise feature set requires further condensation of the identified spectral region. In this study the additional data reduction is performed using a discordancy measure - the Mahalanobis squared-distance (MSD) - between the newly-presented (and therefore potentially-damaged) state and the previously-recorded undamaged state. The resulting features are the discordancy values associated with damage sensitive regions of the transmissibility spectra. The MSD is a multivariate extension of the univariate discordancy measure. Discordancy measures allow deviations from normality to be quantitatively evaluated. A brief summary of the technique is given here: the technique is described in [5] and validated for an experimental structure in [6]. For a multivariate data set consisting of n observations in p variables, the MSD may be used to give a measure of the discordancy of any given observation. The scalar discordancy value Dξ is given by (1) where the p-vector covariance matrix.

is the potential outlier,

is the mean of the sample observations and the (p x p) matrix

is the sample

For this study a guided, manual approach was taken to feature selection, which proceeded as follows. First, discordancy plots were generated for all possible 20-spectral line windows using Dataset A. As each spectrum contains 4097 lines, 4078 such windows existed for each of the 13 spectra, resulting in a total of 53014 candidate features. Plotting the discordancy values

199 across the feature set for each of the structural states allows the sensitivity of the features to be visualised, and a greatly reduced set of candidate features to be specified. Next, the transmissibility spectra corresponding to each remaining candidate feature were inspected. This allowed a degree of ‘engineering insight’ to be exercised. Decisions on the final feature set were made using considerations relating, for example, to the perceived ‘robustness’ of each feature – a feature window displaying similar behaviour to other features in its immediate vicinity would be preferred to one that did not. By applying such considerations, a final set of 50 features was selected on the basis of Dataset A. Ten features were selected for each of the 5 panels and were labelled F1-F50. 4.

VISUAL FEATURE EVALUATION: SINGLE-SITE AND MULTI-SITE DATA

The performance of the features selected using the single-site training damage observations of Dataset A when applied to the multi-site damage observations of Dataset B were first evaluated visually. An example of the visual evaluation of feature F32 is given in Figure 4.

Figure 4. Feature evaluation results for feature F32 Figure 4 illustrates feature F32, which performed well in identifying both single-site and multiple-site removals involving panel P4. The first plot illustrates the clear distinction between the removal of panel P4 (in red) and the other states included in Dataset A (other single-site damage states in green and normal states in black) and this distinction lead to the selection of the feature. The second plot illustrates the same feature window for the transmissibility spectra of Dataset B. It is observed that the spectra behave in a very similar fashion to that found for Dataset A, both for the removal of panel P4 alone (shown as a solid line) and where panel P4 was one of several removed (shown as dashed lines). The clear distinction between the removal of panel P4 and other states is maintained. This is reflected in the discordancy values illustrated in the third plot. Observation numbers 1-2300 refer to normal state data; observations 2301-3300 are of single-site damage; and observations

200 3301-4600 refer to multiple-site damage. The feature ‘fires’ strongly for the observations of single panel removal, and similarly strongly for each of the states in which it was one of several panels removed (highlighted with a green background). This very encouraging level of performance was observed for the vast majority of the 50 identified features, and supports the stated hypothesis. The visual assessment of the selected feature windows suggests that there is similarity in the behaviour of spectra due to single-site and multi-site panel removal over some portions of the frequency ranges, and that this behaviour is reflected in the returned feature values. However, the visual assessment of a large number of features is a somewhat onerous task. A concise method for presenting the quantitative evaluation results for the selected features is instead introduced. 5.

QUANTITATIVE FEATURE EVALUATION: SINGLE-SITE AND MULTI-SITE DATA

The evaluation process generates a large amount of graphical data, and a summary statistic was sought with which to evaluate the discriminatory effectiveness of the selected features. Receiver Operating Characteristic (ROC) curves offer such a measure, and the area under the ROC curve (AUC) is adopted as a summary statistic for quantifying the discriminative ability of the selected features. An example of the ROC curve, and the area under it, for feature F46 is presented in Figure 5.

Figure 5. Area under the ROC curve for feature F46 ROC curves allow a simple benefit (true-positive rate) vs. cost (false-positive rate) analysis for two-class data. The AUC characteristic takes a value in the range [0 1], where a perfect classifier would return an AUC value of 1. The AUC has a useful interpretation as the probability that the classifier will rank a randomly selected positive instance higher than a randomly chosen negative instance [7]. This AUC is in fact the same quantity that is estimated when calculating the Wilcoxon statistic [8]. For the example of feature F46 given in Figure 5 the AUC value when discriminating between panelon and panel-off data for Dataset B was 0.976. This is taken to represent an excellent level of discrimination between classes. For each of the 50 features identified using Dataset A, two ROC curves were calculated. The first curve illustrates the ability of the feature to separate normal data from single-site damage data for Dataset B. The second illustrates the ability of the feature to separate normal data from multi-site damage data for Dataset B. The area under each of the ROC curves was calculated to give an AUC value. The desired outcome is that the individual features would display a good degree of separability, evidenced by an AUC value of close to 1, for both cases. The AUC values calculated for the discrimination of single-site and multi-site damage for each of the five panels are summarised in Table 1.

201

Panel

Mean Area Under Curve Single-Site Multi-Site

P1 P2 P3 P4 P5

0.970 0.999 0.991 0.993 0.933

0.970 1.000 0.993 0.997 0.867

Overall

0.977

0.966

Table 1. Summary of areas under ROC curves, Dataset B Perfect (i.e. AUC=1) classification was achieved by 18 of the 50 individual features for single-site panel removal, and 19 of the individual features for multi-site panel removal. Some have worked exceptionally well, others less so. Some of the features ‘fired’ unexpectedly when presented with normal condition observations. A small degree of inter-test variability between Datasets A and B is observed through comparison of normal condition spectra for these features. This finding serves to reiterate the importance of gathering a training set that is truly representative of the conditions that may be encountered. 6.

CONCLUSIONS

Overall, it was found that the individual features selected on the basis of the training Dataset A (comprising normal and single-site panel removal data) performed very well when presented with single-site panel removal data from a previously unobserved dataset. This result, while expected, serves to vindicate the focus on test sequencing in §2. Of primary interest, however, is the finding that the features also performed very well when presented with multi-site panel removal data, despite no multi-site data being used for feature selection. The multi-site performance of the classifiers averaged over the 50 selected features is very close to that when presented with single-site data, and is evidenced both through visual inspection and through applying quantitative measures. The conclusion drawn is that the study supports the hypothesis stated at the outset: for the structural states investigated, it has been possible to find features that offer a high degree of discrimination between multiple-site damage states despite only the single-site damage states having been observed. The significance of this result is the suggestion that the problem of the explosion in the number of damage states that must be observed in order to build a classifier capable of identifying multiple-site damage may be circumvented in some circumstances. It is, of course, not possible to draw general conclusions on the basis of one case study, and further investigation into the validity of the approach is warranted. In this paper, focus has been on the statistical performance of the selected features. In order to assess the broader validity of the approach, a better understanding of the physical basis for similarities in feature behaviour between single- and multi-site damage states must be sought. REFERENCES [1] Rytter A. Vibration Based Inspection of Civil Engineering Structures [PhD Dissertation]: Aalborg University, Denmark; 1993. [2] Ruotolo R, Surace C. Damage assessment of multiple cracked beams: Numerical results and experimental validation. Journal of Sound and Vibration. 1997;206(4):567-88. [3] Lin R-J, Cheng F-P. Multiple crack identification of a free-free beam with uniform material property variation and varied noised frequency. Engineering Structures. 2008;30(4):909-29. [4] Contursi T, Messina A, Williams EJ. A multiple-damage location assurance criterion based on natural frequency changes. Journal of Vibration and Control. 1998;4(5):619-33. [5] Worden K, Manson G, Fieller NRJ. Damage detection using outlier analysis. Journal of Sound and Vibration. 2000 Jan;229(3):647-67. [6] Worden K, Manson G, Allman D. Experimental validation of a structural health monitoring methodology: Part I. Novelty detection on a laboratory structure. Journal of Sound and Vibration. 2003 Jan;259(2):323-43. [7] Fawcett T. An introduction to ROC analysis. Pattern Recognition Letters. 2006;27(8):861-74. [8] Hanley JA, McNeil BJ. The Meaning and Use of the Area under a Receiver Operating Characteristic (Roc) Curve. Radiology. 1982;143(1):29-36.

Assessment of Nonlinear System Identification Methods using the SmallSat Spacecraft Structure

G. Kerschen Space Structures and Systems Laboratory (S3L), Structural Dynamics Research Group Department of Aerospace and Mechanical Engineering, University of Lige, Belgium

L. Soula, J.B. Vergniaud Astrium Satellites, Toulouse, France

A. Newerla European Space Agency (ESTEC), Noordwijk, The Netherlands

ABSTRACT In this paper, several techniques for nonlinear system identification are applied to a real-world structure, the SmallSat spacecraft structure developed by EADS-Astrium. This composite structure comprises two vibration isolation systems, one of which possesses mechanical stops. The loading case considered in the present study is a random (local) excitation. A careful progression through the different steps of the system identification process, namely detection, characterization and parameter estimation, is carried out. Different methods are applied to data resulting from numerical experiments, without having access to the finite element model which generated these data.

1

INTRODUCTION

Nonlinear structural dynamics has been studied for a relatively long time, but the first contributions to the identification of nonlinear structural models date back to the 1970s. Since then, numerous methods have been developed because of the highly individualistic nature of nonlinear systems [1, 2] . A large number of these methods were targeted to single-degree-of-freedom (SDOF) systems, but significant progress in the identification of multi-degree-of-freedom (MDOF) lumped parameter systems has been realized during the last ten or twenty years. To date, simple continuous structures with localized nonlinearity are within reach. Among the well-established methods, there exist • time-domain methods such as the restoring force surface (RFS) exogeneous input (NARMAX) methods [4, 5] ;

[3]

and nonlinear auto-regressive moving average with

• frequency-domain methods such as the conditioned reverse path (CRP) of the output (NIFO) methods [7] ;

[6]

and nonlinear identification through feedback

• time-frequency analysis methods such as the wavelet transform (WT) [8] . Nonlinear system identification is an integral part of the model validation process, and, as outlined in Figure 1, it can be viewed as a succession of three steps, namely detection, characterization and parameter estimation. Once nonlinear behavior is detected,

T. Proulx (ed.), Advanced Aerospace Applications, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 4, DOI 10.1007/978-1-4419-9302-1_18, © The Society for Experimental Mechanics, Inc. 2011

203

204 a nonlinear system is said to be characterized after the location, type and functional form of all the nonlinearities throughout the system are determined. The parameters of the selected model are then estimated using linear least-squares fitting or nonlinear optimization algorithms depending upon the method considered. The objective of this paper is twofold. First, numerical experiments of a complex real-world example, the SmallSat spacecraft structure, are considered, and a careful progression through the different steps of the system identification process is carried out. Second, different methods are used, and their performance are compared.

1. Detection

Y es or N o ?

Aim: detect whether a nonlinearity is present or not (e.g., Yes)

? 2. Characterization

W hat ? W here ? How ?

Aim: a. determine the location of the non-linearity (e.g., at the joint) b. determine the type of the non-linearity (e.g., Coulomb friction) c. determine the functional form of the non-linearity [e.g., fN L (y, y) ˙ = α sign(y)] ˙

? 3. Parameter estimation

How much ?

Aim: determine the coefficient of the non-linearity (e.g., α = 5.47)

? fN L (y, y) ˙ = 5.47 sign(y) ˙ at the joint

Figure 1: Nonlinear system identification process.

2

SMALLSAT SPACECRAFT STRUCTURE

The SmallSat structure has been conceived as a low cost structure for small low-earth orbit satellite [9] . It is a monocoque tube structure which is 1.2m long and 1m large. It incorporates eight flat faces for equipment mounting purposes, creating an octagon

205 shape, as shown in Figure 2(b). The octagon is manufactured using carbon fibre reinforced plastic by means of a filament winding process. The structure thickness is 4.0mm with an additional 0.25mm thick skin of Kevlar applied to both the inside and outside surfaces to provide protection against debris. The interface between the spacecraft and launch vehicle is achieved through four aluminium brackets located around cut-outs at the base of the structure. The total mass including the interface brackets is around 64kg. Telescope dummy

6 Top floor

6

(a)

(b)

U

WEMS fixed part

6

WEMS mobile part

6

-

NC1

q ?

- NC2

NC4

R NC3 (c)

(d)

Figure 2: SmallSat structure. (a) Finite element model; (b) real structure without the WEMS module; (c) WEMS module mounted on a bracket and supporting a dummy reaction wheel and (d) close-up of the WEMS module; NC stands for nonlinear connection.

The SmallSat structure supports a telescope dummy composed of two stages of base-plates and struts supporting various concentrated masses; its mass is around 140kg. The telescope dummy plate is connected to the SmallSat top floor via three shock attenuators, termed SASSA [10] , the behavior of which is considered as linear in the present study. The top floor is a 1 square meter sandwich aluminium panel, with 25mm core and 1mm skins. Finally, as shown in Figure 2(c), a support bracket

206 Spring Axial Lateral

Clearance cz =1 cx =1.27

Stiffness of the mechanical stop 1 0.39

TABLE 1: Nonlinear spring characteristics (adimensional values for confidentiality).

connects to one of the eight walls the so-called wheel elastomer mounting system (WEMS) device which is loaded with an 8 kg reaction wheel dummy. The purpose of this device is to isolate the spacecraft structure from disturbances coming from reaction wheels through the presence of a soft interface between the fixed and mobile parts. In addition, mechanical stops limit the axial and lateral motion of the WEMS mobile part during launch, which gives rise to nonlinear dynamic phenomena. Figure 2(d) depicts the WEMS overall geometry, but details are not disclosed for confidentiality reasons. The finite element (FE) model in Figure 2(a) was created in Nastran and is used in the present study to conduct numerical experiments. The comparison with experimental measurements revealed the good predictive capability of this model. The WEMS mobile part was modeled as a rigid body, which is connected to the WEMS fixed part through four nonlinear connections, labeled NC1-4 in Figure 2(d). Each nonlinear connection comprises two nonlinear springs, namely one axial spring and one lateral spring, possessing piecewise linear characteristics to model the mechanical stops. The springs characteristics are listed in Table 1. Random excitation was applied locally at the reaction wheel dummy in the axial direction. It is a white noise sequence band-limited into the 0-60 Hz range, as illustrated in Figure 3. This frequency range encompasses 12 elastic modes of the structure, including the first lateral bending modes, the first axial mode and local modes of the WEMS device. Three different excitation amplitudes were considered, namely low (LL, 200N), intermediate (IL, 1000N) and high (HL, 2000N) levels.

Power spectral density (dB)

50

0

−50

−100

−150 0

100

200

300

Frequency (Hz)

400

500

Figure 3: Power spectral density of the random excitation at low level.

3 3.1

NONLINEARITY DETECTION Homogeneity Test

The detection of structural nonlinearity is the first step toward establishing a structural model with a good predictive accuracy. Various concepts and analytical constructions for the analysis of linear systems do not directly apply to nonlinear theory. The breakdown of the principle of superposition is a possible means of detecting the presence of a nonlinear effect. Let y1 (t) and y2 (t) be the responses of a structure to the input forces x1 (t) and x2 (t), respectively. The principle of superposition is violated if αy1 (t) + βy2 (t) is not the structural response to the input αx1 (t) + βx2 (t). The test for homogeneity, which is a restricted form of the principle of superposition (i.e., β is set to 0), is one of the most popular detection techniques. Homogeneity violation is best visualized in the frequency domain through comparison of frequency response functions (FRFs) for different excitation levels. This test was applied to the SmallSat spacecraft, and the FRFs corresponding to LL, IL and HL are superposed in Figure

207 4. Substantial distortions in the FRF at HL can be observed, which is an indication of nonlinear behavior. There are also some slight differences between the FRFs at LL and IL.

−20

−10 −20

FRF (dB)

FRF (dB)

−40 −60 −80 −100

LL IL QL

(a) −120

10

20

30

40

Frequency (Hz)

50

−30 −40 −50 LL IL QL

−60

(b) −70

60

10

20

30

40

Frequency (Hz)

50

60

Figure 4: Frequency response functions for three different excitation levels. (a) Top floor (axial response); (b) WEMS NC4 mobile part (axial response).

3.2

Ordinary Coherence Checks

The ordinary coherence function γ is normally used for assessing the quality of data measured under random excitation γ 2 (ω) =

|Syx (ω)|2 H1 = Sxx (ω)Syy (ω) H2

with

H1 =

Syx (ω) Syy (ω) , H2 = Sxx (ω) Syx (ω)

(1)

where Syy (ω), Sxx (ω) and Syx (ω) contain the power spectral density (PSD) of the response (e.g., acceleration signal), the PSD of the applied force and the cross PSD between the response and the applied force, respectively; H1 and H2 represent the so-called H1 and H2 FRF estimators. The coherence function is required to be unity for all accessible ω if and only if the system is linear and noise-free. It can also be utilized as a detection tool for nonlinear behavior, because it is a rapid indicator of the

1

0.8 0.6 0.4 0.2

LL IL QL

(a) 0

10

20

30

40

Frequency (Hz)

50

60

Ordinary coherence (Hz)

Ordinary coherence (Hz)

1

0.8 0.6 0.4 0.2

LL IL QL

(b) 0

10

20

30

40

Frequency (Hz)

50

60

Figure 5: Ordinary coherence for three different excitation levels. (a) Top floor (axial response); (b) WEMS NC4 mobile part (axial response).

208 presence of nonlinearity in specific frequency bands or resonance regions. It is arguably the most often-used test, by virtue of the fact that almost all the commercial spectrum analyzers allow its calculation; however, it does not distinguish between the cases of a nonlinear system and noisy signals. The coherence functions measured on the SmallSat spacecraft at LL, IL and HL are depicted in Figure 5. A clear departure from a unit coherence can be observed for the high excitation level, which reveals the presence of nonlinear dynamic phenomena.

3.3

Wavelet Transform

The WT can be viewed not only as a basis for functional representation, but at the same time as a useful technique for timefrequency analysis. In contrast to the Fast Fourier Transform (FFT) which assumes signal stationarity, the WT involves a windowing technique with variable-sized regions. Small time intervals are considered for high-frequency components, whereas the size of the interval is increased for lower-frequency components, thereby giving better time and frequency resolutions than the FFT. The Morlet mother wavelet, which is a Gaussian-windowed complex sinusoid, is considered herein.

Frequency (Hz)

Through a direct exploitation of the frequency-energy (or frequency-amplitude) dependence of nonlinear oscillations, the WT can be an extremely efficient detection method, particularly for transient responses. The WT at NC4 for the three excitation levels is shown in Figure 6. The plots shown represent the amplitude of the WT as a function of frequency (vertical axis) and time (horizontal axis). Heavy shaded areas correspond to regions where the amplitude of the WT is high whereas lightly shaded regions correspond to low amplitudes. Such plots enable one to deduce the temporal evolutions of the dominant frequency components of the signals analyzed. The WTs at LL and IL are similar; at HL, the sudden appearance and disappearance of high-frequency components can be interpreted as the symptom of nonsmooth nonlinearity.

200

(a)

Frequency (Hz)

100 0 0

5

10

15

20

25

Time (s)

30

35

40

45

50

200

(b)

Frequency (Hz)

100 0 0

5

10

15

20

25

Time (s)

30

35

40

45

200

50

(c)

100 0 0

5

10

15

20

25

Time (s)

30

35

40

45

50

Figure 6: Wavelet transform of the axial acceleration at the mobile part of WEMS NC4. (a) LL; (b) IL and (c) HL.

209 3.4

Subspace Angles

Because the previous three methods rely on a qualitative (and possibly subjective) assessment, another method, which is quantitative by essence, is also used herein. This method was initially developed for sensor fault detection [11] and was recently extended to nonlinearity detection [12] . Given a response matrix Y containing the accelerations over n sensors during m time samples   y1 (t1 ) · · · y1 (tm )  Y =  ··· ··· ··· yn (t1 ) · · · yn (tm )

(2)

the singular value decomposition of the response matrix can be computed: Y = USVT

(3)

where U is an n x n orthonormal matrix containing the left singular vectors; S is a n x m pseudo-diagonal and semi-positive definite matrix with diagonal entries containing the singular values, and V is an m x m orthonormal matrix containing the right singular vectors. As detailed in [12] , the method consists in comparing the subspace spanned by the first r columns of matrix U at a specific excitation level (termed the reference data) to the subspace spanned by the same columns of U at another excitation level (termed the current data). Because of the superposition principle, if the system is linear, the subspaces should be identical for different input levels. However, for a nonlinear system, as the level of excitation increases, the subspace spanned by the vectors retained in U is distorted. The quantification of these distortions can be carried out using the concept of principal angles between subspaces introduced by Jordan [13] . The principal angles between two subspaces are a generalization of an angle between two vectors, and their number is equal to the dimension of the smallest subspace. A numerical algorithm for the computation of the angles involving a QR factorization and the singular value decomposition was proposed by Bjorck and Golub [14] and can also be found in [15] .

Reference data

?

?

θ1

θ2

θ= q σθ =

1 N

1 N

PN

PN k=1

k=1 (θk

Current data

? θ3

θk (mean angle)

θ(data cur , data ref ) ? > LoL = θ + 3σθ Yes : nonlinear system No : linear system

−

θ)2

(standard deviation)

Figure 7: Comparison of the principal angles of the current and reference data.

Before moving to detection, the reference data are partitioned into several data sets. The principal angle between the subspace spanned by each of these data sets and the subspace spanned by the whole data set is computed, which yields a collection of

210

Subspace angle (degree)

50

LoL IL HL

40 30 20 10 0

2

4

6

# data set

8

10

Figure 8: Subspace angles between LL data and IL/HL data.

different angle values. This is illustrated on the left side of Figure 7. Based on this collection of angles, an upper control limit, termed limit of linearity (LoL), is defined as the mean angle plus three times its standard deviation. This corresponds to an 99.7 % confidence interval for a normal distribution. A nonlinearity is detected once the principal angle between the current data and the reference data exceeds the LoL. Considering the SmallSat spacecraft, the reference data are those collected at LL. The number of vectors retained in matrix U is equal to 3, because these vectors concentrate more than 99.5 % of the total energy in the system. The LL data are partitioned into 10 data sets, and the resulting LoL corresponds to an angle of 1.47 degrees. As shown in Figure 8, the LoL is barely exceeded at IL, whereas the subspace angles between the LL and HL are well above the LoL. These results confirm the qualitative assessment provided by the previous methods.

4 4.1

NONLINEARITY CHARACTERIZATION Restoring Force Surface Method

According to the scheme in Figure 1, nonlinearity characterization is the second step of the nonlinear system identification process and is of paramount importance. Without a precise understanding of the nonlinear mechanisms involved, the third step of the identification process, i.e., parameter estimation, is bound to failure. Characterization is also a very challenging step because nonlinearity may be caused by many different mechanisms and may result in plethora of dynamic phenomena. The RFS method has ‘built-in’ characterization capabilities. It is based on Newton’s second law, which, for a single-degree-offreedom system, is: m¨ y (t) + f (y(t), y(t)) ˙ = x(t) (4) where x(t) is the applied force and f (y(t), y(t)) ˙ is the restoring force, i.e., a non-linear function of the displacement and velocity. From equation (4), it is possible to find the restoring force defined as fi = xi − m¨ yi

(5)

where subscript i refers to the ith sampled value. Thus, for each sampling instant a triplet (yi , y˙ i , fi ) is found, i.e., the value of the restoring force is known for each point in the phase plane (yi , y˙ i ). By representing the restoring force as a function of the displacement and velocity in a three-dimensional plot, the nonlinearity can be conveniently visualized. A characterization of the elastic and dissipative forces can be obtained by taking a cross section of this three-dimensional plot along the axes where either the velocity or the displacement is equal to zero, respectively. The resulting plots are termed stiffness and damping curves, respectively. For the SmallSat spacecraft, the situation is more complex because the interface between the fixed and mobile parts of the

211 WEMS device comprises 8 nonlinear springs. However, qualitative information about the nonlinear behavior can still be obtained by representing the inertia force of one nonlinear component (NC1, NC2, NC3 or NC4) in one particular direction (x,y or z) against the relative displacement and relative velocity across this component in the same direction. Displacement and velocity signals were not available in the context of this study and were estimated from the acceleration signals through integration using the trapezium rule and high-pass filtering [16, 17] . Figure 9 shows the resulting stiffness curves. Several interesting conclusions can be reached from this figure:

• The piecewise linear characteristics is evident for NC3 in the z direction. It can also be distinguished for other nonlinear components in the x and z directions. • Linear behavior is observed in the y direction. This is consistent with the real behavior, because no mechanical stop constraints the motion in this direction. • A first estimation of the clearance can be obtained from these plots. It is estimated to be cz = 1.03 in the axial direction and cx = 1.31 in the lateral direction.

Inertia force (adimensional)

NC1

NC2

NC3

NC4

1.5

1.5

1.5

1.5

1

1

1

1

0.5

0.5

0.5

0.5

0

0

0

0

−0.5

−0.5

−0.5

−0.5

−1 −2

0

2

−1 −2

0

2

−1 −2

0

−1 −2

2

0.1

0.1

0.1

0.15

0.05

0.05

0.05

0.1

0

0

0

0.05

−0.05

−0.05

−0.05

0

−0.1 −1

0

1

2

−0.1 −1

0

1

1

−0.1 −1

0

1

−0.05 −1

1

0

y

0

1

z

0 −0.5

−1 −2 −2

2

0.5

0

0

0

1

0.5

1

x

−1

−0.5

−1 0

2

−1.5 −2

0

2

−2 −2

0

2

−1 −2

0

2

Relative displacement (adimensional) Figure 9: Characterization of the nonlinear behavior using the RFS method. Inertia forces and relative displacements have been adimensionalized for confidentiality reasons. The mentioned directions correspond to local WEMS axes, with z axis consistently aligned with the axial direction.

Now that an estimation of the clearances is available, one can count the number of impacts in the relative displacement signals across the nonlinear connections. Table 2 summarizes the results and indicates that the third axial nonlinear connection is by far the dominant nonlinearity in the response.

212

x z

NC1 141 149

NC2 307 139

NC3 207 1820

NC4 207 509

TABLE 2: Number of impacts.

200 180 160

160

140

140

120 100 80

120 100 80

60

60

40

40

20

20

0 15

15.5

16

16.5

17

17.5

Time (s)

18

18.5

19

(b)

180

Frequency (Hz)

Frequency (Hz)

200

(a)

19.5

0 15

20

15.5

16

16.5

200

17

17.5

Time (s)

18

18.5

19

19.5

20

(c)

180 160

Frequency (Hz)

140 120 100 80 60 40 20 0 15

15.5

16

16.5

17

17.5

Time (s)

18

18.5

19

19.5

20

Figure 10: Correlation between the WT and the impacts. (a) cx = cz = 0.5; (b) cx = 1.28 and cz = 0.96; and (c) cx = cz = 1.8.

4.2

Wavelet Transform

Because the relative displacements across the different components were computed for the RFS method, one can construct a vector, and, for each time instant, assign it a value of 1 when a single or several relative displacements exceeds the assumed clearances cx and cz and a value of 0 when all relative displacement signals are below the assumed clearances. This binary information can then be correlated with the appearance and disappearance of the high-frequency components in the WT of Figure 6, which is performed in Figure 10. The parameters cx and cz can be adjusted until satisfactory correlation is obtained, which occurs for cx = 1.28 and cz = 0.96.

213

Nonlinear coherence function

1 0.95 0.9 0.85 0.8 0.75

0.5

1

1.5

Clearance (adimensional)

2

Figure 11: Averaged nonlinear coherence function as a function of the axial clearance cz .

4.3

Conditioned Reverse Path Method

The CRP method, which is described in detail in Section 5.1, is a parameter estimation method per se, but it has also characterization capabilities. In [18] , the ordinary coherence function is extended to nonlinear systems through the concept of cumulative coherence function. This nonlinear coherence function indicates the extent of uncorrelated noise present and the accuracy of assumed mathematical models employed for describing nonlinear systems. As the ordinary coherence function, it is always between 0 and 1. Figure 11 shows the cumulative coherence (averaged over all sensors and the frequency range of interest) as a function of the axial clearance. This latter can be determined by considering the clearance which maximizes the coherence function; i.e., cz = 1. Repeating the same process for the lateral clearance yields cx = 1.23.

5

PARAMETER ESTIMATION IN THE PRESENCE OF NONLINEARITY

5.1

Conditioned Reverse Path Method

Now that adequate functional forms for the nonlinear springs have been selected, parameter estimation can be safely carried out. To this end, the CRP method [6] generalizes the measurements of FRFs to nonlinear systems. In the presence of nonlinear forces, the classical H1 and H2 methods for FRFs measurements in Equation (1) cannot be used, because the nonlinearities corrupt the underlying linear characteristics of the response. In the CRP method, spectral conditioning techniques are exploited to remove the effects of nonlinearities before computing the FRFs of the underlying linear system. Once the FRFs are known, the nonlinear coefficients may then be identified.

5.1.1

THEORY

Estimation of the underlying system properties The vibrations of a nonlinear system are governed by the following equation M¨ y(t) + Cy(t) ˙ + Ky(t) +

n X

Aj zj (t) = x(t)

(6)

j=1

where M, C and K are the structural matrices; y(t) is the vector of displacement coordinates; zj (t) is a nonlinear function

214 vector; Aj contains the coefficients of the term zj (t); x(t) is the applied force vector. For example, in the case of a grounded cubic stiffness at the ith DOF, the nonlinear function vector is z(t) = [0 ... yi (t)3 ...0]T

(7)

In the frequency domain, equation (6) becomes B(ω)Y(ω) +

n X

Aj Zj (ω) = X(ω)

(8)

j=1

where Y(ω), Zj (ω) and X(ω) are the Fourier transform of y(t), zj (t) and x(t), respectively; B(ω) = −ω 2 M + iωC + K is the linear dynamic stiffness matrix. Without loss of generality, let us assume that a single nonlinear term Z1 is present. The spectrum of the measured responses Y can be decomposed into a component Y(+1) correlated with the spectrum of the nonlinear vector Z1 through a frequency response matrix L1Y , and a component Y(−1) uncorrelated with the spectrum of the nonlinear vector; i.e., Y = Y(+1) + Y(−1) . In what follows, the minus (plus) sign signifies uncorrelated (correlated) with. Likewise, the spectrum of the external force X can be decomposed into a component X(+1) correlated with the spectrum of the nonlinear vector Z1 through a frequency response matrix L1X , and a component X(−1) uncorrelated with the spectrum of the nonlinear vector; i.e., X = X(+1) + X(−1) . Since both vectors Y(−1) and X(−1) are uncorrelated with the nonlinear vector, they correspond to the response of the underlying linear system and the force applied to this system, respectively; as a result, the path between them is the linear dynamic stiffness matrix B X(−1) (ω) = B(ω)Y(−1) (ω) (9) The whole procedure is presented in diagram form in Figure 12.

Y(ω)

¾

Y(+1) (ω)

¾

Σ

-

L1Y

-

Z1 (ω)

-

Z1 (ω)

6

Y(−1) (ω) ?

B(ω) X(−1) (ω) ¾ ?

X(ω)

¾

Σ

¾

X(+1) (ω)

-

L1X

Figure 12: Decomposition of the force and response spectra in the presence of a single nonlinearity .

The generalization to multiple nonlinearities is straightforward. In this case, the spectra of the response and the force need to be uncorrelated with all n nonlinear function vectors  Pn Pn  Y(−1:n) = Y − j=1 Y(+j) = Y − j=1 LjY Zj(−1:j−1) (10) P  X(−1:n) = X − n L Z jX j(−1:j−1) j=1 Y(−1:n) and X(−1:n) are both uncorrelated with the nonlinear function vectors; the path between them is the linear dynamic stiffness matrix B X(−1:n) (ω) = B(ω)Y(−1:n) (ω) (11)

215 By transposing equation (11), premultiplying by the complex conjugate of Y (i.e., Y ∗ ) taking the expectation E[•] and multiplying by 2/T , the underlying linear system can be identified without corruption from the nonlinear terms Syx(−1:n)

= =

2 2 E[Y ∗ XT(−1:n) ] = E[Y ∗ (BY(−1:n) )T ] T T 2 T E[Y ∗ Y(−1:n) BT ] = Syy(−1:n) BT T

(12)

where Syx(−1:n) and Syy(−1:n) are conditioned PSD matrices. Calculation of these matrices is laborious and involves a recursive algorithm. For the sake of conciseness, only the final formulae are given herein. In [?] , it is shown that

where

Sij(−1:r) = Sij(−1:r−1) − Sir(−1:r−1) LTrj

(13)

LTrj = S−1 rr(−1:r−1) Srj(−1:r−1)

(14)

It follows from equation (12) that the dynamic compliance matrix H which contains the FRFs of the underlying linear system takes the form Hc2 : HT = S−1 (15) yx(−1:n) Syy(−1:n) This expression is known as the conditioned Hc2 estimate. If relation (11) is multiplied by the complex conjugate of X instead of Y, the conditioned Hc1 estimate is obtained Hc1 : HT = S−1 (16) xx(−1:n) Sxy(−1:n) Estimation of the nonlinear coefficients Once the linear dynamic compliance H has been computed by solving equation (15) or (16) at each frequency, the nonlinear coefficients Aj can be estimated. By applying to equation (8) the same procedure as the one used for obtaining equation (12) from equation (11), the following relationship is obtained Six(−1:i−1) = Siy(−1:i−1) BT +

n X

Sij(−1:i−1) ATj

(17)

j=1

£ ¤ It should be noted that Sij(−1:i−1) = E Z∗i(−1:i−1) ZTj = 0 for j < i since Z∗i(−1:i−1) is uncorrelated with the spectrum of the T nonlinear function vectors Z1 through Zi−1 . If equation (17) is premultiplied by S−1 ii(−1:i−1) , the first term in the summation is Ai . Equation (17) is then transformed into à ! n X T −1 T T Ai = Sii(−1:i−1) Six(−1:i−1) − Siy(−1:i−1) B − Sij(−1:i−1) Aj (18) j=i+1

Because the expression of the linear dynamic compliance has been computed, equation (18) is rewritten in a more suitable form T ATi HT = S−1 ii(−1:i−1) (Six(−1:i−1) H − Siy(−1:i−1) −

n X

Sij(−1:i−1) ATj HT )

(19)

j=i+1

The identification process starts with the computation of An working backwards to A1 . The nonlinear coefficients are imaginary and frequency dependent. The imaginary parts, without any physical meaning, should be negligible when compared to the real parts. On the other hand, by performing a spectral mean, the actual value of the coefficients should be retrieved. Coherence functions The ordinary coherence function can be used to detect any departure from linearity or to detect the presence of uncorrelated noise on one or both of the excitation and response signals. For a multiple input model with correlated inputs, the sum of ordinary coherences between the inputs and the output may be greater than unity. To address this problem, the ordinary coherence function has been superseded by the cumulative coherence 2 function γM i n X 2 2 2 2 2 γM γjx(−1:j−1) (ω) (20) i (ω) = γyi x(−1:n) (ω) + γzx (ω) = γyi x(−1:n) (ω) + j=1

216 γy2i x(−1:n) is the ordinary coherence function between the ith element of Y(−1:n) and excitation X γy2i x(−1:n) =

¯ ¯ ¯Sy x(−1:n) ¯2 i Syi yi (−1:n) Sxx

(21)

2 It indicates the contribution from the linear spectral component of the response of the ith signal. γjx(−1:j−1) is the ordinary coherence function between the conditioned spectrum Zj(−1:j−1) and excitation X ¯ ¯ ¯Sjx(−1:j−1) ¯2 2 γjx(−1:j−1) = (22) Sjj(−1:j−1) Sxx

and

n X

2 γjx(−1:j−1) indicates the contribution from the nonlinearities.

j=1

The cumulative coherence function is always between 0 and 1 and may be considered as a measure of the model accuracy; it is a valuable tool for the selection of an appropriate functional form for the nonlinearity. Application to the SmallSat spacecraft structure The CRP method was already successfully applied to low-dimensional systems in the literature first application of the method to a real-life structure.

[6, 19, 20]

, but this is arguably the

To facilitate parameter estimation, nonlinearities are introduced one by one in the model (i.e., in a sequential manner). Table 3 lists the value of the cumulative coherence (averaged over all sensors and the frequency range of interest) when one nonlinear spring is introduced. This table shows that a very substantial increase in the coherence is obtained when including the NC3 spring in the z direction in the model. This nonlinearity therefore seems to be the dominant one in the system response, which was also clear from Table 2 and Figure 9. The same process is repeated for a second nonlinearity. Table 4 shows that the cumulative coherence can be increased from 0.968 to 0.978 when considering the NC4 spring in the z direction. Taking into account a third nonlinearity in the model, the NC2 spring in the x direction (see Table 5), results in a very modest improvement of the coherence function (0.978 vs. 0.984). The coherence function has now a very satisfactory value, which indicates the accuracy of the system identification process, and no more nonlinearities are included in the model.

x z

NC1 0.778 0.800

NC2 0.789 0.780

NC3 0.785 0.968

NC4 0.785 0.810

TABLE 3: Averaged nonlinear coherence function when one nonlinearity is incorporated in the model (HL excitation).

x z

NC1 0.970 0.977

NC2 0.972 0.976

NC3 0.972 —

NC4 0.972 0.978

TABLE 4: Averaged nonlinear coherence function when two nonlinearities (including NC3-z) are incorporated in the model (HL excitation).

x z

NC1 0.982 0.983

NC2 0.984 0.983

NC3 0.983 —

NC4 0.983 —

TABLE 5: Averaged nonlinear coherence function when three nonlinearities (including NC3-z and NC4-z) are incorporated in the model (HL excitation).

Because the CRP method is a frequency-domain method, the identified coefficients, i.e., the stiffness of the axial and lateral mechanical stops, are frequency-dependent, as illustrated in Figure 13. However, by taking their spectral mean, the actual

217

NC3−z

1.5 1 0.5 0

10

15

20

25

30

35

40

45

50

55

60

10

15

20

25

30

35

40

45

50

55

60

10

15

20

25

30

35

40

45

50

55

60

NC4−z

10 5 0 −5

NC2−x

2 1 0 −1

Frequency (Hz)

Figure 13: Frequency variations of estimated nonlinear coefficients (HL excitation).

NC3-z NC4-z NC2-x

clearance 1 1 1.23

estimated coefficient 0.94-0.01i 0.86-0.14i 0.38-0.01i

TABLE 6: Identified nonlinear coefficients which are averaged over the frequency range of interest (HL excitation).

values should be retrieved [6] . The coefficients averaged over the frequency range of interest are given in Table 6, and they are found to be in good agreement with the actual coefficients in Table 1. The coefficients are also complex, but their imaginary part, without any physical meaning, is smaller than the corresponding real part. Figure 14(a) displays two different estimations of the FRF at reaction wheel dummy. Clearly, the FRF computed using the H2 estimator is strongly distorted by the nonlinearities, whereas a much smoother FRF is computed through the CRP method. The ordinary and cumulative coherence functions in Figure 14(b) confirm this finding. The FRFs at two different locations and for the different excitation levels (H2 at LL and IL; CRP at HL) are compared in Figure 15. A very good agreement between the FRFs is obtained, which is further evidence of the accuracy of the identification process

6

CONCLUSION

Nonlinear system identification of a real-world example, the SmallSat spacecraft structure, was considered in this paper. A careful progression through the different steps of the identification process, namely detection, characterization and parameter estimation, was achieved. Even though nonlinearity detection was fairly straightforward, the application of different techniques proved very useful. For instance, the wavelet transform highlighted the presence of nonsmooth nonlinearities, whereas the detection method based on subspace angles avoided a subjective interpretation. The importance of nonlinearity characterization was also emphasized in

218

−10

1 0.95

−20

Coherence

FRF (dB)

0.9 −30 −40

0.85 0.8 0.75

−50 −60

CRP HL H2 HL

(a) 10

20

30

40

Frequency (Hz)

50

(b)

Ordinary coherence HL Nonlinear coherence HL

10

20

0.7 0.65

60

30

40

Frequency (Hz)

50

60

Figure 14: Axial response at reaction wheel dummy (HL excitation). (a) FRF; (b) coherence function.

−10

−20 −25 −30

FRF (dB)

FRF (dB)

−20 −30 −40

−60

(a) 10

20

30

40

Frequency (Hz)

50

−40 −45 −50

H2 LL H2 IL CRP HL

−50

−35

H2 LL H2 IL CRP HL

−55 60

−60

(b) 10

20

30

40

Frequency (Hz)

50

60

Figure 15: FRF. (a) Reaction wheel dummy; (b) NC3 mobile part (axial response).

the paper, and an estimation of the axial and lateral clearances using three different methods was proposed. Finally, parameter estimation was performed using the conditioned reverse path method, and the coefficients of the dominant nonlinearities in the system response were satisfactorily assessed.

ACKNOWLEDGMENTS This paper has been prepared in the framework of the ESA Technology Research Programme study ”Advancement of Mechanical Verification Methods for Non-linear Spacecraft Structures (NOLISS)” (ESA contract No.21359/08/NL/SFe)

219 REFERENCES [1] K. Worden, G.R. Tomlinson, Nonlinearity in Structural Dynamics: Detection, Identification and Modelling. Institute of Physics Publishing, Bristol and Philadelphia, 2001. [2] G. Kerschen, K. Worden, A.F. Vakakis, J.C. Golinval, Past, present and future of nonlinear system identification in structural dynamics, Mechanical Systems and Signal Processing 20 (2006), 505-592. [3] S.F. Masri, T.K. Caughey, A nonparametric identification technique for nonlinear dynamic problems, Journal of Applied Mechanics 46 (1979), 433-447. [4] I.J. Leontaritis, S.A. Billings, Input-output parametric models for nonlinear systems, part I: deterministic nonlinear systems, International Journal of Control 41 (1985), 303-328. [5] I.J. Leontaritis, S.A. Billings, Input-output parametric models for nonlinear systems, part II: stochastic nonlinear systems, International Journal of Control 41 (1985), 329-344. [6] C.M. Richards, R. Singh, Identification of multi-degree-of-freedom non-linear systems under random excitations by the reverse-path spectral method, Journal of Sound and Vibration 213 (1998), 673-708. [7] D.E. Adams, R.J. Allemang, A frequency domain method for estimating the parameters of a non-linear structural dynamic model through feedback, Mechanical Systems and Signal Processing 14 (2000), 637-656. [8] M. Ruzzene, A. Fasana, L. Garibaldi, B. Piombo, Natural frequencies and dampings identification using wavelet transform: application to real data, Mechanical Systems and Signal Processing 11 (1997), 207-218. [9] A.G. Russell, Thick skin, faceted, CFRP, monocoque tube structure for smallsats, Proceedings of the European Conference on Spacecraft Structures, Materials and Mechanical Testing, Noordwijk, The Netherlands, 2000. [10] P. Camarasa, S. Kiryenko, Shock attenuation system for spacecraft and adaptor (SASSA), Proceedings of the European Conference on Spacecraft Structures, Materials and Mechanical Testing, Toulouse, France, 2009. [11] G. Kerschen, P. De Boe, J.C. Golinval, K. Worden, Sensor validation using principal component analysis, Smart Materials and Structures 14 (2005), 36-42. [12] A. Hot, G. Kerschen, E. Foltte, S. Cogan, Detection and quantification of non-linear structural behavior using principal component analysis, Mechanical Systems and Signal Processing, in review. [13] C. Jordan, Essai sur la gomtrie n dimensions, Bulletin de la Socit mathmatique 3 (1875), 103-174. [14] A. Bjorck, G.H. Golub, Numerical methods for computing angles between linear subspaces, Mathematics of Computations 27 (1973), 579-594. [15] G.H. Golub, C.F. Van Loan, Matrix Computations, The Johns Hopkins University Press, Baltimore, 1996. [16] K. Worden, Data processing and experimental design for the restoring force surface method, part 1: integration and differentiation of measured time data, Mechanical Systems and Signal Processing 4 (1990), 295-319. [17] G. Kerschen, J.C. Golinval, K. Worden, Theoretical and experimental identification of a non-linear beam, Journal of Sound and Vibration 244 (2001), 597-613. [18] C.M. Richards, R. Singh, Feasibility of identifying non-linear vibratory systems consisting of unknown polynomial forms, Journal of Sound and Vibration 220 (1999), 413-450. [19] S. Marchesiello, Application of the conditioned reverse path method, Mechanical Systems and Signal Processing 17 (2003), 183-188. [20] G. Kerschen, V. Lenaerts, J.C. Golinval, Identification of a continuous structure with a geometrical non-linearity, part I: conditioned reverse path method, Journal of Sound and Vibration 262 (2003), 889-906.

Ground Vibration Testing Master Class: modern testing and analysis concepts applied to an F-16 aircraft

Jenny Lau1, Bart Peeters1, Jan Debille1, Quentin Guzek1, Willam Flynn1, Donald S. Lange2, Timo Kahlmann3 1

LMS international, Interleuvenlaan, 68, B-3001 Leuven, Belgium, [email protected]

2

Air Force Flight Test Center, 307 East Popson Ave., Edward AFB CA 93524, U.S.A.

3

AICON 3D Systems GmbH, Biberweg 30 C, 38114 Braunschweig, Germany

ABSTRACT Ground Vibration Testing (GVT) of aircraft is performed very late in the development process. The main purpose of the test is to obtain experimental vibration data of the whole aircraft structure for validating and improving its structural dynamic models. Among other things, these models are used to predict the flutter behavior and carefully plan the safety-critical in-flight flutter tests. Due to the limited availability of the aircraft for a GVT and the fact that multiple configurations need to be tested, an extreme time pressure exists in getting the test results efficiently. The aim of the paper is to discuss modern testing and analysis concepts for performing a GVT that are able to help realize an important testing and analysis time reduction without compromising the accuracy of the results. For the past several years, LMS has organized so-called Master Classes on the GVT topic. The aim of the class is to introduce an integrated approach to handle the test preparation, modal testing, modal analysis, numerical model correlation, model updating, and model exploitation, to the industry by means of using a full-scale aircraft. This paper illustrates this approach by presenting testing and analysis results of an F-16 aircraft.

Introduction Over more than 80 years [1, 4, 5, 6, 7], methodology and technology used to understand and provide insight into the dynamic characteristics of an aircraft have significantly evolved. Ground vibration testing on a prototype of a new aircraft is often considered to be one important step used to describe the structural dynamics of the aircraft and for updating the finite element model. As a result, greater confidence on flutter behavior can be achieved, leading to mitigation of the hazards related to this inflight testing. Testing methods have evolved from normal mode testing (also referred to as sine dwell, or phase resonance method), which identifies one resonance frequency at a time, to modern multiple input multiple output random excitation methods, where measured transfer functions are analyzed with experimental modal analysis. The challenges behind a ground vibration test are not only providing a qualified model based on high quality test data, but also providing these results within a tight schedule and with limited budgets. Various directions and improvements have already been tested and/or applied such as different excitation methods for the optimization of quality of data and test time. Instrumentation is one of the time consuming tasks during a ground vibration testing campaign since the number of locations to measure increases in proportion with the size of aircraft. For example, nearly 900 sensors are used to test the A380 [7]. Alternative techniques like digital sensors, wireless sensors, and optical measurement systems have been investigated throughout the last 5 years in order to reduce the instrumentation time [6]. At the same time, in order to easily identify sensor locations, photogrammetry is studied and tested on a A737 and considered to be satisfactory in both accuracy and cost expectation [2].

T. Proulx (ed.), Advanced Aerospace Applications, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 4, DOI 10.1007/978-1-4419-9302-1_19, © The Society for Experimental Mechanics, Inc. 2011

221

222

LMS Master Class For the past several years, LMS international has organized worldwide, several Ground Vibration Testing Master Classes. The classes were conducted by our experienced engineers in the field. Lectures are given by professors and professionals in the aircraft development field. Topics ranged from testing to simulation, from equipment to instrumentation, from process explanation, to onsite test execution. Each time, we are challenged by availability of equipment, test object, test facility, etc. The test article ranged from a wind tunnel scaled aircraft model (Fig. a) up to an F-16 (Fig. b and Fig. c). The test facilities started from military base hangar to a testing hall of an aircraft manufacturer. It was, of course, not as stressed as a real ground vibration test campaign, since we didn’t have the objective to deliver precise test data as input for finite element models and with the inherent tight time constraints. Nevertheless, due to availability of the test facility, test article, test equipment and sometimes the engineers themselves, we typically only had 2-3 days for instrumentation. Besides, test articles are different for each master class and our engineers have to be well prepared and as efficient as possible during a ground vibration test campaign, in order to extract the maximum information from the test article in the short time allowed. During the ground vibration testing master class, our engineer involved the participants on encountered difficulties and tried out solutions to quickly solve the problems. In parallel, we also get use of these fabulous opportunities to validate new theories, new concepts and new testing capabilities in order to improve the whole procedure with our research engineers on a real case.

Fig. a Wind tunnel model used during LMS Master class in Russia, 2008

Fig. b F-16 used during LMS Master Class in Belgium, 2008

Fig. c F-16 used during LMS Master class in U.S.A., 2010, source Edwards AFB No. AFFTC-PA-10268

223

Master class in the past In the summer of 2009, at one of the military bases of Belgium, Campus Saffrannberg at Sint-Truiden, we had the opportunity to perform a ground vibration test on an F-16 with a wingspand around 15m. A ground vibration test was conducted with an LMS SCADAS III frontend with 132 channels. As explained in [2, 3], identification of sensors location is one of the key challenges of a ground vibration testing campaign. During the master class, we had an opportunity to validate the use of photogrammetry techniques for identification of geometry coordinates. The photogrammetry technique, as explained in [2], implemented in AICON DPAPro solution, and is fully integrated in LMS Test.Lab Geometry. We decided to take the opportunity to validate the real-life use case of photogrammetry on a 15m by 10m aircraft.

Fig. d A picture used to process the coordinate of coded markers

Fig. e This picture shows some of the identified targets

The procedure consists of putting coded markers on the structure. By means of using coded markers the data processing is automated to a maximum degree. After this, pictures were taken in various angles and from different positions. Fig. d is one of the pictures taken with some coded markers (in black and white). One important remark is that these coded markers are standard available without using printer to print in correct format, thus speeding up the processs and more reliable in scale. Information such as camera charactersitics and scale information are introduced into the software, and pictures are imported in the AICON DPA-Pro software for processing. During the measurement process in situ, the camera was calibrated by the software. For the F-16, we placed around 300 coded markers and took 320 pictures with a Nikon D300. It took us 40 minutes to place the markers, and 30 minutes to take the pictures. We used 5 minutes for processing the pictures and 15 minutes to analyse the data. In total, less than 1 hour and 30 minutes, we obtained the geometry of cloud points directly in the Test.Lab software without any file importing or export manipulations. The accuracy is better than 2 mm for an object of 11m x 10m x 3m. In this case, we are satisfied with the result. In the future, we can also use automatic blunder detection which will guarantee even better results. Unfortunately, we instrumented the aircraft a few days prior to testing. Thus the locations that we measured are not those of the sensors. Nevertheless, we expected that the result can give us an overall image of the aircraft. Fig. e shows some coded markers (identified by yellow number labels) identified by one of the pictures. Fig. f shows the calculated coordindate imported into LMS Test.lab Geometry. These cloud points are named with the number of the coded markers. With less than 2 hours work, we obtained very good data accuracy with a relatively low cost solution. As usual, in the modal analysis context, we use the geometry to animate the mode shape in order to check the validity of mode shape. First of all, we performed random excitation and measured the transfer functions of 132 degrees of freedom, experimental modal analysis was performed with LMS PolyMAX solver. Although, these points didn’t correspond to

224 sensors location, we have superimposed the test geometry with the cloud points in Test.Lab Geometry workbook such that we can use the automatic mode shape expansion to visualize more easily mode shapes in a denser geometry. The advantage of automatic mode shape expansion is that there is no need to describe the relationship between the degree of freedom of the test data and the cloud points. LMS Test.lab will determine, based on a geometrical based algorithm, the closer test points to the corresponding cloud points in order to animate the unmeasured cloud points, thus generating “slave degrees of freedom”. Fig. g shows the first symmetrical mode animated with automatic mode shape expansion. The lines are connected between the measured degree of freedom and the cooresponding points resulting from photogrammetry. The automatic mode shape expansion technique can also be used on geometry defined by CAD or finite element model.

Fig. f Cloud points of the coded target directly imported in Test.Lab Fig. g Symmetric mode animated with automatic mode shape expansion

Master class in USA In May 2010, a ground vibration testing master class has been organized at the Air Force Flight Test Center (AFFTC) in Edwards, California, U.S.A. The test article was an F-16. Two current control electrodynamic shakers of MB Dynamics (Fig. h) are provided and instrumented by AFFTC personnel. The shakers are located at the wing tips and excited in the Zdirection (vertical direction). A 60 channel SCADAS III system was used to measure 2 force cells, 47 laser sensors in velocity, 10 acceleration sensors and send out 2 excitation signals to shakers. The laser sensors are measured in the Z-direction at the wing and horizontal tail and in the Y-direction at the vertical tail. Laser sensors in both the Z- and Y-directions measured the fuselage dynamics. The ten accelerometers are located at the left wing at the same locations as laser sensors in order to compare velocity data to acceleration data (Fig. i).

Velocity measurements Through a Small Business Innovative Research (SBIR) project, CSA Engineering worked with AFFTC to evaluate a new way of instrumentation using non-contact laser sensors (Fig. j). These small lasers are mounted to a frame and that can be

225 built well in advance of the test. When the test article is available, engineers simply position the sensors frame to the test article before the ground vibration test and do a quick check on the distance between the measured surface and the laser sensor. Fig. k shows the setup of a laser sensor on the frame. The main advantage is that most of the set up work is accomplished before the aircraft is available. Another advantage results when an aircraft configuration needs to be made such as fueling or defueling. Typically, the aircraft instrumentation is disconnected from the aircraft and the aircraft is towed outside the test hangar and is fueled. When the aircraft is back in the hangar, again the sensor stands are simply re-positioned and testing resumes with minimal time expended. A separate technical paper discussing these sensors has been written and will be presented at IMAC 2011.

Fig. h Shaker instrumented in the wing tip, source Edwards AFB No. AFFTC-PA-10268

Fig. i Laser sensor and the corresponding accelerometer instrumented in Left wing, source Edwards AFB No. AFFTC-PA-10268 For the GVT Master class, the setup of the test object was mostly coordinated by AFFTC since the test will mainly be conducted with these small lasers. Before the ground vibration testing master class, AFFTC had already evaluated the results measured with the CSA laser system with their own LMS Ground vibration testing system. The measured quantity of the laser is “velocity”, whereas quite often accelerometers are used instead. Nevertheless, experimental modal analysis is conducted without any problem in Test.Lab modal analysis even when acceleration and velocity data are combined in the data set.

Excitation methods and analysis tools During the master class, several excitation methods, such as multiple input multiple output random excitation, multiple input multiple output sweep sine excitation, multiple input multiple output stepped sine excitation and multiple input multiple output normal mode testing, are used to illustrate the difference between each other. Random excitation measured up to 64 Hz bandwidth and with 32 % overlap. It provides a quick method to identify on a wide frequency range the overall dynamic behavior of the structure. Fig. l and Fig. m illustrate the driving point FRFs and coherence, which shows that the reciprocity between driving points is achieved and the response at the driving points are noisy after 40 Hz. Fig. n compares the FRF measured at one of point of the left wing in Z direction; the red curve represents the measurement of laser in velocity/force format in -Z direction and green curve represents the measurement of accelerometerin acceleration/force format in +Z direction. The green curve is integrated in velocity/force for comparison. The amplitude of the measurement corresponds quite well whereas the velocity data seems to be a bit noisier than the

226 acceleration data. We also observed the phase difference which can easily explained by +/- Z direction of the sensors. Overall, below 40 Hz, velocity and acceleration data correspond quite well with each other. Stepped sine excitation are used to illustrate excitation around certain resonance frequency with different excitation levels in order to identify non-linearity. During the master class, sine excitation is conducted from 5 to 25 Hz with 0.125 Hz resolution. The test took 19 minutes to achieve 2 linear sweeps. Compared to random excitation, the measurement time is more important but this method allows higher excitation level. Another alternative is to use sweep sine excitation which will provide a similar effect but shorter transient time is applied between each sine spectrum. It’s quite often used today to replace random excitation to achieve higher excitation level and signal to noise ratio with less measurement time than stepped sine excitation.

Fig. j Small laser sensors are attached in a frame for velocity measurement, source Edwards AFB No. AFFTC-PA-10268

Fig. k Laser sensors setup, source Edwards AFB No. AFFTC-PA-10268

The result of these excitation methods are FRFs which were analyzed with experimental modal analysis. Another excitation method is the normal mode excitation which excite one mode at a time, meaning the structure is in resonance. The vibration response of the structure is measured directly and animated in a geometry display. This provides a very good feel for the dynamics of the structure. The operator modifies excitation frequency, excitation level and phase of the shakers in real time. The objective is to obtain the 90° phase difference between acceleration and force, or 0° phase difference between velocity and force. Another important aspect is the positioning of the shakers. For example, when exciting the first bending mode of the wing, we place the shakers in Z direction close to the wing tip where useful energy is injected in the wing tip in order to better excite the structure. In other words, for each mode, an optimum shaker position is necessary. Thus normal modes testing is used only whenever more understanding is needed at certain frequency such as non-linearity phenomena or very closely spaced modes. Fig. o and Fig. p show the symmetric bending wing mode animated on the test frame and on the CAD model. Animation on the CAD model is achieved with the automatic expansion of measured mode shape. The master/slave degree of freedom relationship is calculated by searching the closest geometrical point (or closest point belonging to the same topology definition such as on the same surface or connection line). The automatic mode shape expansion can easily be used for the test geometry and the CAD geometry (or finite element model). This capability provides better visualization and hence interpretation of the mode shapes. Fig. q and Fig. r show the anti-symmetric bending wing mode animated on the test frame and on the CAD model.

227

Conclusion In this paper, we have illustrated a few concepts on real-life tests. First of all, photogrammetry as a cost and time efficient solution for identification of sensor locations is applied on a large structure without compromising accuracy. Also, noncontact laser sensors are considered to be convincing in reducing setup time during a GVT, and the data is seamlessly integrated for modal curve fitting. Automatic mode shape expansion is used for better visualization of mode shape animation. Future work and experience on improvement of ground vibration testing can be conducted during these master classes.

Fig. l shows the driving points FRFs

Fig. m shows the coherence of the driving points

Fig. n show the velocity FRF (red) and acceleration FRF integrated in velocity format (green)

228

Fig. o Symmetric mode shown in test geometry

Fig. p Symmetric mode shown in CAD geometry

Fig. q Anti-symmetric mode shown in test geometry

Fig. r Anti-symmetric mode shown in CAD geometry

Acknowledgments

I would like to thank Donald S. Lange, Structural Technical expert of Edwards Air Force Base who helped us to organize the LMS Ground Vibration Testing Master Class in 2010 and especially on providing his advice and experience on ground vibration testing. I would like to thank Timo Kahlmann of AICON 3D Systems GmbH who showed and helped us with the photogrammetry measurement during the LMS Ground Vibration Testing Master Class in 2009. References [1] B. Peeters, Modern solution. In Proc. IMAC XXVI International Modal Analysis Conference, Orlando, Florida, USA, Feb 2008 [2] G. Foss, Photogrammetry of aircraft structures for sensors location. In Proc. IMAC XXIV International Modal Analysis Conference, 2004 [3] S. Pauwels, Experimental Modal Analysis: Efficient geometry model creation using optical techniques, In Proc. 75th Shock 1 vibration Symposium, 2004 [4] R. J. Dieckelman, The ground vibration Test - A Boeing IDS perspective, In Proc. IMAC XXIII International Modal Analysis Conference, 2005 [5] D. L. Hunt, A comparison of methods for aircraft ground vibration testing, In Proc. IMAC III International Modal Analysis Conference, 1985 [6] C.R. Pickrel, New concepts GVT, In Proc. IMAC XXIV International Modal Analysis Conference, 2006 [7] D. Göge, Ground Vibration Testing of Large Aircraft - State-of-the-Art and future perspectives, In Proc. IMAC XXV International Modal Analysis Conference, 2007

Advanced shaker excitation signals for aerospace testing Bart Peeters1, Alex Carrella2, Jenny Lau1, Mauro Gatto3, Giuliano Coppotelli3 1 2

LMS International, Interleuvenlaan 68, B-3001 Leuven, Belgium

University of Bristol, Department of Aerospace Engineering, Queens Building, Bristol BS8 1TR, UK 3

Università di Roma “La Sapienza”, Dipartimento di Ingegneria Aerospaziale e Astronautica, Via Eudossiana 18, 00184 Roma, Italy

ABSTRACT The need to reduce testing time without diminishing the quality of the data is an important driver for innovation in the aerospace testing industry. In this paper, the use of advanced, flexible shaker excitation signals will be investigated with the aim (1) to obtain improved Frequency Response Function (FRF) estimations and (2) to assess the non-linearities of the excited system / structure. Pseudo-random and more general multisine signals, rather than the more traditional pure or burst random signals, will be used to increase the accuracy of the FRF estimate. Moreover, special multisine data acquisition and processing methods to identify the level of non-linearity will be illustrated by means of Ground Vibration Testing data of an F-16 aircraft. The presented methods allow assessing the non-linearities at a single excitation level, which is in contrast to the more traditional method of repeating the test at multiple excitation levels and observing the FRF differences. In addition, a new perspective will be given on the post-processing of stepped sine FRFs. Stepped sine shaker excitation signals are traditionally used to highlight and study non-linear behaviour. In this paper, a curve-fitting method based on FRF data at fixed response levels is applied to identify and quantify the non-linearities of the structure. Again, the approach will be illustrated by means of F-16 aircraft data. 1. INTRODUCTION It is clear that an improvement of the accuracy of the estimate of the frequency response functions (FRFs) and a better exploitation of the available test data, e.g. by including non-linearity assessment without increasing the test time, will contribute to the efficiency increase of the whole testing phase. This is particularly true when dealing with actual systems disturbed by high noise levels and strongly affected by non-linear contributions that could be referenced, at least partially, as additional noise [1][2]. Typical non-linear behaviour could originate from control chains, friction, or free-plays of junctions [2]. In traditional modal testing, (burst) random noise sources are sent to the different shakers attached to the structure. Some recent developments allow the use of more sophisticated signals which offer much more flexibility and control during a vibration test. In this paper, frequency response functions will be estimated from pseudo random, user defined amplitude and random phase, excitation signals. These excitation signals are periodic and if the signals are synchronously acquired, the leakage distortions are limited to the initial transients [3][4]. Also, the whole time block is used to excite the structure, reducing the required testing time or increasing the Signal-to-Noise Ratio (SNR). In [5], both numerical simulations of a nonlinear system and laboratory experimental investigations are carried out in order to validate the proposed techniques. The non-linearities considered in the numerical model are introduced as quadratic and cubic stiffness terms, whereas the experimental data are gained from tests carried out on an aircraft scale model. This paper complements the work in [5] in the sense that multisine excitation is applied to a full-scale F-16 aircraft in order to obtain improved FRF estimates and to assess the influence of non-linearities on the FRF estimates. In addition, stepped sine measurements are acquired on the same aircraft and an alternative non-linear identification method is applied in Section 4. 2. THEORETICAL BACKGROUND This section provides the theoretical background. An overview will be given of shaker excitation signals aiming to get improved FRF estimates. The advantages of multisine and pseudo random signals compared to pure and burst random will be discussed. The Schröder multisine can be used to achieve a higher SNR. Furthermore, a special sequence of pseudo random excitation can be used for both FRF estimation and non-linearity assessment. The odd-odd multisine is useful in non-linearity detection [1].

T. Proulx (ed.), Advanced Aerospace Applications, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 4, DOI 10.1007/978-1-4419-9302-1_20, © The Society for Experimental Mechanics, Inc. 2011

229

230

2.1. Overview of shaker excitation signals A brief overview of shaker excitation signals is provided. More information can be found in [1]-[7]. Note that the list below is non-exhaustive. Broadband shaker signals are discussed. Beyond the scope of the paper are excitation signals like impacts, stepped-sine, normal modes, swept sine. A discussion on improved FRF estimation in case of (MIMO) sine sweep data can be found in [8][9]. •

•

•

Pure random: A pure random signal is typically generated in time domain by a random number generated. The bandwidth and spectrum can be shaped using digital filters. In frequency-domain, it has random amplitudes and random phases. It provides a very good linear approximation in the presence of non-linearities and is characterized by a good SNR. However, data are strongly affected by leakage effects and the use of windows is necessary. Averaging is needed to cancel out non-coherent noise. Burst Random: Burst random is a random signal that is only active for a user defined percentage of the acquisition block, and with no excitation during the remaining time. A very good linear approximation in the presence of nonlinearities is obtained with burst random and no windowing is needed because leakage is minimized, provided that the response of the structure has died out in the observation window. The SNR is lower than in the random case because only a percentage of the time block is used to effectively excite the structure. Multisine: A source signal u (t ) that is the sum of multiple sine waves (1): NS

u (t ) = ∑ Ak cos(2πkf 0 t +φ k )

(1)

k =1

•

where N s is the total number of sine components, Ak , φ k are the amplitude and phase of sine component k and f 0 is the fundamental frequency. For Discrete Fourier Transform (DFT) processing, it is important that the sine components lie on a “Fourier grid”. This makes a multisine periodic with respect to the observation period. All components can have arbitrary amplitude and phases. A multisine is repeated different times to allow the transient response to decay. The main advantage of using multisines as excitation signals is that the vibration phenomena are periodic and do not suffer from leakage when using DFT processing. Some special cases of a multisine are discussed in the following. Pseudo random: Special case of a multisine, with constant amplitudes of all components and phases randomly selected from a uniform distribution between –180° and 180°. The time series of a pseudo random signal is obtained by applying the inverse DFT to the generated frequency-domain representation of the signal. Transient effects are very light if one or more delay blocks (i.e. block during which the structure is excited, but actual acquisition only starts after repeating the same block a number of times) are used. The FRF is not distorted by leakage or windowing and due to the continuous excitation, the signal has a high SNR. An example scheme of pseudo random excitation is shown in Fig. 1.

Fig. 1 Example scheme of a pseudo random signal (the number of realizations is user defined as well as the number of blocks for each realization). •

•

Periodic random: Special case of a multisine, of which the frequency spectrum has random amplitude and random phase distribution. Historically, this signal was generated in the time domain and consists of a pure random time block which is sent out repeatedly. When the transient response has decayed, the input and response time histories become periodic and only frequency components at the spectral lines exist. Schröder multisine: For a broadband signal (such as a multisine), it can be important to have a low crest factor (ratio of peak amplitude to RMS value) to optimise the use of the shaker/amplifier combination and to improve the SNR. A pseudo random signal has a crest factor of 3-4. One particular method to decrease the crest factor consists of using Schröder phases (instead of random phases) of the different sine waves:

231 φ k = φ1 −

πk (k − 1) Ns

(2)

This equation is valid for a multisine with constant amplitudes. Other more general methods exist for optimizing the crest factor such as the swapping method or more complex optimization methods [10]. A Schröder multisine has the aspect of a sine sweep signal that sweeps through all frequencies within 1 acquisition block. No windows are needed, but the structure is excited always in the same way and, therefore, stochastic non-linearities are not averaged out. 2.2. Non-linearity assessment In industrial practice, the degree of non-linearity is assessed by (visually) comparing FRFs measured with different excitation amplitudes. For linear systems, all FRFs should coincide. For example Fig. 2 (Left) illustrates the PSD of the excitation (increasing level from red to cyan) and the corresponding FRF. The system under investigation is quite linearly behaving from 10 Hz to 50 Hz, and shows strong non-linearities from 50 Hz to 90 Hz and less strong non-linearities from 90 Hz to 200 Hz. Also coherence functions are inspected for assessing the non-linearity, but these FRF quality indicators combine effects of leakage, noise and non-linearity. More powerful techniques to assess structural non-linearity exist that rely upon the use of multisines as shaker excitation signals [1][4][11][12]. In this paper one such alternative technique to assess the presence of non-linearity is presented and applied. The technique (for which the overall procedure is presented schematically in Fig. 2 (Right)) consists of a special measurement sequence of pseudo random signals. A pseudo random signal is sent out periodically allowing assessing the noise level. Additionally, multisines with different realisations of random phases are sent out allowing assessing the combined influence of noise and non-linear distortions. A detailed mathematical background of this method can be found in [11].

Fig. 2 Non-linearity assessment with the conventional method (Left) and with pseudo random (Right). For a certain realization i, the variance σ i2 due to noise is given by the following (single-input case, H 1 FRF estimate) [4]: σ i2 =

1 M

 1 − γ i2   γ2  i

  Hi  

2

(3)

where M is the number of repeated blocks (without delay blocks) for each realization, γ i2 is the estimated coherence and H i is the estimated FRF. The overall variance only due to noise is then computed as σ 2Noise =

1 R2

R

∑σ i =1

2 i

(4)

where R is the number of realizations and σ i2 is the variance of the ith realization. On the other hand, by taking classical “ H 1 ” averages over all experiments ( M repeated blocks and R realizations), the variance due to both noise and distorting non-linearities can be estimated in a similar way as in (3):

232 σ 2Noise + NL =

1 1− γ 2  MR  γ 2

 H  

2

(5)

in which H and γ 2 have been estimated using all available data. Any difference between σ 2Noise (4) and σ 2Noise+ NL (5) can be attributed to non-linearities. Some further insights on the use of multisines and the application to vibration analysis of mechanical structures can be found in [13] and [14]. 3. APPLICATION OF MULTISINE EXCITATION TO AN F-16 AIRCRAFT 3.1. Linear identification and classical non-linearity assessment In this section, the F-16 aircraft case study is introduced by means of the results of a linear identification (modal analysis) and some classical non-linearity assessment methods. More background on the F-16 Ground Vibration Testing activities carried out by LMS in recent years can be found in [15]. Fig. 3 shows the stabilization diagram obtained by applying PolyMAX [16] to single-shaker (at the right wing) multisine excitation data. Starting from such a clear diagram, modal parameter estimation is a rather straightforward task. Fig. 4 shows the instrumented F-16 aircraft and the first symmetric wing bending mode of the aircraft identified using PolyMAX. It is interesting to note that the measured mode shape components have been automatically expanded by means of a CAD model. The unmeasured “slave” Degrees-of-Freedom (DOFs) of the CAD model are obtained by relating them to the closest (or closest AND belonging to the same topology definition such as a surface or connection line) measured “master” nodes. This capability may enhance the visual interpretation of the mode shape [15][17]. Some additional (but not expanded) mode shapes are shown in Fig. 5.

Fig. 3 LMS PolyMAX stabilization diagram using single-shaker pseudo-random excitation. Fig. 6 represents the classical non-linearity assessment method: a random excitation test is repeated at different force levels and the differences in FRFs are observed. Linear structures do not show a force level dependent FRF behaviour. It can be seen from Fig. 6 (Right) that with an increase of force amplitude, the eigenfrequencies of the F-16 aircraft are decreasing (i.e. softening behaviour). In Fig. 7 (Left), an FRF of a low-level pseudo-random test is compared to the FRFs of stepped-sine tests. Again the non-linear behaviour (softening) is clearly observed. In Fig. 7 (Right), the stabilization diagram of a stepped sine measurement is shown. Fitting a non-linear peak of an FRF using a linear modal model results in the identification of 2 modes: one with low damping and the other with high damping. A similar observation is made in [18], where the GVT of the A400M is discussed. Apart from frequency shifts, a non-linear response can be associated to significant FRF peak skewness at affected resonances. Certain highly damped, highly force-dependent structural resonances showing positive skewness were systematically fitted 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. Sometimes such a “mode splitting”, 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 [18].

233

Fig. 4 (Left) Instrumented F-16 aircraft; (Right) measured symmetric wing bending mode expanded to CAD

Fig. 5 Estimated mode shapes: (Left) symm. wing bending + torsion; (Right) Anti-symm. wing bending + torsion -10.00

-35.00

N dB

g/N dB

FRF ALDP:-Z/ALDP:+Z Low Force FRF ALDP:-Z/ALDP:+Z Medium Force FRF ALDP:-Z/ALDP:+Z High Force

-60.00 180.00 ° Phase

AutoPow er ALDP:+Z AutoPow er ALDP:+Z AutoPow er ALDP:+Z

-180.00

-40.00 0.00

Hz

100.00

15.00

Hz

23.00

Fig. 6 Classical non-linearity assessment using pure random excitation at 3 different force levels (Left): the influence on the FRF is clearly visible (Right).

234 0.04 Ps eudoRandom Stepped Sine 5 N Stepped Sine 10 N Stepped Sine 15 N

g/N

Amplitude

s

s s v

s s s s s s s s s v s s s s s v s s s v v v v v s v o

v v v o

1.00e-3

°

Phase

180.00

6.51

-180.00 6.50

Hz

v v v s o

v v v o o

Linear

o

15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

7.51

Hz

7.50

Fig. 7 (Left) Zoom of FRFs at resonance around 7 Hz. Non-linearity is clearly visible in comparison between broadband level, stepped sine 5 N, stepped sine 10 N, and stepped sine 15 N. (Right) PolyMAX stabilization diagram of stepped sine FRF data: “mode splitting” due to non-linearity. Fig. 8 shows the eigenfrequency and damping ratio variations as a function of increasing force level. These modal parameters have been obtained by applying PolyMAX to the measured FRFs. Although changes are relatively small, significant trends can be observed. With increasing force level, all (but one) eigenfrequencies decrease and all (but one) damping ratios increase. In [19], an interesting approach to non-linear Experimental Modal Analysis is discussed: linear modal fits at different levels are interpolated and extrapolated for predicting the resonances at other levels.

Fig. 8 Modal parameter estimation from random excitation FRFs at 3 different force levels: (Left) eigenfrequncy shifts; (Right) damping ratio shifts. 3.2. Analysis of the multisine measurements In this section, the results from the multisine measurements will be analysed in detail. Fig. 9 compares the quality of the FRF estimates from different broadband excitation signals: pure random, burst random, multisine. The FRF quality is both expressed as standard deviation and coherence. For the single-input case, the relation between variance (square of standard deviation) and coherence is given by the following (see also (5)): σ2 =

1 N avg

1− γ2  2  γ 

 H  

2

(6)

where N avg is the number of spectral averages used in the FRF estimation. It can be seen that at the resonances, burst random signals represent a significant improvement with respect to pure random signals. Indeed, when using pure random signals, the

235

resonance regions are heavily affected by leakage, whereas a burst random signal tries to suppress leakage by assuming that the response sufficiently decayed at the end of the observation window. The multisine signal represents a further improvement at the resonances (remaining residual leakage that may still be present when using burst random signals reduced) and at other locations (a multisine signal results in a better overall SNR). -20.00

FRF ALDP:-Z/ALDP:+Z FRF ALDP:-Z/ALDP:+Z FRF ALDP:-Z/ALDP:+Z FRF ALDP:-Z/ALDP:+Z

Multisine FRF std Random std Burst Random std Multisine

g/N dB

g/N dB

-20.00

FRF ALDP:-Z/ALDP:+Z FRF ALDP:-Z/ALDP:+Z FRF ALDP:-Z/ALDP:+Z FRF ALDP:-Z/ALDP:+Z

Multisine FRF std Random std Burst Random std Multisine

-120.00

-120.00 Hz

100.00

1.00

Hz

1.00

g/N dB

Amplitude

-20.00

20.00

FRF ALDP:-Z/ALDP:+Z Coherence ALDP:-Z/ALDP:+Z Coherence ALDP:-Z/ALDP:+Z Coherence ALDP:-Z/ALDP:+Z

-90.00 1.00

Hz

Multisine FRF Random Burst Random Multisine

/

0.00

0.00

20.00

Fig. 9 FRF quality assessment from different broadband excitation techniques: pure random, burst random, multisine. (Top) FRF quality is expressed as standard deviation on the FRF estimate (Left: full band 0-100 Hz; Right: 1-20 Hz); (Bottom) FRF quality is expressed as coherence. In order to highlight another benefit of using multisine excitation, the non-linearity assessment procedure outlined in Section 2.2 is applied with following parameters: M = 5 (Number of cyclic averages), R = 10 (Number of realizations), 1 delay block. The results for the driving point FRF are synthesized in Fig. 10. In the traditional coherence-based FRF quality assessment method, the influence from noise and non-linearities cannot be separated (Fig. 10 - Top). Using the new multisine averaging method, the standard deviations due to noise (4) and the standard deviations due to noise and non-linearities (5) can both be calculated. From Fig. 10 (Bottom-Right), it can for instance be observed that at the resonance around 7 Hz, the combined uncertainty from noise and non-linearity is about 20 dB higher than the noise-only uncertainty. At the resonance around 4.8 Hz, the difference is 15 dB. Even at the relatively low force levels from the multisine excitation, the influence of the non-

236

linearities is already clearly visible. Interesting side remark: the multisine excitation started at 1 Hz; it is therefore not unexpected to observe that the standard deviations bellow 1 Hz have about the same magnitudes as the estimated FRF.

g/N dB

-20.00

g/N dB

-20.00

FRF ALDP:-Z/ALDP:+Z Coherence ALDP:-Z/ALDP:+Z

FRF ALDP:-Z/ALDP:+Z Coherence ALDP:-Z/ALDP:+Z -120.00

-120.00 0.00

-20.00

Hz

100.00

Hz

20.00

-20.00 FRF ALDP:-Z/ALDP:+Z FRF FRF ALDP:-Z/ALDP:+Z Noise FRF ALDP:-Z/ALDP:+Z Noise + Non-Lin

g/N dB

g/N dB

FRF ALDP:-Z/ALDP:+Z FRF FRF ALDP:-Z/ALDP:+Z Noise FRF ALDP:-Z/ALDP:+Z Noise + Non-Lin

0.00

-120.00

-120.00 0.00

Hz

100.00

0.00

Hz

20.00

Fig. 10 Quality assessment of estimated driving point FRF obtained using multisine excitation. (Left) full band 0-100 Hz; (Right) zoom from 0-20 Hz; (Top) traditional method using the coherence function combining the influence of noise and non-linearity; (Bottom) new method using standard deviations on the estimated FRFs allowing to separate influences from noise and non-linearities Fig. 11 represents the multisine results at some other response locations (nose and horizontal tail plane) of the F-16 aircraft. It can be concluded that the noisy character of the nose FRF is entirely due to noise (added influence of non-linearity is negligible). However, at the horizontal tail plane, the standard deviation due to noise and non-linearity exceeds the noise-only standard deviation by 15 dB. Fig. 12 represents some right wing responses, showing non-linear disturbances at the resonances (e.g. modes at 4.8 Hz and 7.1 Hz).

237 -20.00

-20.00

FRF RHT:TA:+Z/ALDP:+Z FRF FRF RHT:TA:+Z/ALDP:+Z Noise FRF RHT:TA:+Z/ALDP:+Z Noise + Non-Lin

g/N dB

g/N dB

FRF FS:Radome:+X/ALDP:+Z FRF FRF FS:Radome:+X/ALDP:+Z Noise FRF FS:Radome:+X/ALDP:+Z Noise + Non-Lin

-120.00

-120.00 1.00

Hz

20.00

1.00

Hz

20.00

Fig. 11 (Left) FRF of response at aircraft nose mainly affected by noise; (Right) FRF of response at horizontal tail plane at which standard deviation due to noise and non-linearity exceeds the noise-only standard deviation by 15 dB -20.00

-20.00

FRF RWng:3F:+Z/ALDP:+Z FRF FRF RWng:3F:+Z/ALDP:+Z Noise FRF RWng:3F:+Z/ALDP:+Z Noise + Non-Lin

g/N dB

g/N dB

FRF RWng:1F:+Z/ALDP:+Z FRF FRF RWng:1F:+Z/ALDP:+Z Noise FRF RWng:1F:+Z/ALDP:+Z Noise + Non-Lin

-120.00

-120.00 1.00

Hz

20.00

1.00

Hz

20.00

Fig. 12 FRFs of responses at the right wing showing non-linear disturbances at the resonances (e.g. modes at 4.8 Hz and 7.1 Hz) 4. NON-LINEAR IDENTIFICATION BY SDOF FRF FITTING AT FIXED RESPONSE AMPLITUDES In this section, a simple method based on the information contained in the Frequency Response Function (FRF) of a structure is applied to identify and quantify the non-linearities of the aircraft. The technique falls within the category of Single-Degreeof-Freedom (SDOF) methods. The basic principle is that, at a fixed amplitude of response, it is possible to extract the dynamic properties of the underlying system. Repeating this linearization for different response amplitudes allows extracting the stiffness and damping as functions of the amplitude of vibration. Furthermore, because of its mathematical simplicity and practical implementation during standard vibration test, is particularly suitable for the engineering community. More details on the fixed-amplitude SDOF FRF fitting method can be found in [20][21]. From burst-random tests performed at different excitation levels, it was observed a shift in natural frequency of the first bending mode. This occurrence has induced to attempt to shed more light on this non-linear phenomenon. In order to do so,

238

the aircraft has been excited with a stepped-sine signal (with a frequency resolution of 0.05 Hz) around the first bending mode resonance (4.81 Hz). This test has been repeated for 2 different levels of force, 6 and 16 Newton. The modulus of the receptance FRF is shown in Fig. 13 with dotted and solid line respectively. Furthermore, in order to prevent the force dropout which occurs at resonance, the force control tool (available with LMS Test.Lab [17]) was activated: the input forces are shown in Fig. 14. It can be seen that in the vicinity of the resonance it is hard to maintain the force at its nominal value of 6N (dotted line) and 16N (solid line). All the measurements were taken on the right-wing only, were the shaker was attached. At a glance, from Fig. 13 it seems clear that the wing shows a softening behaviour. It is also interesting to notice that the first bending mode which was found at 4.81 Hz with a burst-random excitation appears at 4.7 Hz with 6N force amplitude and 4.65 when the force is increased to 12N. x 10

-4

RWng:1A:+Z RWng:1F:+Z RWng:2A:+Z RWng:2F:+Z RWng:3A:+Z RWng:3F:+Z RWng:4A:+Z RWng:4F:+Z RWng:1A:+Z RWng:1F:+Z RWng:2A:+Z RWng:2F:+Z RWng:3A:+Z RWng:3F:+Z RWng:4A:+Z RWng:4F:+Z

..... 6 N

Receptance FRF [m/N]

___ 16 N

2

1

0

4

4.5

5

Frequency [Hz]

5.5

Fig. 13 Modulus of receptance FRF to a stepped-sine excitation of all the points on the right wing for two different ampltude of force: 6N (dotted lines) and 16N (solid lines) 22 20 Excitation Force [N]

18 16 14 12 10 8 6

4

4.5

Frequency [Hz]

5

5.5

Fig. 14 Excitation force measured with the load cell attached to the wing: 6N (dotted lines) and 16N (solid lines) Following the observation of the shift in natural frequency for different excitation levels, the non-linear identification method presented in [20][21] is applied in the following. In brief, having measured and saved both FRF and force, the algorithm extracts the natural frequency (and damping ratio) of the structure at different amplitudes of displacement response. In fact, by fixing this value the stiffness (which is assumed to be a function of the displacement) is a constant and it is therefore possible to determine the natural frequency of the system at that amplitude. Repeating this calculation at different response

239

amplitudes enables one to derive the natural frequency as function of the response displacement amplitude. This is shown in Fig. 15 for all the 8 measurement points and for both levels of excitation. Albeit plotting all the curves on the same graph might make the figure too busy, some features can be observed: At small displacement all the curves have the same natural frequency. As the displacement of the response increases, there seems to be a general trend of decreasing natural frequency (softening effect) but there are some points which show stronger local effects. The point with maximum amplitude of displacement (point RWng:1F which is at the tip of the wing) shows a marked softening behaviour with a decrease in natural frequency of almost 7.5%. This is shown in Fig. 16 (Left). The sudden drop in natural frequency which can be observed in Fig. 16 (Right) for the point RWng:3F as the force level is increased indicates a local steep change in stiffness, which can be caused by a joint opening for example.

• • •

5 4.8 natural frequency [Hz]

RWng:1A:+Z RWng:1F:+Z RWng:2A:+Z RWng:2F:+Z RWng:3A:+Z RWng:3F:+Z RWng:4A:+Z RWng:4F:+Z RWng:1A:+Z RWng:1F:+Z RWng:2A:+Z RWng:2F:+Z RWng:3A:+Z RWng:3F:+Z RWng:4A:+Z RWng:4F:+Z

..... 6 N ___ 16 N

4.6 4.4 4.2 4 3.8 3.6

0

0.1

0.2

0.3 0.4 Displacment [mm]

0.5

0.6

0.7

Fig. 15 Non-linear identification results obtained with the described SDOF FRF fitting method for all the points on the right wing and for two levels of force: 6N (dotted line) and 16N (solid line) 5 4.8

___ 16 N

4.6

X: 0.582 Y: 4.434

4.4 4.2 4

RWng:1F:+Z RWng:1F:+Z

3.8 3.6

0

0.1

0.2

0.3 0.4 Displacment [mm]

0.5

0.6

..... 6 N ___ 16 N

4.8 natural frequency [Hz]

natural frequency [Hz]

5

..... 6 N

0.7

4.6 4.4

RWng:3F:+Z RWng:3F:+Z

4.2 4 3.8 0

0.02

0.04 0.06 Displacment [mm]

0.08

0.1

Fig. 16 Non-linear identification results for: (Left) the point on the wing with maximum displacement (RWng:1F which is located at the wing tip): a 7.5% decrease in natural frequency is measured when the level of force is increased; (Right) the point RWng:3F: there seems to be a local effect whereby the stiffness changes dramatically with level of excitation (i.e. displacement)

240

The non-linear analysis shown here points out that, in order to have a model which faithfully reproduces the measured data, local and non-linear effects cannot not be ignored. Although there is a global mode (1st wing bending) at about 4.8 Hz, there are phenomena which require more detailed tests and analyses in order to be fully characterised. The scope of the identification technique is, at this stage, only to provide a first indication of the type and location of the non-linearities. Further and more-in-depth test should be carried out in order to fully characterise the system non-linear behaviour. Also, the damping has not been considered as investigation of the dissipation mechanism(s) go beyond the scope of this test and would require specific measurements. It should be also pointed out that the method is based on the two main assumptions that: (i) there is one mode dominating the dynamic response, i.e. it is a SDOF method; and (ii) the non-linearity is only amplitudedependent, which is a limiting assumption as non-linear effects may be dependent on other variables (e.g. frequency). Further studies are being conducted to account for Multiple-Degree-of-Freedoms and other sources non-linear phenomena. 5. CONCLUSIONS This paper discussed the use of advanced, flexible shaker excitation signals will with the aim (1) to obtain improved Frequency Response Function (FRF) estimations and (2) to assess the non-linearities of the excited system / structure by nonconventional techniques. The presented methods allow assessing the non-linearities at a single excitation level, which is in contrast to the more traditional method of repeating the test at multiple excitation levels and observing the FRF differences. FRF measurements’ results with random and burst random have been compared with results obtained with multisines. Measurements on a full-scale F-16 aircraft have demonstrated that multisine excitation can improve the accuracy of FRF measurements by reducing typical problems such as leakage and transient effects. In addition, it is very useful in detecting if and where, in the frequency bandwidth, non-linear distortions occur. Finally, the fixed-amplitude SDOF FRF fitting method for identifying and quantifying structural non-linearities was applied to stepped-sine F-16 data. The method provided a first indication of the type and location of the non-linearities. ACKNOWLEDGEMENTS This paper was the result of collaboration between the University of Rome “La Sapienza”, the University of Bristol and LMS International. Part of the research leading to these results has also received funding from the European Community’s Seventh Framework Programme FP7/2007-2013 under grant agreement nr. 213371 (www.maaximus.eu). REFERENCES [1] J. Schoukens, Y. Rolain, J. Swevers, and J. De Cuyper, Simple methods and insights to deal with non-linear distortions in FRF-measurements, Mechanical Systems and Signal Processing, 14(4):657-666, 2000. [2] D.J. Ewins, Modal testing: theory, practice and applications, Second Edition, Research Studies Press, Baldock, UK, 2000. [3] W. Heylen, S. Lammens, and P. Sas, Modal analysis theory and testing, Department of Mechanical Engineering, Katholieke Universiteit Leuven, Leuven, Belgium, 1997. [4] R. Pintelon and J. Schoukens, System identification: a frequency domain approach, IEEE Press, New York, 2001. [5] M. Gatto, B. Peeters, and G. Coppotelli, Flexible shaker excitation signals for improved FRF estimation and nonlinearity assessment, In Proceedings of the ISMA 2010 International Conference on Noise and Vibration Engineering, Leuven, Belgium, 20-22 September 2010. [6] D.L. Brown, G. Carbon, and R.D. Zimmerman, Survey of excitation techniques applicable to the testing of automotive structures, SAE Paper No. 770029, 1977. [7] A.W. Phillips, A.T. Zucker, and R.J. Allemang, Comparison of MIMO-FRF excitation/averaging techniques on heavily and lightly damped structures, In Proceedings of IMAC 17, the International Modal Analysis Conference, Kissimmee (FL), USA, February 1999. [8] S. Orlando, B. Peeters, and G. Coppotelli. Improved FRF estimators for MIMO Sine Sweep data. In Proceedings of the ISMA 2008 International Conference on Noise and Vibration Engineering, Leuven, Belgium, 15-17 September 2008. [9] G. Gloth and M. Sinapius, Analysis of swept-sine runs during modal identification, Mechanical Systems and Signal Processing, 18:1421–1441, 2004. [10] P. Guillaume, Multi-input multi-output systems using frequency-domain models, PhD Thesis, Dept. ELEC, VUB, 1992. [11] J. Schoukens, J. Swevers, R. Pintelon, and H. Van der Auweraer. Excitation design for FRF measurements in the presence of nonlinear distortions. In Proceedings of the ISMA 2002 International Conference on Noise and Vibration Engineering, Leuven, Belgium, September 2002. [12] K. Vanhoenacker, T. Dobrowiecki, and J. Schoukens, Design of multisine excitations to characterize the nonlinear distortions during FRF-measurements, IEEE Trans. Instrum. Meas., 50:1097-1102, 2001. [13] P. Guillaume, P. Verboven, S. Vanlanduit, and E. Parloo. Multisine excitations – new developments and applications in modal analysis. Proc. IMAC 19, Kissimmee (FL), Feb 2001.

241

[14] P. Verboven, P. Guillaume, S. Vanlanduit and B. Cauberghe. Assessment of non-linear distortions in modal testing and analysis of vibrating automotive structures. In Proceedings of IMAC 22, Dearborn (MI), USA, 26–29 January 2004. [15] J. Lau et al., Ground Vibration Testing Master Class: modern testing and analysis concepts applied to an F-16 aircraft, In Proceedings of IMAC 29, Jacksonville, FL, USA, 31 January – 3 February 2011. [16] 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. [17] LMS International. LMS Test.Lab, Leuven, Belgium, www.lmsintl.com, 2011. [18] J. Rodríguez Ahlquist, J. Martinez Carreño, H. Climent, R. de Diego, and J. de Alba, Assessment of nonlinear structural response in A400M GVT, In Proceedings of IMAC 28, Jacksonville, FL, USA, 1-4 February 2010. [19] M. Link, M. Böswald, S. Laborde, M. Weiland, and A. Calvi, An approach to non-linear Experimental Modal Analysis, In Proceedings of IMAC 28, Jacksonville, FL, USA, 1-4 February 2010. [20] A. Carrella, D.J. Ewins, A. Colombo, and E. Bianchi, Identifying and quantifying structural non-linearities from measured Frequency Response Functions, In Proceedings of IMAC 28, Jacksonville, FL, USA, 1-4 February 2010. [21] A. Carrella and D.J. Ewins, Identifying and quantifying structural nonlinearities in engineering applications from measured frequency response functions, Mechanical Systems and Signal Processing, article in press, 2011.

System and Method for Compensating Structural Vibrations of an Aircraft caused by Outside Disturbances by W. Luber1 and J. Becker2 CASSIDIAN - Air Systems - an EADS Company 85077 Manching, Germany [email protected]

Abstract A method for compensating lateral - anti-symmetric and longitudinal - symmetric structural vibrations of an aircraft caused by turbulence, gust, wind blasts, wake penetration and buffeting in flight is presented. The method includes the steps of detecting the structural vibrations by a novel measurement technology using pitch, roll and yaw rates determined by an inertial sensing system. The noval measurement supplying the determined disturbing values to a flight control system, producing phase- and amplitude-correct control flap movements by generating appropriate control signals in respective control drives to counteract the phases and amplitudes of the excited vibrations. The design requirements especially the elastic mode stability criteria are described. For the symmetric mode alleviation additional means for improvement using high frequency trailing edge split-flap flight control system control laws are outlined. Also lateral vibration alleviation using rudder and trailing edge flaps is described.

Introduction A method for compensating structural vibrations of an aircraft caused by external disturbances acting on the aircraft in flight is described. The method comprising the acts of detecting the structural vibrations via measurement technology using pitch, roll and yaw rates determined essentially by an inertial sensing system of the aircraft without requiring additional sensors. The intellectual property of the method from the EADS Deutschland GmbH is documented in Ref. 1 to 6. The very complicate measurement of local accelerations which would cause severe problems for the aircraft instrumentation and the flutter with the influence of the active system and aero-servo-elastic proof of stability and the certification of such a system is therefore avoided. The disturbing parameters of the detected structural vibrations are fed into a flight control system of the aircraft. The control laws of the flight control system produce phase- and amplitude-correct control flap movements of the aircraft by generating appropriate control 1 2

Wolfgang Luber, Chiefengineer Structural Dynamics and Aeroelasticity Dr. Ing. Jürgen Becker, Retired from EADS

T. Proulx (ed.), Advanced Aerospace Applications, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 4, DOI 10.1007/978-1-4419-9302-1_21, © The Society for Experimental Mechanics, Inc. 2011

243

244

signals at respective drives to counteract phases and amplitudes of excited structural vibrations. The external disturbances are wind blasts, gusts, turbulence and buffeting on the aircraft. The system comprises a plurality of drive units operatively associated with respective aircraft control surfaces, said drive units receiving the respective appropriate control signals from the flight control system to correctly move the associated control surfaces to counteract the excited structural vibrations. The implications of the system associated with the aero-servo-elastic stability problems are outlined. Dynamic response calculations using a coupled rigid and flexible aircraft model including the flight control law dynamics has to be applied for the design of the vibration alleviation system see Fig. 1. For the longitudinal control special load redistribution by different inboard and outboard trailing edge control surface split control laws are described in order to reduce aero-servoelastic stability problems especially for multiple wing store configurations. Finally examples of vibration alleviation are demonstrated.

Figure 1: Aeroservoelastic problems combined with active vibration alleviation

Description of the Vibration Alleviation Method The described system relates to a method for compensating structural vibrations of an aircraft caused by external influences; especially wind gusts, turbulence in particular compensating structural vibrations of a manned or remote pilot fixed-wing aircraft, a missile, or a helicopter, as well as a system for performing this method. Another viewpoint of the vibration alleviation is the so-called buffeting or "shaking," in other words the response of the behaviour of the aircraft as a result of separated air flow ("buffet")

245

and vortices at high angles of attack. The dynamic loads from the process of separated flow are high and result in design criteria for the wings and empennage of aircraft, especially manned or remote pilot fighter aircraft. Weakening of the effects of these disturbances is therefore very important in designing aircraft and in increasing and monitoring the load limits of aircraft. The behaviour of an aircraft in flight in a turbulent atmosphere should be as calm as possible, but should lie at least in the range of known rules. This is especially true of passenger aircraft for which there are special requirements regarding passenger comfort, also called "ride comfort." External disturbances, especially wind blasts and turbulence, act directly on structural parts of the aircraft which are excited to vibrate as a result of the shapes of the structures. Because of this effect, the behaviour during flight, and especially the comfort of the passengers and pilots, is unfavourably influenced and the structure of the aircraft is subjected to loads. Thus far, to reduce the effects of such external atmospheric disturbances on the aircraft, sensor devices, especially acceleration sensors, have been mounted in the past at various locations on the aircraft. Preferably, acceleration sensors were placed at the tips of the wings, on the fuselage, and at the tips of the vertical and horizontal stabilizers. The sensor signals are processed in a controller and considered in determining the adjustments of the drives of the rudder surfaces. Such systems are described for example in "Proceedings of the 26th Aircraft Symposium," Sendai, Japan, October 19-21, "Japan Publications Trading Company," Tokyo, 1988, pages 160-163 or in "DGLR Paper" 084-094 of the Deutsche Gesellschaft fur Luft- und Raumfahrt, Annual Meeting, Oct. 1-3, 1984, Hamburg. One disadvantage of these sensor devices or systems is that the cost of measurement technology and the expense for regulating and adjusting devices is very high. Especially in controlled aircraft, the cost is further increased by the fact that in the event of an error in such sensor devices, appropriate error recognition and system reconfiguration functions must be provided to at least limit the effects of such errors or to compensate for them. In addition, methods and systems for compensating disturbances in aircraft structures are known from U.S. Pat. Nos. 4,905,934; 5,072,893; 5,186,416 and 4,725,020. These patents are intended to reduce the loads on the structure caused by these disturbances. Acceleration signals or these signals combined with stress values and attack angles are used to move the control surfaces of the aircraft so that the load distribution in the structure is reduced. Higher frequency forms of vibration and structural couplings are not compensated however. Phase delays, which play an important role particularly at higher frequency vibrations, are not taken into account so that the method according to the prior art is suitable only for very low frequencies. Hence, the present system provide a method and device for compensating structural vibrations of an aircraft caused by wind gusts, wake penetration and buffeting that is also suitable for compensating higher frequency vibrations. This goal is achieved by a method for compensating structural vibrations of an aircraft caused by wind blasts and buffeting on the aircraft in flight, the method including the steps of: 1. detecting structural vibrations by measurement technology using pitch, yaw and roll rates determined in an inertial sensing system; 2. feeding the disturbing parameters detected to a flight control system; and

246

3. producing phase- and amplitude-correct control flap movements by generating appropriate control signals at the respective drives to counteract the phases and amplitudes of the excited vibrations In the method the cost of the equipment is extremely low and, depending on the flight control system in question, comprises additional connections between the inertial sensing system of the flight control system and the actual flight control systems. In addition, there is a functional cost for in-phase processing of the values fed back from the inertial sensing system for damping the behaviour in flight. This expense is much greater in the method according to the prior art because devices must be provided to supply the signals of the acceleration sensors inphase and to monitor them. If it is assumed that an inertial sensing system and a flight control system are already aboard the aircraft in question, a flight control system is only functional in feeding back additional values from the inertial sensing system for damping the behaviour of the aircraft in flight. Surprisingly, it has been found that the effects of atmospheric disturbances and buffet on an aircraft in flight can essentially be detected with only the pitch, roll and yaw rates from an inertial sensing system designed for a flight control system and that these effects can be damped by appropriately supplying and processing these values in the flight control system. The system is described with reference to the figure 2 that shows a block diagram of a closed regulating circuit with a flight control system, control drives, and an inertial sensing system, see Ref. 1.

Figure 2: Bolck diagram of the vibration alleviation system which is integrated in a existing flight control system The figure 2 shows an electronic flight control system 1 connected by at least one data connection 11 with one or more control drives (actuators) 12 for the horizontal stabilizer or fore-plane, with a data connection 13 with control drives 14 for ailerons, and with a data connection 15 for the drives 16 of the vertical stabilizer (rudder) and wing trailing edge flaps. The data connections 11, 13, 15 in turn can comprise a plurality of electrical analogue or digital lines. The effects of the control drives 12, 14, 16 on aircraft behaviour represented symbolically by function block 30 are likewise displayed symbolically with connecting lines 17, 18, and 19.

247

The aircraft behaviour 30 is detected by an inertial sensing system 21 connected with the electronic flight control system 1, with the inertial sensing system being represented symbolically and being coupled by connecting line 20. Inertial sensing system 21 in turn is connected by analogue signal lines or digital data connections 22 with electronic flight control system 1. Only the connection for vertical acceleration 23, the pitch rate 24, lateral acceleration 25, roll rate 26, and yaw rate 27 are shown as signal or data connections 22. The angular rates are essential. These values are determined by inertial sensing system 21 and consolidated (21a) there or in the electronic flight control system. Both the inertial sensors 21 and the electronic flight control system 1 preferably have a plurality of redundant and similar or dissimilar components (21b) that exchange data with one another to identify and to recognize the appearance of errors and to guarantee suitable reconfiguration mechanisms for maintaining all of the flight control functions. The functions of the inertial sensing system 21, which are connected functionally downstream of the actual inertial sensor components (also not shown), are integrated with the functions of electronic flight control system 1 in a data processing device or they are functionally separate from them. The signal and data connections 22 can therefore also be in the form of software. For damping the lengthwise movement of the aircraft, the pitch rate 24 is fed to flight control system 1. Following processing therein, they are taken into account in the control signal for those control drives which, depending on the aircraft configuration, are responsible for influencing the lengthwise movement of the aircraft. In most fixed-wing aircraft, this is the vertical stabilizer control drive 14. For damping aircraft behaviour especially during lateral movement, the values determined in inertial sensing system 21 for roll rate 26, or yaw rate 27 (or a combination of two of these values) is supplied to electronic flight control system 1. Here, these values are processed and taken into account in the control signals for the drives responsible for the lateral movement of the aircraft for damping interference. This takes place in most fixed-wing aircraft through suitable lines 13, 15 by means of the aileron drive 14 and the rudder drive 16. Depending on the type of aircraft, a plurality of drives is preferably supplied for each elevator provided on the aircraft. This is also true for the ailerons, the rudders, or other control surfaces of the aircraft depending on the configuration of the aircraft together with a control drive. In suitable fashion and also as a function of the safety concept of flight control system 1, one or more data or signal connections 11 is/are provided for each drive 12. Signal or data processing can also be associated with each control drive 12 provided either physically on the control drive or at a distance from it in the aircraft. The same is true of the drives of other control surfaces of the aircraft and their corresponding signal and data connections. The data 23, 24, 25, 26, and/or 27 supplied by the inertial sensing system 21 to flight control system 1 are filtered in flight control system 1 preferably in filter stages (also not shown), including avoiding the excitation of elastic mode frequencies of the aircraft structure by drives 12, 14, 16. To control the aircraft, the latter, other sensor data, and stored data are processed with one another using control technology. Relative to the damping of atmospheric disturbances, the data provided for this purpose 23, 24, 25, 26, and/or 27 are compared with the values of these data in previous calculation cycles and the resultant phase differences are transmitted following suitable amplification as control signals over lines 11, 13, and 15 to drives 12, 14, and 16. These produce corresponding deflections of the control flaps that

248

counteract the atmospheric disturbances and/or buffet on the aircraft. Damping of the disturbances acting on the aircraft is achieved when data 23, 24, 25, 26, and/or 27 are processed in-phase in the aircraft control system. Then, by means of appropriate control signals, the effects of atmospheric disturbances and buffet on the aircraft relative to the highfrequency shapes of the aircraft structure are reduced as a damping action. The entire system from the inertial sensing system 21 through the flight control system 1 and the control drives 12, 14, and 16 must be sufficiently fast, in other words they must exhibit a suitably slow phase shift and the control drives 12, 14, and 16 must have a suitable response performance and especially control rates in order to produce a sufficiently rapid movement of the control flaps of the aircraft and to achieve timely damping of the effects of disturbances on the aircraft.

Effects of the Vibration Alleviation System on the Aircraft Stability of an Automatically Controlled Airplane Process for designing flight controllers with respect to elastic aircraft stability The process for designing flight controllers is described, which is based upon Ref. 2, in which first for the rigid airplane and then for the elastic airplane the damping and the phase delay for each excitation frequency is determined, and the flight controller is adapted in such a manner that the structural responses to each excitation frequency for both the rigid airplane and the elastic airplane in the open control circuit outside two design fields, applicable to the elastic airplane, are laid around the instability points in the data field comprising damping and phase delay, whereby for the design of the elastic airplane between the phase delays of −270 degrees and −495 degrees, a damping exceeding −6 dB is allowed.

Figure 3 Nichols diagram – dB and phase requirements for open loop frequency response functions (From US Patent US 6,816,823 B2) A process for designing a flight controller is described, the process comprising the acts of: determining a damping and phase delay for each excitation frequency first for a rigid airplane and then for an elastic airplane; adapting the flight controller such that structural responses to each excitation frequency for both the rigid airplane and the elastic airplane in an open control circuit outside of two design fields, applicable to the elastic airplane, are located around

249

instability points in a data field comprising the damping and phase delay; and wherein for the design of the elastic airplane a damping exceeding -6 dB is allowed between the phase delays of -270 degrees and -495 degrees. The process for designing a flight controller, wherein a design curve for the rigid and elastic airplane must be located below and/or aside of a first and a second of said two design fields due to a corresponding adaptation of the flight controller; further wherein the first design field is formed by data points determined by a gain and the phase delay arising from an excitation of the airplane: (+c, -270 degrees), (+c, -180 degrees), (+c/2, -135 degrees), (-c/2, -135 degrees), (-c, -180 degrees), (-c, -240 degrees), (-2/3 c, -270 degrees); and further wherein the second design field is determined by the following data points: (+c, -630 degrees), (+c, -540 degrees), (+c/2, -495 degrees), (-c/2, -495 degrees), (-c, 540 degrees), (-c, -600 degrees), (-2/3, -630 degrees); where edges of the two design fields are chamfered by 45 degrees, starting from at least a damping of (+/-2/3 d1); and where the value c is at least 4 dB and the value d1 is at least 4 dB.

Background and Summary This application claims the priority of German Application No. 100 12 517.4, filed Mar. 15, 2000, the disclosure of which is expressly incorporated by reference herein. In modern controlled airplanes there exists the requirement to compensate for severe structural vibrations, especially with the use of ever softer airplane structures. German Patent document DE 198 41 632.6 discloses a process to compensate for the structural vibrations of an airplane arising from turbulence and buffeting, where the structural vibrations are detected, according to a measuring method, by means of the rotational speeds (pitch rate, yaw and roll rate) that are determined by an inertial sensor system. The detected rotational speeds are fed to the flight control system, and control flap movements are produced, according to phase and amplitude, for the purpose of minimizing the phases and amplitudes of the excited vibrations. The design of flight control systems is known from the military specification MIL-A-8868B for the elastic airplane and from MIL-F-8785B for the rigid airplane, where the open circuit diagrams for phase delay and damping generally specify an amplitude margin of 6 dB and, independently thereof, a phase margin of at least +/-45 degrees. The object is to provide a process for designing a flight controller that is suitable especially for airplane structures. This problem is solved by a process for designing flight controllers, in which first for the rigid airplane and then for the elastic airplane the damping and the phase delay for each excitation frequency is determined. The flight controller is adapted in such a manner that the structural responses to each excitation frequency for both the rigid airplane and the elastic airplane in the open control circuit outside two design fields, applicable to the elastic airplane, are located around the instability points in the data field comprising damping and phase delay, whereby for the design of the elastic airplane between the phase delays of -270 degrees and -495 degrees, a damping exceeding -6 dB is allowed.

Brief Description of the Drawings

250

Figure 2 depicts an electronic flight control system 1, which is connected by means of at least one data connection 11 to one or more regulating drive mechanism(s) 12 for the airplane's elevators, by means of a data connection 13 to the regulating drive mechanisms 14 for the airplane's ailerons, and by means of a data connection 15 to the rudder regulating drive mechanisms 16. The effects from these control surfaces arising from the airplane's behaviour are represented symbolically with the functional block 10. The airplane behaviour is detected of an inertial sensor system 21, assigned to the flight control system, a state that is represented symbolically with the connecting line 20. The inertial sensor system 21 in turn is connected by means of analogous signal lines 22 to the flight control system 1. To dampen the structural vibrations arising from turbulence and buffeting, the rotational speeds, detected in the inertial system 21, are used. In the flight controller, individual notch filters or structural filters, phase derivative filters and regulating amplifiers are used, in order to dampen those elastic shapes that are not considered in the vibration damping. So-called inverse notch filters can also be provided that reinforce the elastic shapes under discussion. These notch filters also enable a phase shift (that is supposed to be optimized in a design) whose purpose is to stabilize the elimination of electric signals over the inverse notch filter. The phase derivative filters serve to reverse the low frequency phase losses, generated by the notch filters, so that the inventive stability criteria for the dynamics of the rigid and controlled airplane can be obtained. The aforementioned components of the controller are used in such a manner, according to the level of skill in the field of control systems, that the design criteria for the flight controller, described below, can be obtained. Figure 3 describes the inventive open circuit design criteria for the flight controller. These criteria are based on the rigid airplane, i.e., the airplane including a controller without any regard for the elastic modal shapes, and the elastic airplane, i.e., the airplane including a controller with regard to the elastic modal shapes. In the diagram, the phase delay is plotted on the abscissa and the gain is plotted in dB on the ordinate. The curve 31 (blue line) of the rigid airplane shows the phase delay and the damping of the rigid airplane with the controller for increasing excitation frequencies. This curve or the response characteristic of the rigid airplane 31 is determined via a simulation using a rigid airplane model. A design field 51 or open circuit criterion 33 that applies to areas of the rigid airplane defines a criterion in terms of area as a function of the damping and phase delay, which states, which pair of values from the damping and phase delay for the rigid airplane model with the controller may not occur. This field is formed around a reference point or instability point 51a, which is defined by the pair of values having a phase delay of -180 degrees and a gain of 0 dB. With respect to the behaviour of the elastic airplane with the controller and the aforementioned filters the method provides, besides the design field 51, another design field 52. The design fields or criteria areas 51, 52 are open circuit criteria, which are applicable to areas of the rigid airplane and defined as a function of the damping and the phase delay in a phase diagram. The curve 53 of the response characteristic of the elastic airplane with controller may not touch or travel through the fields 51, 52. The fields 51, 52 are generally designed in such a manner - that is, with respect to their minimum limits - that their minimum limits represent parabolic closed curves that are symmetrical to the damping zero line. These curves start from the respective instability point, that is, from the instability point 51a with a phase delay of -180 degrees and a damping of 0

251

dB or around the instability point 52a with a phase delay of -45 degrees and damping of 0 dB. The boundaries of the fields 51, 52 are defined by a uniform direction of curvature along the circumferential lines of the same and by the corner points -90 degrees and +45 degrees with respect to the phase delay and +/-4 dB with respect to the damping. The fields are formed preferably in the shape of a polygon in order to have a clear definition of all of the limits of the same at each point. Thus, the method involves a process for designing flight controllers, where first for the rigid airplane and then for the elastic airplane the damping and the phase delay for each excitation frequency is determined; and the flight controller is adapted in such a manner that the structural responses to each excitation frequency outside the design fields (51, 52), applicable to both the rigid and the elastic airplane, are laid around the instability points in the data field of damping and phase delay, thus permitting for the design of the controlled airplane between the phase delays of -270 degrees and -495 degrees for damping values that exceed -6 dB. Thus, the curve 53 can exhibit a bulge between the fields 51, 52, as depicted by the segment 63 in Fig. 3. In the preferred configuration of the fields as polygons 51, 52, the first design field 51 is formed around the instability point 51a with a phase delay of -180 degrees and damping of 0 dB, whereas the second design field forms around the instability point 52a with a phase delay of -540 degrees and gain of 0 dB. Both design fields 51, 52 form around the respective instability points, first with a constant upper phase boundary 54a or 54b, which lies at a phase distance p3 from the respective instability point 51a or 52a. The phase distance p3 is at a minimum +30 degrees so that the upper phase boundary 54a, 54b is at -150 degrees or -510 degrees phase delay. Preferably, p3 is 45 degrees and a maximum of 60 degrees. Furthermore, the design fields 51, 52 are defined by a bottom phase boundary, which, starting from the respective instability point 51a, 52a, lies at a phase distance p4 in the negative direction. Preferably, the value p4 is 90 degrees so that the bottom phase boundaries in the phase diagram exhibit a phase delay of -270 degrees or -630 degrees. The value p4 is a minimum of 60 degrees and a maximum of 120 degrees. The suitable values for p3 and p4 depend on the application, that is, on the airplane to be designed and the softness of its structure. Furthermore, the design fields 51 and 52 are defined with respect to damping by means of the horizontal boundaries in the open circuit diagram. A damping d2 as the upper damping boundary 56a or 56b and a damping -d2 as the bottom damping boundary 57a or 57b is specified. The value of d2 is preferably 6 dB and is a minimum of 4 dB. The damping boundaries can vary depending on the application. The design fields 51, 52 are decreased, for example, with 45 degree slopes at their corners in order to prevent the edge areas from lying disproportionately far from the instability points 51a, 52a and to prevent a design line 53 from lying unnecessarily far from the instability points. Thus, the design field 51 exhibits for the preferred values for p3, p4, and d2 a first chamfering 61 with the corner point 61a, defined by the pair of values (+d2 dB, -180 degrees) and with the corner point 61b, defined by the pair opf values (+0.5d2 dB, -180 degrees +p3). Furthermore, the design field 51 exhibits a second chamfering 62 with the corner point 62a, defined by the pair of values -(0.5d2 dB, -180 degrees +p3), and by the corner point 62b, defined by the pair of values (-d2 dB, -180 degrees). A third chamfering 63 exhibits a corner point 63a, defined by the pair of values (-d2 dB, -180 degrees, -0.75p4), and a corner point 63b, defined by the pair of values (+0.5 d2 dB, -180 degrees -p4). The chamfering 61, 62, 63 can also be formed by curves.

252

Similarly the design field 52 exhibits at the edge areas reductions, which are formed preferably as the chamfering 64, 65, and 66. Hence, the edges of the design fields are sloped by 45 degrees at least starting from the damping of (+/-2/3 d1). In detail, a first chamfering 64 is provided with the corner point 64a, defined by the pair of values (+d2 dB, -540 degrees), and the corner point 64b, defined by the pair of values (+05 d2 dB, -540 degrees+p3). Furthermore, the design field 52 exhibits a second chamfering 65 with the corner point 65a, defined by the pair of values -(0.5 d2 dB, -540 degrees+p3), and with the corner point 65b, defined by the pair of values (-d2 dB, -540 degrees). A third chamfering 66 exhibits a corner point 65a, defined by the pair of values (-d2 dB, -540 degrees -0.75 p4), and a corner point 65b, defined by the pair of values (+0.5 d2 dB, -540 degrees -p4). The chamfering 64, 65, 66 can also be formed by curves. The dynamic design of the controller for the elastic airplane is implemented at this stage in such a manner that the design line 53 for all excitation frequencies does not touch or cross the design fields 51, 52. Between the design fields 51, 52, the design curve 53 can also exhibit a bulge 63 as long as it does not touch or cross the design fields 51, 52. Of course, the design curve 63 can only be designed in such a manner that the dynamic loads of a control surface and an adjoining point are not exceeded. However, the goal is to have the design curve 53, 63 moves as far as possible upward, that is, to the highest possible gains, since the results are then a higher reduction that is vibration damping. By allowing amplitudes and phase states exceeding a damping of 6 dB in certain areas, the dynamic design requirements are significantly reduced so that significantly fewer measures with respect to the structure or the airplane systems are required as compared to the design process according to the state of the art. The notch filters, phase derivative filters and regulating amplifiers as well as inverse notch filters are adapted in such a manner according to the prior art methods that the design curves 31 or 53, 63 do not enter the related design fields 33 or 51, 52 and do not touch them. Correspondingly, the notch filters and the inverse notch filters are set up individually as the numerator and denominator polynomials in the analogous frequency area. To incorporate into the flight computer, they are modified. The individual frequency response is optimized in such a manner at the intersection in the entire open circuit signal that the stability criteria are guaranteed.

Improvement of the longitudinal stability of the controlled aircraft The longitudinal open loop transfer functions for all flight conditions and aircraft configurations of the aircraft defined at the feedback breakpoint, which is the design input for the control feedback design for the rigid and elastic aircraft can be modified and reduced in amplitude and phase by application of a flight control device described in Ref. 3. Originally the wing inboard and outboard trailing edge flaps of a delta canard configuration are used as a full span flap for the pitch stabilisation. For improvement of the total aircraft stability however a special inboard outboard flap split law can be applied which can be used in a different way for high frequency elastic mode frequencies and low frequency of the rigid Aircraft.

253

The method of flight control of an aircraft, Fig. 4 is characterized by: 1. The flight control device for the longitudinal control of an automatically controlled airplane having adjusting surfaces which, viewed from the longitudinal axis of the airplane, are on the outside and on the inside and are provided for the longitudinal control surfaces 2. The flight control device wherein forward control slats are provided on the airplane are additionally controlled by the automatic flight control system for controlling and stabilizing the airplane in the longitudinal axis. 3. An automatically controlled airplane comprising: a fuselage a pair of wings at opposite lateral sides of the fuselage, inner adjusting surface members disposed at a trailing edge of the respective wings adjacent the fuselage, outer adjusting surface members disposed at the trailing edge of the respective wings at positions laterally outside the inner adjusting surface member with respect to the fuselage, and an automatic control system operable to provide longitudinal control to the airplane by controlling movements of said inner and outer adjusting surface members with an adjusting velocity of said inner adjusting surface members being substantially greater than the adjusting velocity of said outer adjusting surface members during flight with adjusting movement of both the inner and outer adjusting surface members. 4. An automatically controlled airplane according to 3, comprising forward control slats disposed forwardly of the respective wings and adjacent the fuselage, wherein said automatic control system is operable to control said forward control slats. 5. An automatically controlled airplane according to 4, wherein the automatic control system is operable to control forward control slats with an adjusting velocity which is faster than the adjusting velocity of outer adjusting surface members during flight with adjusting movement of both the inner and outer adjusting surface members. 6. An automatically controlled airplane according to 3, wherein the automatic control system operable controls the velocity of adjustment of the inner adjusting members to be between 150% and 200% of the velocity of adjustment of the outer adjusting members during flight with adjusting movement of both the inner and outer adjusting surface members. 7. An automatically controlled airplane according to 6, comprising forward control slats disposed forwardly of the respective wings and adjacent the fuselage, wherein said automatic control system is operable to control said forward control slats. 8. An automatically controlled airplane according to 7, wherein said automatic control system is operable to control said forward control slats with an adjusting velocity which is faster than the adjusting velocity of said outer adjusting surface members during flight with adjusting movement of both the inner and outer adjusting surface members. 9. An automatically controlled airplane according to 3, wherein the automatic control system operable controls the maximal velocity of adjustment of the outer adjusting members to be at least 25% lower than the maximal velocity of adjustment of the inner

254

adjusting surface members during flight with adjusting movement of both the inner and outer adjusting surface members 10. An automatically controlled airplane according to 9, comprising forward control slats disposed forwardly of the respective wings and adjacent the fuselage, wherein said automatic control system is operable to control said forward control slats. 11. An automatically controlled airplane according to 10, wherein said automatic control system is operable to control said forward control slats with an adjusting velocity which is faster than the adjusting velocity of said outer adjusting surface members during flight with adjusting movement of both the inner and outer adjusting surface members. 12. An automatically controlled airplane according to 3, wherein said automatic control system operable controls the velocity of adjustment of the inner adjusting surface members to be at least 50% larger than the velocity of adjustment of the outer adjusting surface members during flight with adjusting movement of both the inner and outer adjusting surface members. 13. A method of operating an automatically controlled airplane comprising: a fuselage, a pair of wings at opposite lateral sides of the fuselage, inner adjusting surface members disposed at a trailing edge of the respective wings adjacent the fuselage, outer adjusting surface members disposed at the trailing edge of the respective wings at positions laterally outside the inner adjusting surface members with respect to the fuselage, and an automatic control system operable to provide longitudinal control to the airplane by controlling movements of said inner and outer adjusting surface members, the method comprising controlling an adjusting velocity of said inner adjusting surface members to be substantially greater than the adjusting velocity of outer adjusting surface members during flight with adjusting movement of both the inner and outer adjusting surface members. 14. A method according to 13, wherein said airplane includes forward control slats disposed forwardly of the respective wings and adjacent the fuselage and wherein the method includes automatic control of forward control slats. 15. A method according to 14, wherein the method includes controlling forward control slats with an adjusting velocity which is faster than the adjusting velocity of said outer adjusting surface members during flight with adjusting movement of both the inner and outer adjusting surface members. 16. A method according to 13, wherein said method includes controlling the velocity of adjustment of the inner adjusting members to be between 150% and 200% of the velocity of adjustment of the outer adjusting members during flight with adjusting movement of both the inner and outer adjusting surface members. 17. A method according to 13, wherein said method includes controlling the maximal velocity of adjustment of the outer adjusting members to be at least 25% lower than the maximal velocity of adjustment of the inner adjusting surface members during flight with adjusting movement of both the inner and outer adjusting surface members.

255

Background This application claims the priority of German application 198 36 191.7, filed in Germany on Aug. 4, 1998, the disclosure of which is expressly incorporated by reference herein. The method relates to a flight control device for improving the longitudinal stability of an automatically controlled airplane. Especially preferred embodiments relate to combat planes in a delta wing configuration which are artificially stabilized by means of an automatic control system. For the longitudinal control of automatically controlled airplanes, the control flaps or adjusting surfaces provided for this purpose, on the one hand, are used for controlling the airplane and, on the other hand, are used for stabilizing the airplane in flying ranges in which the airplane is at or beyond the stability limit. In known state of the art airplanes, the feedback signals of the automatic airplane control system have been switched with essentially the same amplification to the actuators of all control edge flaps and particularly to the actuators for trailing edge flaps responsible for the longitudinal control. For example, because of changed load conditions, such as outside loads, or because of a changed demand on the manoeuvrability of the airplane, control concepts of future airplanes will be required to automatically control airplanes with a greater rearward gravity centre position and thus with a lower stability. In the case of airplanes which were unstable from the beginning or become unstable only because of such shifts of the centre of gravity, it will then be required to stabilize the occurring instability by means of the adjusting surfaces. These results in higher adjusting speeds and larger deflections of the adjusting surface, which, however, excites the airplane, structure more and, as a result, higher structural couplings will occur. This causes an increase of the aeroservoelastic instabilities which are to be avoided. Since there is also a tendency toward softer wing structures, structure filters must be provided in the autopilot in order to avoid that the wing structure is excited to an unacceptable degree. However, such structure filters impair the efficiency of the autopilot so that these may have the result that the demands on the automatic control of the airplane can be met only in a limited manner. It is therefore an object to provide a flight control device for improving the longitudinal stability of an automatically controlled airplane which supplies improved possibilities for meeting the demands on the control and the stabilization of the airplane. This object is achieved by providing a light control device for the longitudinal control of an automatically controlled airplane having adjusting surfaces which, viewed from the longitudinal axis of the airplane, are on the outside and on the inside and are provided for the longitudinal control, wherein the inner adjusting surfaces experience an amplification which is by one half or more larger than that of the outer adjusting surfaces for the stabilization and control of the airplane in its longitudinal axis. This object is also achieved by providing a flight control device for the longitudinal control of an automatically controlled airplane having adjusting surfaces which, viewed from the longitudinal axis of the airplane, are on the outside and on the inside and are provided for the longitudinal control, wherein for the stabilization and control of the airplane in its longitudinal axis, a maximal velocity is provided for the outer adjusting surfaces which is at least by one fourth lower than that for the inner adjusting surfaces.

256

The above mentioned control features can be combined with the provision of controlled forwardly positioned slats. The control systems have the advantage that only by means of an increase of the superimposing of the amplification in the forward branch of the automatic flight control system and thus of the actuating signals to the actuators of the adjusting surfaces situated close to the fuselage, a higher degree of manoeuvrability can be achieved and a greater amount of instability can be stabilized. As a result, the phase delay which occurs in the automatic flight control system because of structure filters to be provided in the automatic flight control system, in the area of the control fraction of the wing flap movements, is at least limited or even reduced so that therefore the unfavourable effects of structure filters onto the low-frequency transmission action of the airplane, particularly the control fractions, can be considerably reduced. Other objects, advantages and novel features will become apparent from the following detailed description when considered in conjunction with the accompanying drawings. Brief Description of the Drawings Fig. 4a is a schematic representation of an adjusting surface configuration of a modern combat airplane, constructed according to preferred embodiments of the present application. Fig. 4b includes a block diagram of the whole automatic flight control system including the feedback signals of an automatically controlled airplane. The airplane illustrated in Fig. 4a has delta-type wings 2. On each delta-type wing 2, two adjusting flaps or adjusting surfaces respectively are provided, specifically in each case an inner adjusting surface 5 close to the fuselage and an outer adjusting surface 6 situated farther away from the fuselage of the airplane. The airplane also has slats 7 positioned forwardly of the wings which can be adjusted for the trimming and also for the control. In Fig. 4b, the transmission block for the transmission action of the interior adjusting surfaces 5 has the reference number 15; the transmission block for the transmission action of the outer adjusting surfaces 6 as the reference number 16 and the transmission block for the transmission action of the slats 7 has the reference number 17, the transmission action of the actuators and of the hydraulic systems in each case being taken into account. Since the outer adjusting surfaces 6 are not at all or only to a limited degree used for controlling and possibly for stabilizing the airplane, the transmission block 16 is illustrated by broken lines. In the model representation according to Fig. 4b, the transmission blocks 15 for the inner adjusting surfaces 5, the transmission block 17 for the slats 7 and, no more than at a significantly lower proportion, the transmission block 16 for the outer adjusting surfaces 6 act upon the transmission block 19 for the airplane and its sensor system relevant to the automatic flight control, particularly the inertial platform. The quantities 20, which are detected by a corresponding sensor system for the automatic flight control, particularly the rotating velocity, the angle of incidence and the vertical acceleration of the airplane, are led into the automatic control module of the flight computer whose transmission block has the reference number 21. The actuating signals 22 generated by these for the actuators of the adjusting surfaces, in the model representation of Fig. 4b, arrive on the transmission element 15 for the inner adjusting surfaces 5, the transmission element 16 for the outer adjusting surfaces 6 and the transmission member 17 for the slats 7.

257

Figure 4a: Control Surfaces of the fighter Aircraft

17 16

19

20

15 22

22

22

21

Figure 4b: Blockdiagramm for longitudinal vibration control As indicated in the representation of Fig. 4b by means of a broken line, the normal highfrequency feedback onto the outer adjusting surfaces 6 block 16 is eliminated or at least considerably reduced. Instead, an increase of the superimposing of the actuating signals takes place onto the inner adjusting surfaces 5 by block 15 and optionally additionally also onto the slats 7 by block 17. A suitable low-pass filter, which admits only the low-frequency actuating signals for the control of the outer flaps, has a stabilizing influence because of the outer flaps control. The superimposition essentially onto the inner adjusting surfaces 5 by block 15 takes

258

place with an amplification increase which is implemented in the flight computer 21 and leads to faster and larger adjusting movements of the inner adjusting surfaces 5 addressed by the flight computer 21 because the control effects of the outer adjusting surfaces 6 occurring according to the prior art must essentially be taken over by the inner adjusting surfaces 5. In this application, the terms adjusting "velocity" or "speed" refer to the angular speed or velocity of the respective adjusting surface members 5, 6, and 7. As an alternative, an adjusting speed limitation can also be provided which becomes operative for the outer adjusting surfaces 6 or control flaps starting at a lower adjusting speed than for the inner adjusting surfaces 5 or for the slats 7, if these are used in the case of the respective airplane. Preferably, the limit speed for the limiting of the outer control flaps 6 amounts to a fourth of the limit speed which is effective for the other control flaps 5. In the flight computer 21, notch filters or structure filters are provided for filtering out fractions of elastic vibrations in the feed back signals. However, the arrangement of this structure filters results in a partly considerable phase delay in the whole automatic control system of the airplane. These structure filters must be effective for the natural frequencies of the airplane structure. Particularly the attaching of outside loads to the wings of the airplane leads to low natural frequencies typically in the range of between 5 and 14 Hz. If, however, structure filters are provided for these frequency ranges in the automatic flight control, in the case of combat airplanes, in the flight-mechanical frequency range of approximately 1 Hz, a phase delay for the actuating quantities takes place because of the structure filters, whereby the characteristics of the automatic control system deteriorate with respect to the stabilizing and controlling of the airplane. By the increase of the amplification in the forward branch of the automatic control with respect to the inner adjusting surfaces 5 in relationship to the amplification with respect to the outer adjusting surfaces 6, the implementation of structure filters in the low frequency range is required to a significantly lower degree because the filtering of the structure coupling caused by the outer adjusting surfaces or flaps causes a phase shift up to 1 Hz to a high degree. Thus, in comparison to the prior art, a lower phase delay occurs in this frequency band. Possibilities are therefore provided of stabilizing configurations of a higher instability or greater rearward centre of gravity position or of improving the stability of today's airplanes. Preferably, the extent of the amplification for the inner adjusting surfaces 5 is twice as high as the amplification for the outer adjusting surfaces 6, but at least one half larger; that is, the movements of the inner adjusting surfaces 5 for stabilizing and controlling the airplane have an adjusting velocity which is at least by one half larger than that of the outer adjusting surfaces 6 in the same function. According to certain preferred embodiments, the automatic control system operable controls the velocity of adjustment of the inner adjusting members to be between 150% and 200% of the velocity of adjustment of the outer adjusting members. This control device is not only advantageous for delta-type wing planes but it can also be provided in the case of all other military and civil airplane configurations whose adjusting surfaces essentially cause the longitudinal control of the airplane. Thus, in the case of civil large capacity air carriers, the adjusting surfaces of the airplane wing can be provided and controlled such that essentially the adjusting surfaces situated on the inside take over the longitudinal control.

259

Description of the effect of the vibration alleviation system on the reduction of vibration levels at the Euro-fighter aircraft Lateral – anti symmetric vibration alleviation

Figure 5: FCS Vibration alleviation control laws for lateral axis: The rudder concept as described for example in Ref. 9 and 10 requires no structural changes to the conventional fin structure except for an eventual modification of the actuator to allow for higher actuation speed. A set of control laws are added to the flight control system to steer the rudder in such a way that aerodynamic forces are excited to counter respectively reduce the dynamic loads. This concept can effectively only be used to damp the first fin bending mode, as the inertia of the rudder would prohibits excitation of the rudder at frequencies of the order of the higher modes. The benefit of this concept is that it can in principle be implemented immediately by changing some of the low frequency Notch Filters in the flight control laws with no or only minor changes to the rudder. Figure 5 demonstrates the FCS block diagram which is valid for vibration alleviation control laws for lateral control and shows the feedback gains and phase advance filters and Notch filters. The active control concept of lateral anti-symmetric vibrations by high frequency activation of the rudder actuator is demonstrated in Fig.6. The required validation of phase and dB roll rate prediction versus frequency at the IMU station through flight test results, which is necessary for the design of the vibration alleviation system, had been performed for the Euro-fighter and an example of the comparison of phase and dB prediction and flight test is demonstrated in Fig. 7. The effect of the lateral vibration alleviation is shown in Fig 8.

260

Figure 6: Rudder Concept for lateral vibration control

Figure 7: Validation of phase and gain lateral response – comparison of prediction and flight test results for IMU roll rate due to rudderexcitation at Mach 0.8; 36kft, 1g

261

The auxiliary rudder concept as described for example in Ref. 9 and 10 requires structural changes to the conventional fin structure and for an implementation of the auxiliary rudder actuator. A set of control laws are added to the flight control system to steer the auxiliary rudder in such a way that aerodynamic forces are excited to counter respectively reduce the dynamic loads. This concept can effectively only be used to damp the first fin bending mode and probably the fin torsion mode, the inertia of the rudder prohibits excitation of the rudder at frequencies of the order of the higher modes. The benefit of this concept is that it can reduce not only the first fin bending mode vibrations. An example of the effects of the lateral vibrations alleviation system on the reduction of the accelerations in the first fin bending and torsion mode is presented in Figure 8.

Figure 8: Active control of fin vibrations by high frequency activation of rudder actuator, from Ref. 9 and 10.

Figure 9: FCS and vibration alleviation control laws for lateral axis: Feedback gains and phase advance filters and notch filters

262

Longitudinal – symmetric vibration alleviation The longitudinal vibration alleviation concept requires no structural changes to the conventional wing structure except for an eventual modification of the actuator to allow for higher actuation speed. A set of control laws are added to the flight control system to steer the inboard and outboard flap in such a way that aerodynamic forces are excited to counter respectively reduce the wing vibration levels, as also applied in the phase stabilisation concept for the minimisation of aeroservoelastic problems in the pitch axis. This concept can effectively only be used to damp the first and second wing mode. The benefit of this concept is that it can in principle be implemented immediately without changing the control system. Figure 9 demonstrates the FCS block diagram which is valid for vibration alleviation control laws for pitch control and shows the feedback gains and phase advance filters and Notch filters. Figure 10 shows the pitch control concept. The required validation of phase and dB pitch rate prediction versus frequency at the IMU station through flight test results, which is necessary for the design of the vibration alleviation system, had been performed for the Euro-fighter, see Ref. 7 and Fig. 11. The effect of the active vibration alleviation could be demonstrated for instance by flight flutter tests, whereby the damping of the first wing bending mode was increased by a factor of 3, which resulted in a high alleviation of the vibration level in this mode.

Figure 10: Active control of symmetric wing vibrations by high frequency activiation of flap actuators

263

Figure 11: Validation of phase and dB prediction versus frequency by flight test results, Ref 7

Conclusions A novel system and method for compensating structural vibrations of an aircraft caused by outside disturbances has been described. The new method uses a sensor system the existing sensor data of the inertia measuring system (IMU) instead of accelerometer signals applied in other systems. The system and method is documented in EADS patents. Some examples of the validation and the effects of the method are demonstrated for the symmetric mode and asymmetric mode vibration alleviation using Euro-fighter flight test results and predictions. The new system represents a very efficient method which would allow a cost effective way for the implementation of a vibration alleviation system in fighter and transport military and civil aircraft.

264

References 1. US Patent 6416017 – Daimler Chrisler AG, Stuttgart, Germany System and method for compensating structural vibrations of an aircraft caused by outside disturbances, July 2002 2. US Patent 6816823 - EADS Deutschland GmbH, Munich, Germany Process for designing flight controllers, Nov. 2004 3. US Patent 6729579 - EADS Deutschland GmbH, Munich, Germany Flight control device for improving the longitudinal stability of an automatically controlled airplane and method of operating same, May 2004 4. EP 1 134 639 A2-EADS Deutschland GmbH, Munich, Germany Verfahren zur Auslegung von Flugreglern, März 2001 5. DE 198 35 191 C1 - EADS Deutschland GmbH, Munich, Germany Flugsteuerungseinrichtung zur Verbesserung der Längsstabilität eines geregelten Flugzeugs, April 2000 6. DE 198 41632 C2 - EADS Deutschland GmbH, Munich, Germany Verfahren zum Kompensieren von Strukturschwingungen eines Flugzeugzeugs aufgrund äußerer Störungen, Juni 2001 7. J. Becker Structural Coupling resulting from interaction of flight control system and aircraft Structure, IFASD, International Forum on Aeroelasticity and Structural Dynamics 17-20 June1997, Rome, Italy 8. J. Becker, W. Luber Flight Control Design Optimization with Respect to Structural Dynamic Requirements, AIAA 96-4047, 6th AIAA/NASA Symposium on Multidisciplinary Analysis and Optimization, MDO; Bellevue, WA, USA, September 1996 9. J. Becker Active Buffeting Vibration Alleviation – Demonstration of Intelligent Aircraft Structure for vibration and dynamic load alleviation, ESF –NFS workshop on Sensor Technology and intelligent Structure, Como, April 2002 10. Johannes K. Dürr, Ursula Herold-Schmidt, and Jürgen Becker Active Fin-Buffeting Alleviation for Fighter Aircraft, RTO 2000 Braunschweig RTO-MP-051: Active Control Technology for Enhanced Performance Operational Capabilities of Military Aircraft, 8-11. May 2000, Braunschweig, Germany

Operational Modal Analysis on a Modified Helicopter E. Camargo+, N-J. Jacobsen*, D. Strafacci+ +

Centro Técnico Aeroespacial, Instituto de Aeronáutica e Espaço, Praça Eduardo Gomes, 50, 12.228-904 São José dos Campos, SP, Brazil *

Brüel & Kjær Sound & Vibration Measurement A/S Skodsborgvej 307, DK-2850 Nærum, Denmark

ABSTRACT Two mounts were added to a helicopter making it possible to carry different payloads. To validate the structural effects of these modifications, modal tests were performed on-ground on the helicopter in its standard configuration as well as in its modified configuration with the added payloads. In addition, an in-flight test was performed to verify the impact on the existing flight envelope. For all tests, Operational Modal Analysis was used. The obtained results allowed for updating the flight procedures and operating profiles for the helicopter and provided added flexibility with respect to the best possible helicopter configuration to obtain the mission objectives, while maintaining optimum safety for the flight crew. 1

Introduction

During flights, a helicopter’s tail structure undergoes significantly dynamic loads from aerodynamic flow and vibrations induced by the engine and rotors. Consequently, when modifications are made to a helicopter, the potential change in structural behavior of the tail section can be critical. This paper presents different Operational Modal Analysis tests that were performed to study the change in the modal parameters on a helicopter after attaching a new payload system to it. The tests are essential to validate and refine the design of payload systems and provide information about internal stresses at critical locations. This information is required to avoid both helicopter tail damage thereby reducing the maintenance time between flights and to impose potential limitations to the flight paths. In Chapter 2, the configurations of the helicopter are described together with the used test setups and instrumentation. In Chapter 3, the various Operational Modal Analysis tests and their assumptions are explained and the results are presented and discussed in Chapter 4. The conclusion is drawn in Chapter 5. 2

Configurations, Test Setups and Instrumentation

2.1

Configurations

Two different configurations were used to perform the measurements and analyses:  Standard configuration of helicopter  Modified configuration with payloads attached The payloads were located on both sides of the helicopter close to the centre of gravity and consisted each of a rectangular plate where cargos were attached. The configurations with accelerometer positions are shown in Fig. 1 together with the standard configuration of the helicopter during flight.

T. Proulx (ed.), Advanced Aerospace Applications, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 4, DOI 10.1007/978-1-4419-9302-1_22, © The Society for Experimental Mechanics, Inc. 2011

265

266

7000 640

2840

18 830

15

1420

1500

17 530

16

19

4400

220

10 11 12 9 8

1 2

Payload

13 14

19 7 6

Payload

62 0

59 0

32 0

32 0

59 0

59 0

220

3

5

39 0

27 28 56 0

59 0

32 31

10 (ref.)

61 0

61 0

5 (ref.)

36 0

20 0

30

29

25

26

22 0

36 0

Fig. 1 Standard helicopter (top) and configurations and accelerometer positions (bottom)

267 The number, positions and directions (DOFs) of the accelerometers were selected from preliminary analyses using simulation while at the same time keeping the number of DOFs down to reduce the instrumentation. The chosen DOFs were sufficient to obtain good mode shape estimates in a single measurement, thereby avoiding the need for using roving accelerometers. This gives the best possible data consistency and significantly reduces expensive test time. For both the standard and modified helicopter configuration, the measurements were done using 26 accelerometers mounted on the tail section of the structure - 16 uniaxial accelerometers measuring in the global z-direction, 6 triaxial accelerometers measuring in the global y- and z-directions and 4 triaxial accelerometers measuring in all three global directions. For the modified helicopter configuration 5 additional triaxial accelerometers were added of which 4 were measuring in global y- and z-directions and one in all three global directions. In Fig. 2, a model of the tail structure of the helicopter is shown, that is, from point 19 and backwards. A model of the payload is shown as well. Note that point 11 was not replicated on the opposite side as just one point was enough to predict the vibration level in the junction of the two structures. Also, the number of measurement DOFs could be reduced for the payloads.

Fig. 2 Models used for the tests Standard Helicopter (left), Modified Helicopter (middle), Payload (right)

2.2

Test Setups

Four different tests were performed:    

An on-ground test of the helicopter in its standard configuration using single shaker excitation An in-flight test of the helicopter in its standard configuration using internal and ambient excitation An in-flight test of the helicopter in its modified configuration using internal and ambient excitation An on-ground test of the payload system using hammer impact excitation

From the first two tests, the change in structural modes from ground and in-flight testing of the same standard helicopter configuration could be observed before making an in-flight test of the modified helicopter configuration. The test on the payload system was performed to predict and validate its structural effects on the modified helicopter. Examples of test setups are shown in Fig. 3.

268

Fig. 3 Examples of test setups Shaker excitation (left), accelerometer mounting (top right) and front-ends (bottom right) 2.3

Instrumentation

For all tests, data acquisition and analysis were performed using a Brüel & Kjær PULSE™ analyzer system consisting of three 17 channel Type 3560-C front-ends connected to a Dell Latitude D610 PC running PULSE™ software. Data acquisition and validation were performed using PULSE Modal Test Consultant™ Type 7753 software and modal parameter extraction using PULSE Operational Modal Analysis Type 7760 software. The responses were measured using 10 Brüel & Kjær Miniature Triaxial Acclerometers Type 4520 and 16 Brüel & Kjær Uniaxial Accelerometers Type 4514 both of the piezoelectric type. For the shaker test, an LDS V455 permanent magnetic shaker with power amplifier PA 1000 was used. The impact testing was performed using a Brüel & Kjær One-pound Head Modal Sledge Hammer Type 8207. 3

Operational Modal Analysis Tests

For all tests, Operational Modal Analysis was used to analyze the acquired time data. Operational Modal Analysis is based on stochastic processes and an underlying assumption for the identification techniques is that the excitation is from broadband random forces distributed over the structure (random in time and space) as described in [1]. Consequently, the excitation should ideally come from multiple broadband sources randomly distributed over the test object. This assumption is clearly fulfilled for the in-flight tests where the ambient excitation from the aerodynamic flow and the vibrations induced by the engine and the rotors by nature is broadband and random in time and space. In addition, harmonic components (deterministic signals) are present due to the rotational parts from the engine and rotors. There are techniques in the PULSE Operational Modal Analysis software that automatically identify and eliminate the influence of harmonic components so that they do not affect the obtained results. For more information on identification and elimination of harmonic components in Operational Modal Analysis see [2].

269 For the shaker test and the hammer impact test, the assumption is violated as regards the random excitation points as, in both cases, only a single excitation point was used. It was, however, considered acceptable to violate the assumption as previous classical modal analysis tests of the standard helicopter configuration showed identical results for the critical modes of interest. For the payload system itself the model was fairly simple and there was no notable limitation in selecting only a single excitation point. Another important parameter in Operational Modal Analysis is the length of the time recording as explained in [2]. The required time recording length depends on various factors such as the spectral shape and duration of the excitation signal, presence of harmonic components, the complexity of the test object and the quality of the data acquisition equipment. However, as a rough rule of thumb, the time histories should be at least 500 times longer than the period of the lowest mode of interest. For the tests performed, the lowest modes of interest – the critical modes – were above 30 Hz requiring a time recording length of at least 17 s which was fulfilled in all tests. Using only Operational Modal Analysis for all tests simplified the complete test procedure and was, as such, an important objective to reduce expensive test time. 3.1

Analysis Parameters

All measurements were done in a 400 Hz frequency range using at least 60 s of time data. The Operational Modal Analyses were performed using the Enhanced Frequency Domain Decomposition (EFDD) peak-picking technique. A frequency resolution of 1 Hz was used in the underlying spectral estimation by using 512 frequency lines. The model estimation was performed using a MAC Rejection Level of 0.9 and the maximum and minimum correlation limits were set to 0.95 and 0.50, respectively. An example of the EFDD technique is shown in Fig. 4 for the modified helicopter configuration during flight. For more information on the frequency domain decomposition techniques for Operational Modal Analysis, see [2] and [3].

Fig. 4 EFDD used on data from the modified helicopter configuration during flight The three main critical modes are picked

270 4

Results

In this chapter the main results from the four different tests are presented and discussed. 4.1

On-ground Test Results of the Standard Helicopter Configuration

Two different tests were performed using single shaker excitation - one using random noise and one using swept sine as excitation signal. Both tests covered the frequency range from 10 Hz to 500 Hz. The swept sine test was, in general, found to give better stability in the estimation of the modal parameter and the results are summarized in Table 1. Mode #

Frequency [Hz]

Damping Ratio [%]

Description

1

13

1.77

Rigid Body Pitch Mode

2

15

1.63

Rigid Body Roll Mode

3

28

1.82

1st Bending Mode of the Vertical Stabilizers (in-phase)

4

36.0

2.24

1st Bending Mode of the Vertical Stabilizers (out-of-phase)

5

57

1.63

1st Torsional Mode of the Vertical Stabilizers (out-of-phase)

6

68.9

0.97

1st Torsional Mode of the Vertical Stabilizers (in-phase)

7

90

0.79

1st Bending Mode of the Horizontal Stabilizer

8

96.9

0.68

2ndBending Mode of the Horizontal Stabilizer

9

106

0.58

1st Bending Mode of the Tail Fin

10

128

1.26

3rd Bending Mode of the Horizontal Stabilizer combined with Torsion of the Vertical Stabilizers (in-phase)

11

190

0.67

3rd Bending Mode of the Horizontal Stabilizer

12

210

0.30

1st Torsional Mode of the System

Table 1 Modes of standard helicopter configuration from shaker testing using swept sine excitation The three main critical modes of interest are shown in italics and with increased accuracy Using the results of this test and comparing it with the results of the standard helicopter during flight made it possible to find the critical modes, that is, modes where the natural frequency is significantly shifted and at the same time modes that significantly influence the structural and aerodynamic behavior of the helicopter. High vibration level at these frequencies can reduce the time between maintenance of the helicopter and are, therefore, important to know. The three most critical modes were found at 36.0 Hz, 68.9 Hz and 96.9 Hz. Fig. 5 shows the mode shapes at these critical natural frequencies.

271

Fig. 5 Mode shapes at the three most critical natural frequencies Left to right: (36.0 Hz; 2.24%), (68.9 Hz; 0.97%) and (96.9 Hz; 0.68%) 4.2

In-flight Test Results of the Standard Helicopter Configuration

Measurements were done using four different flight path configurations.    

Hovering at 100 m altitude (no influence from aerodynamic flow) Ascendant flight with increasing speed until reaching the speed limit of the helicopter Descendent flight Cruising at 3000 ft at different speeds

The flight time for each path was 30 minutes with 60 s of data acquisition every 5 minutes. The mode shapes at the critical natural frequencies are shown in Fig. 6.

Fig. 6 Mode shapes at the three most critical natural frequencies Left to right: (34.0 Hz; 2.01%), (65.2 Hz; 0.95%) and (97.5 Hz; 0.61%)

Comparing the results to the measurements made on-ground, the same mode shapes were observed but with a general downward shift in the natural frequencies as expected due to the added mass (flight crew) of the helicopter and the aerodynamic effects imposed on the tail structure.

272 It was observed that an increase in speed did not change the mode shapes, but caused increased vibration levels. Maneuvers were performed during cruising and apart from increased vibration levels some shifts in natural frequencies - especially at helicopter’s speed limit - were observed that are believed to be caused by the interference of the aerodynamic flow with the structure. These observations were important to predict the structural behavior and thereby be able to increase the time between maintenance. 4.3

In-flight Test Results of the Modified Helicopter Configuration

The mode shapes at the critical natural frequencies are shown in Fig. 7.

Fig. 7 Mode shapes at the three most critical natural frequencies Left to right: (33.9 Hz; 0.43%), (64.7 Hz; 0.21%) and (96.9 Hz; 0.15%) With payloads attached to the helicopter, an increase in the vibration levels was observed, but no significant reduction in the natural frequencies despite the added mass. Also more significant changes to the natural frequencies could have been expected due to interference of the aerodynamic flow through the stabilizers as the payloads were attached just on the same line as the horizontal stabilizers. This is, however, considered to be a likely explanation for the significant decrease in the observed damping ratios, but more studies need to be conducted to completely understand this phenomenon. As shown in next section, the natural frequencies of the payloads were not coinciding with the beat frequencies of the helicopter’s rotor and they were different to the natural frequencies of the helicopter structure. This leads us to conclude that the payloads can be attached to the helicopter without critically reducing the flight performance. Attention should, however, be paid when maneuvers are done close to the helicopter’s speed limit if payloads are attached. 4.4

Test Results of the Payload

The analysis of the payload system was done using hammer impact excitation. The purpose was to understand how the payload system influences the aerodynamic flow along the tail section and to see if there were some coupling effects between the payload modes and the modes of the tail section. The modes found are shown in Table 2 and it can be seen that only the first mode of the payload is close to the modes of the helicopter. During flight no interferences were observed.

273

Mode #

Frequency [Hz]

Damping Ratio [%]

Description

1

29

1.77

1st Bending Mode

2

77

2.80

Torsional Asymmetric Mode

3

120

0.42

Combined Asymmetric Torsional with Bending Mode

4

199

1.13

Torsional Symmetric Mode

Table 2 Modes of the payload system from hammer impact testing

5

Conclusion

Operational Modal Analysis on a helicopter has been performed during flight with the helicopter in its standard configuration and in a modified configuration with payloads attached. The standard configuration of the helicopter was also tested onground. The results showed expected downward shifts in the natural frequencies during flight both for the standard and modified configuration. This was especially the case when the dynamics loads were increased with maneuvers at helicopter’s speed limit. The observed natural frequencies were presented and discussed. Lower damping ratios were found during flight and a large reduction was seen when the payloads were attached. This is expected to be caused by interference of the aerodynamic flow through the stabilizers but more studies have to be conducted. The critical mode shapes of the tail structure during flight were found to be identical to those found at the on-ground test. The maximum vibration levels were observed during flight with payloads attached due to the forces induced by aerodynamic air flows. The vibration levels measured were, however, found to be within the limits established. Maintenance time can be reduced as the attached payloads did not result in any observable reduction of the helicopter’s flight performance. Finally, it was found that by applying Operational Modal Analysis during flight the time used for testing can be reduced.

References [1]

Herlufsen, H; Andersen, P; Gade, S; Møller, N Identification Techniques for Operational Modal Analysis – An Overview and Practical Experiences 1st IOMAC Conference, 2005

[2]

Jacobsen, N.J.; Andersen, P Operational Modal Analysis on Structures with Rotating Parts 23rd ISMA Conference, 2008

[3]

Gade, S; Møller, N; Herlufsen, H; Konstantin-Hansen, H Frequency Domain Techniques for Operational Modal Analysis 1st IOMAC Conference, 2005

Development of New Discrete Wavelet Families for Structural Dynamic Analysis

Jason R. Foley* and Jacob C. Dodson *

Air Force Research Laboratory AFRL/RWMF; 306 W. Eglin Blvd., Bldg. 432; Eglin AFB, FL 32542-5430, [email protected]

Andrew J. Dick, Quan M. Phan, and Pol D. Spanos Rice University

Jeffrey C. Van Karsen Keweenaw Research Center, Michigan Technological University

Gregory L. Falbo LMS Americas, Inc.

ABSTRACT Wavelet analysis is a powerful method for analyzing the time histories of signals. A discrete wavelet family is developed for structural dynamics by using the dilation equation to calculate scaling function coefficient values for arbitrary waveforms. The performance of this formula is verified by analyzing the scaling functions of multiple Daubechies wavelets. To assure the new discrete wavelet families have the characteristics of a specific system, the formula is applied to analytical and experimental response data. The relationship between the number of coefficients and their ability to successfully capture the characteristics of the signal is studied and a method is developed for determining the number of coefficients to be used when applying the formula. The resulting new families of discrete wavelets are based upon the nominal characteristics of a given system for use in signal processing and model discretization applications. The impulse response of a structure is proposed as a tuned baseline for structural health monitoring applications; the corresponding Wavelet Analysis of Structural Anomalies using Baseline Impulse-Response (WASABI) method is presented and discussed in the context of the wavelet development. NOMENCLATURE

⁄2 ,

 

A, H, L, W E I K N

Wavelet function Wavelet scaling function Damping ratio Density Natural frequency (nth mode) Damped angular frequency Wavelet filter banks Mass Beam area, height, length, and width Elastic modulus Moment of inertia Modal index Number of coefficients

INTRODUCTION Wavelets are waveforms with localized properties which are used in a number of signal analysis (e.g. [1]) and model discretization applications (e.g. [2]). The energy within a wavelet is distributed over a finite domain, unlike the sinusoids used in Fourier-based signal analysis which have their energy distributed from negative to positive infinity. This allows for

T. Proulx (ed.), Advanced Aerospace Applications, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 4, DOI 10.1007/978-1-4419-9302-1_23, © The Society for Experimental Mechanics, Inc. 2011

275

276

wavelet-based signal analysis to be utilized much more successfully for signals which contain transient oscillations or other localized properties [3]. The use of wavelets has become increasingly popular in the analysis of signals from physical systems in recent decades. These studies include the application of wavelet analysis to various mechanical [4] and electrical [5, 6] systems. A number of recent studies have focused on the use of wavelets to analyze the impulse responses of these systems. Since there are numerous wavelets available to serve as a basis for an analysis, research efforts have sought to develop means to identify the best wavelet for a specific signal. Various techniques for evaluating wavelets in order to determine which would enable the best approximation of a given signal have been developed [7, 8]. These methods include processes that use statistical analysis, cost function optimization, and the evaluation of the signal projected into the wavelet subspaces. However, in some applications it is desirable to use a wavelet which has characteristics that are more closely related to those of a system or signal than available wavelets are able to provide. To this end, a number of efforts have been made toward the development of methods to create new wavelets based upon the characteristics of a given signal. Chapa and Rao [9] developed a methods for deriving equations for continuous wavelets and their corresponding scaling function with band-limited orthonormal properties. In the work of Shark and Yu, a method was developed to create orthonormal wavelet filter banks by using the genetic algorithm [10]. These wavelets were developed in order to avoid issues associated with the influence of time shifts on the values of the transform coefficients. The performance of wavelets prepared by using this method was compared to that of Daubechies wavelets for denoising applications and it was found to out-perform the Daubechies wavelets by 24%. Gupta et al. also proposed methods for the creation of new discrete wavelets based upon a given signal by using statistical methods and by minimizing the energy of a signal projected into the wavelet subspace [11, 12]. Given the amount of information about a system which is present within an impulse response, a number of recent efforts have focused on creating wavelets from these signals. Suárez and Montejo utilized the method developed by Chapa and Rao in order to derived a new continuous wavelet based upon the impulse response of a single degree-of-freedom underdamped oscillator. This wavelet was used in order to accurately simulate the frequency characteristics of an earthquake [13]. By using the impulse response reflected about the origin as the wavelet function, a scaling function was derived to provide complementary frequency characteristics [14, 15]. In the recent work of Yan and Gao, a set of recursive equations were derived from the dilation equation in order to calculate the coefficients for a discrete wavelet from impact response data [16]. The calculated coefficient values are then refined by minimizing an error function. Within this study, a new formula is proposed which will provide the foundation for the development of a method to create new wavelet families based upon a given signal. WAVELETS Two different categories of wavelets have been developed: continuous wavelets and discrete wavelets. Continuous wavelets, defined by continuous (analog) functions, are widely used in signal analysis and include wavelets such as the Morlet wavelet /2   cos 5 . Discrete wavelets are defined by a pair of filter banks and which serve to separate a exp discrete (digital) signal into low frequency content and high frequency content, respectively. The waveform corresponding to these filter banks are obtained by using the dilation equation and the wavelet equation. The dilation equation, also known as the recursive equation and defined in Eq. (1), relates the scaling function to the half-scaled version of the scaling function 2 through the scaling function coefficients . By applying multiple iterations of the dilation equation with the 2 scaling function coefficients of a specific discrete wavelet and an initial vector consisting of a one followed by zeros, the vector will converge to the corresponding scaling function.  

√2

2

 

(1) 

Once a sufficient number of iterations have been performed and the scaling function has converged, the scaling function is substituted into the wavelet equation, defined by Eq. (2). By using the coefficients of the high-pass filter bank , the waveform for the discrete wavelet is produced. The coefficients of this filter bank are obtained from the coefficients of the low-pass filter bank through Eq. (3) when the wavelet is orthogonal.  

√2

2

 

(2) 

  1 . (3) In addition to being more readily implemented than continuous wavelets in digital signal processing applications, some discrete wavelets have additional properties such as compliance with quadrature mirror filter (QMF) requirements. These wavelets, which include the Daubechies wavelets [17], are able to be used to decompose a signal into multiple levels of approximation and detail coefficients and then perform a perfect reconstruction of the original signal. These properties also

277

allow for some discrete wavelets to be used as a basis for the discretization of systems modeled with partial differential equations [2]. Whether discrete wavelets are being used in multi-resolution analysis or for model discretization, it is important to select a wavelet with characteristics in common with the signal or system being studied. DERIVATION OF WAVELET FORMULATION In order to calculate the scaling function coefficients for an arbitrary waveform, we begin with Eq. (1). The right side of the dilation equation corresponds to the convolution of the scaling function coefficients and the half-scale scaling function 2 ,   (4) 2 . In order to solve for the coefficients of the scaling function, a discrete version of this equation is prepared in matrix form. The convolution operation is represented by the product of a vector and a matrix: .    (5)  / In the matrix form of the convolution operation, the coefficients of the scaling function ( ) are represented as a row vector of length . The half-scale scaling function /2, which is then used to populate a /  is represented as a vector of length matrix with rows and columns. The half-scaled scaling function is indexed in each successive row in order to place the vector along the diagonal of the matrix and the remainder of the matrix is filled with zero vectors: 0 / 0 /

 

/

0

/

0

0

/

0

/

. 

(6) 

/

It is also possible to represent the convolution operation by preparing the matrix with the scaling function coefficients ( ) along the diagonal and multiplying the matrix by a row vector corresponding to the half-scale scaling function. However, the goal of the derivation is to solve the equation for the coefficients and the form of the convolution in Eq. (5) is used. In order to complete the derivation, it is necessary to further manipulate Eq. (6). The transpose of Eq. (5) is   .  /

(7) 

The final action in the derivation is to pre-multiply both sides of Eq. (7) by the inverse of / . However, this matrix is generally not square, , so the standard inverse operation cannot be applied. In order to complete the derivation, the Moore-Penrose pseudoinverse is used. The result of this operation,  

/

, 

(8) 

provides the means to calculate the vector of coefficients for the scaling function representing an arbitrary waveform. Thus, in order to calculate these coefficients only two inputs are required: the discrete waveform of length M and the number of coefficients N. VERIFICATION OF WAVELET FORMULATION In order to verify the performance of Eq. (8), the discrete scaling functions of known discrete wavelets are examined. Scaling function coefficients are calculated and compared to the true values which were used to prepare the discrete waveforms. Three Daubechies wavelets are used for this purpose. By using Eq. (1), the scaling function coefficients of the Daubechies 2 (DB2), Daubechies 3 (DB3), and Daubechies 4 (DB4) wavelets are used to prepare the discrete waveforms of these wavelets for between 4 and 10 iterations. Equation (8) is then applied by using the appropriate number of coefficients, N = 4, N = 6, and N = 8, respectively. The scaling function for the Daubechies 2 wavelet after 8 iterations is compared with the waveform produced by using the true coefficient values in Figure 1(a.). Excellent agreement is observed. The calculated coefficients are then compared with the true values and the norm of the difference between the two vectors is determined. These values are plotted in Figure 1(b.). The data plotted within this figure indicates that by increasing the number of iterations of Eq. (1), the accuracy of the calculated coefficients is improved. With each successive iteration, the norm of the error is reduced by a factor of two. Additionally, when comparing the norm of the error for the calculated coefficients of the three different Daubechies wavelets, higher order wavelets are determined to require additional iterations of Eq. (1) in order to attain the same level of accuracy as the coefficients calculated for lower order wavelets.

278

(a.) (b.) Figure 1. (a.) Comparison of the scaling function obtained from the coefficient calculated by using (8) to the waveform produced by using the true coefficient values and Equation (1). (b.) Plot of the norm of the error in the calculated scaling function coefficients versus the number of iterations of the dilation equation used to produce the discrete waveform of the scaling function for three different Daubechies wavelets. ANALYSIS The method developed in the previous section is now applied to response data from different systems. By using the model of a single degree-of-freedom, underdamped linear oscillator, an impulse response is derived. This response, 1   sin .  (9)  is a function of the natural frequency and the damping ratio of the system. The parameter corresponds to the mass of the system and the damped natural frequency is represented by 1 . Three sets of parameter values are used to prepare the impulse response signals. These parameter sets are presented in Table 1 with the mass parameter held constant. The length of the data set is selected to be an integer number of periods which is greater than five time constants in order to allow for the response to decay. Table 1. Parameter sets for impulse response data. Damping Ratio Parameter set Natural Frequency 1 2 3

2

0.3 0.5 0.3

While the number of coefficients is already known for the Daubechies wavelets, this information must be determined for arbitrary waveforms. By using Eq. (9) with the values of parameter set 1 from Table 1, impulse response data is prepared and Eq. (8) is applied in order to calculate the coefficients of the scaling function. The coefficients are calculated for different values of N. Without the benefit of a vector of true coefficient values, the accuracy of the calculated coefficients is determined by using them with Eq. (1) and the half-scale impulse response signal to calculate the scaling function. The calculated scaling functions are compared to the impulse response signals for three values of N in Figure 2. In Figure 2(a.), the scaling function calculated with N = 4 is compared to the impulse response signal and poor agreement seen. With too few coefficients to successfully capture the information of the signal, the half-scale components are assembled in such a way that the individual components are still identifiable within the calculated scaling function. When the number of coefficients is increased to N = 20 in Figure 2(b.), the agreement between the calculated scaling function and the impulse response signal is greatly improved. The individual half-scale components are significantly less prominent within the calculated scaling function. The improvement continues with N = 30 in Figure 2(c.); while better agreement is observed in the first period of oscillation, a greater amount of error develops in the latter half of the scaling function. The error between the calculated scaling function and the original data set is calculated by integrating the absolute value of the difference between the two data sets for a total time normalized to one. This calculation provides a single numerical value which characterizes the relative error of the coefficients in capturing the characteristics of the original signal. This cumulative error value is plotted versus a range of different values of N for the impulse response prepared with the values for parameter set 1 in Figure 2(d.).

279

(a.)

(b.)

(c.) (d.) Figure 2. Comparison of the scaling functions derived by using the coefficients calculated for the impulse response signal for parameter set 1 for (a.) N = 4, (b.) N = 20, (c.) N = 30, and (d.) the error between calculated scaling function and original data versus the number of coefficients for the three parameter sets. The error data for parameter set 1 in Figure 2(d.) has smaller values as the number of coefficients is increased. By increasing the value of N beyond twenty, the error is decreased further but by only a small amount. For example, when the number of coefficients is increased to N = 30, the error is decreased by only 0.0018. In order to further study the influence of the number of scaling function coefficients on the accuracy of the calculated coefficients, impulse responses are calculated by using Eq. (9) with values from parameter sets 2 and 3. Again coefficients are calculated for different values of N and used to calculate the scaling function from the half-scale response data in order to compare it to the impulse response. The error calculated for these parameter sets are plotted in Figure 2(d.). Based upon the error data in this figure, the number of coefficients are selected as N = 16 and N = 20 for parameter sets 2 and 3, respectively. In order to further study the influence of the number of coefficients when using Eq. (8), the number of coefficients required for each of the three parameter sets is examined. When the value of the damping ratio is increased from parameter set 1 to parameter set 2, fewer coefficients are required. When the frequency is increased from parameter set 1 to parameter set 3, the error remains unchanged and the same number of coefficients is used. This occurs since the length of the signal is adjusted based upon the system parameters. When the lengths of the signals are normalized, they are identical. These trends suggest that signals of greater complexity require an increased number of coefficients. EXPERIMENT The performance of the formula is then evaluated by analyzing experimentally collected impact response data. The system from which the data is collected is a cantilevered aluminum beam, shown in Figure 3. The geometry of the beam is 360 mm × 45 mm × 4.05 mm (L×W×H). The cantilever is instrumented with a PCB Piezotronics accelerometer with a sensitivity of

280

103.6 mV/g. The response data is collected by using a National Instruments CompactDAQ data acquisition system. Ten seconds of data is collected with a sampling rate of 50 kSa/s. A signal is sent to the shaker in order to provide base excitation for the aluminum beam. The oscillations of the free end of the beam are sufficiently small that there are no interactions with the adjacent fixture which is used in other studies.

Figure 3. Demonstration experiment using a cantilevered aluminum beam with single accelerometer to measure the resulting impact response. The cantilevered beam is struck and the impact response is recorded. In order to minimize the effects of signal noise, a lowpass filter is applied to the signal. The time history data is shown in Figure 4(a.) and shows a characteristic damped resonant response dominated by the fundamental mode of the system. The autopower spectrum in Figure 4(b.) confirms the fundamental resonance is the strongest relative to the higher frequency peaks starting at 115 Hz. The experimental data is evaluated against analytic results. In this simple geometry, the first few modes of a clamped-free beam are found using tabulated results [18]. The nth mode is  

2

, 

(10) 

65 GPa is the assumed elastic modulus of aluminum, 1.8 10  m is the cross-sectional area, ⁄12 2.4 10  m is the moment of inertia of the beam, 2700 kg/m is the density, and is a modespecific constant as defined in Table 2. Since √ / 5.78 m2 /s from the properties of the beam, the natural frequencies resulting from Eq. (9) can be calculated and are shown in Table 2. The predicted modal frequencies of the beam are higher than the experimentally observed values, which can be attributed to mass loading effects of the accelerometer. (The third experimentally observed mode at 175 Hz is not due to bending.) where

281 80

10

10

115 Hz

60 40

175 Hz

392 Hz

8

10 Linear power spectrum [g2]

Acceleration [g]

20 0 -20 -40 -60

22.5 Hz

6

10

4

10

-80 -100 -120

2

10 0

0.1

0.2 Time [s]

0.3

0.4

1

10

2

10 Frequency [Hz]

3

10

4

10

(a.) (b.) Figure 4. The experimental (a.) acceleration time history and (b.) linear autopower spectrum of the output from the cantilevered beam. The modes of the beam are indicated as peaks in the autopower spectrum. Table 2. Analytic bending modes of a cantilevered (clamped-free) beam. Modal constant Natural Frequency Mode Number n 1 1.8751 24.97 Hz 2 4.694 156.5 Hz 3 7.855 438.1 Hz 4 10.996 858.6 Hz 1.419 kHz 2 1 n>4 2.120 kHz 2 In order to focus on the response data within the signal, a subset of 20,001 data points of filtered data is used. Processing of the recorded response data is performed by using the fir1 command in MATLAB. This method uses a window and lowpass filter with cut-off frequency set to 125 Hz, allowing for the response of only the first two experimental modes. The filtered signal and Eq. (8) are used in order to calculate scaling function coefficients. The scaling function is calculated with N = 60 and compared with the impact response data in Figure 5(a.). The calculated scaling function displays very good agreement with the original signal—the absolute error is calculated for a range of numbers of coefficients and plotted in Figure 5(b.). Due to the increased complexity of the experimental data, the error values in this figure do not display the same exponential convergence as for the impulse response data. Generally, a much larger number of coefficients is required to achieve the same accuracy when compared to the analytical response.

282

(a.) (b.) Figure 5. (a.) Comparison of calculated scaling functions with experimentally collected impact response data for N = 60. (b.) Plot of error between calculated scaling function and experimental data versus the number of coefficients. THE WASABI METHOD We now introduce the Wavelet Analysis of Structural Anamolies using Baseline Impulse-Response (WASABI) method. This health-monitoring concept is based on defining new wavelet families based upon system responses and arbitrary waveforms. In the WASABI construct, the mother wavelet incorporates the dynamics of the system, specifically the impulse response of the system. The dynamic response chosen can be obtained from computational estimates or experimental observations of a healthy system at a given output location under impulsive loading. This baseline impulse response (BIR) can also be generated from synthesized or experimental frequency response functions. The hypothesis is the BIR will remain constant while the system is undamaged (i.e., “healthy”) but changes in the system response will create signatures in the wavelet decomposition’s amplitude at specific scales, making it possible to detect and identify temporary or permanent changes in the structure. For both continuous (CWT) and discrete (DWT) wavelet transforms, the analysis of a given impulse-excited output from a healthy system yields a highly localized amplitude at the onset of the event (time) and at the appropriate scale/detail level. For CWT, the result is concentrated at unity scale. For the DWT, the approximation at the first detail level will be highly accurate. A damaged system, in contrast, will demonstrate a de-localization of the wavelet intensity as the wavelet compensates for changed response. This is shown in the continuous wavelet scalogram in Figure 6. The analyzed case is a hypothetical time history with three damped impulse responses. The first two are identical while the third impulse has the same natural frequency but a reduced damping to simulate a subtle structural change. The scalogram shows the time localization is decreased. Similarly, the scale localization is also slightly decreased, shifted towards higher scale to capture the changes with finer resolution.

283 Signal: Three Impulse Responses

Signal

2 1 0 -1

0

2

4

6

8

10 Time

12

14

16

18

20

2

350

1.8

300

1.6 250

1.4

Scale

1.2

200

1 150 0.8 100

0.6 0.4

50

0.2 0

2

4

6

8

10 Time

12

14

16

18

20

0

Figure 6. Continuous wavelet scalogram of a multiple-impulse response. The change in damping coefficient in the third impulse of the time history (top) creates a non-localized, asymmetric time response in the scalogram. FUTURE WORK Future work will continue the analysis started in this effort on a discrete wavelet for analyzing impulse response data from dynamic systems. Some remaining technical challenges include preprocessing of the signal with the goal of ensuring compliance with the orthogonality conditions while maintaining the characteristics of the original data set. This will allow for the scaling function coefficients to be used to calculate the wavelet coefficients and apply discrete wavelet analysis. The new wavelet families will be used to study additional data from the original system and provide the means to distinguish between data for nominal system conditions from data for altered system conditions. The use of these wavelet families for reduced order models, e.g., wavelet spectral element models, and other applications will also be studied. SUMMARY By using the dilation equation, a formula was derived for calculating a vector of scaling function coefficients for a given discrete waveform and a specified length of the coefficient vector. The formula was applied to data sets corresponding to the scaling functions of Daubechies wavelets and excellent agreement was observed between the calculated coefficient values and the true values when a sufficient number of iterations of the dilation equation are used to prepare the data sets. By using response data from an analytical model and an experimental system, the performance of the formula was studied and a method for selecting the number of coefficients is developed. The impulse response of a structure was proposed as a basis for the wavelets. The Wavelet Analysis of Structural Anomalies using Baseline Impulse-Response (WASABI) method was introduced using the wavelet development approach and evaluated using a series of impulses with varying damping ratios.

284

ACKNOWLEDGEMENTS The authors would like to thank AFOSR (Program Manager: Dr. David Stargel) and AFRL for the support of this research effort in the form of the Air Force Summer Faculty Fellowship Program and Cooperative Agreement No. FA8651-10-2-0006. The Vietnamese Ministry of Education and Training is thanked for supporting QP. Opinions, interpretations, conclusions and recommendations are those of the authors and are not necessarily endorsed by the United States Air Force. REFERENCES [1] S. G. Mallat, "A theory for multiresolution signal decomposition: the wavelet representation," Pattern Analysis and Machine Intelligence, IEEE Transactions on, vol. 11, pp. 674-693, 1989. [2] K. Amaratunga, J. R. Williams, S. Qian, and J. Weiss, "Wavelet–Galerkin solutions for one-dimensional partial differential equations," International Journal for Numerical Methods in Engineering, vol. 37, pp. 2703-2716, 1994. [3] A. K. Louis, P. Maass, and A. Rieder, Wavelets : theory and applications. Chichester; New York: Wiley, 1997. [4] R. Bussow, "Bending wavelet for flexural impulse response," The Journal of the Acoustical Society of America, vol. 123, pp. 2126-2135, 2008. [5] S. N. Fernando and M. R. Raghuveer, "Application of wavelets to identify faults during impulse tests," in Electrical Insulation and Dielectric Phenomena, 2005. CEIDP '05. 2005 Annual Report Conference on, 2005, pp. 581-584. [6] S. N. Fernando, M. R. Raghuveer, and W. Ziomek, "Optimal wavelet selection to identify faults during impulse tests," in Electrical Insulation and Dielectric Phenomena, 2006 IEEE Conference on, 2006, pp. 77-80. [7] A. H. Tewfik, D. Sinha, and P. Jorgensen, "On the optimal choice of a wavelet for signal representation," Information Theory, IEEE Transactions on, vol. 38, pp. 747-765, 1992. [8] E. L. Schukin, R. U. Zamaraev, and L. I. Schukin, "The optimisation of wavelet transform for the impulse analysis in vibration signals," Mechanical Systems and Signal Processing, vol. 18, pp. 1315-1333, 2004. [9] J. O. Chapa and R. M. Rao, "Algorithms for designing wavelets to match a specified signal," Signal Processing, IEEE Transactions on, vol. 48, pp. 3395-3406, 2000. [10] L.-K. Shark and C. Yu, "Design of optimal shift-invariant orthonormal wavelet filter banks via genetic algorithm," Signal Processing, vol. 83, pp. 2579-2591, 2003. [11] A. Gupta, S. D. Joshi, and S. Prasad, "A new approach for estimation of statistically matched wavelet," Signal Processing, IEEE Transactions on, vol. 53, pp. 1778-1793, 2005. [12] A. Gupta, S. D. Joshi, and S. Prasad, "A new method of estimating wavelet with desired features from a given signal," Signal Processing, vol. 85, pp. 147-161, 2005. [13] L. E. Suárez and L. A. Montejo, "Generation of artificial earthquakes via the wavelet transform," International Journal of Solids and Structures, vol. 42, pp. 5905-5919, 2005. [14] M. Duval-Destin, M. A. Muschietti, and B. Torresani, "Continuous Wavelet Decompositions, Multiresolution, and Contrast Analysis," SIAM Journal on Mathematical Analysis, vol. 24, pp. 739-755, 1993. [15] S. Mallat, A Wavelet Tour of Signal Processing, Third Edition: The Sparse Way: Academic Press, 2008. [16] Y. Ruqiang and R. X. Gao, "Design of an impulse wavelet for structural defect identification," in Prognostics and Health Management Conference, 2010. PHM '10., 2010, pp. 1-6. [17] I. Daubechies, Ten lectures on wavelets: Society for Industrial and Applied Mathematics, 1992. [18] C. M. Harris and A. G. Piersol, "Shock and Vibration Handbook," 5th ed New York: McGraw-Hill, 2002.

Model Updating with Neural Networks and Genetic Optimization

M. Ersin Yumer1, Ender Cigeroglu, H. Nevzat Özgüven

Department of Mechanical Engineering, Middle East Technical University, Ankara, TR ABSTRACT In dynamic analysis of structures, the accuracy of the mathematical model plays a crucial role. However, because of several uncertainties like local nonlinearities, welding points, bolted joints, material properties and geometric tolerances, the mathematical model will contain differences compared to the manufactured product. Hence, it is essential to update mathematical models by using vibration test data taken from the structure. This paper presents a new approach to model updating via utilizing neural networks and genetic optimization algorithms. The key point in this new approach is that the model updating capability of neural networks is improved by a genetic optimization algorithm by guiding the optimization problem with results obtained from neural network identification. Employing the nominal mathematical model created for a particular structure, a data set of selected mode shapes and natural frequencies is created by a number of simulations performed by perturbing selected updating parameters randomly. A neural network is then created and trained with this data set. Upon training the network, it is used to update the initial model with the test data. The results are then improved further by using the “network updated mathematical model” as an initial model and updating it again by employing a genetic optimization algorithm. The most important advantage of the proposed approach is the possibility of using different number of degrees of freedom for each mode shape; as a result, additional flexibility is introduced to the approach, since the proposed method can be used with incomplete test data. The application and capabilities of the proposed approach is illustrated via real test data taken from a GARTEUR test bed, where it is seen that the proposed method updates mathematical models associated with such complex structures efficiently. Keywords: model updating, neural networks, genetic optimization, modal testing, garteur. 1. INTRODUCTION It is of primary importance that dynamic analysis of certain structures, like aerospace systems, are carried out with accurate mathematical models. Whichever tool an analyst incorporates, such as finite element modeling (FEM), direct analytical formulations, or a mixture of these, there are always assumed and/or unknown features that the actual structure will have over the model. For example, it is a well-known fact that FEM is based upon material and physical properties of the actual system. The models used are generally based on the assumption that material properties are homogeneous over particular areas; however in a real case, a material property, for example the Young’s modulus would not be the same at any location throughout the structure because of the distribution associated with the material production process and different material supplies. This is a typical situation where we introduce error into our models. Connection points constitute a major source of inaccuracy, since local irregularities such as bolted and riveted joints are very difficult to model accurately. Another source of modeling error is due to the manufacturing tolerances. Physical dimensions of the real structure will never exactly match those of the mathematical model. Hence, it is essential to update mathematical models by making use of the test data taken from actual structure. Model updating has been extensively studied and is an unavoidable tool for correcting mathematical dynamic models to have accurate and reliable representations. Model updating literature can be studied in two different groups; direct model updating and indirect model updating. 1

Currently at the Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, PA

T. Proulx (ed.), Advanced Aerospace Applications, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 4, DOI 10.1007/978-1-4419-9302-1_24, © The Society for Experimental Mechanics, Inc. 2011

285

286 In direct model updating field, one of the earliest important works is the ‘Method of Lagrange Multipliers’ [1-2], aiming to correct stiffness and mass matrices of theoretical model. Sidhu and Ewins [3] determined error of stiffness and mass matrices using experimental modal data. In one of the recent works by Carvalho et al. [4], authors do not use any model reduction or expansion during the model fitting process, and they also prevent spurious modes in the frequency range of interest. The main disadvantage of the direct model updating methods is that the connectivity information is modified during the model updating process. Because of this reason, indirect model updating techniques are of more interest to practicing engineers. Indirect updating methods, generally aim on updating several selected parameters of the mathematical model to achieve an accurate mathematical model. In this case, since it is not the system matrices altered directly, connectivity information is not lost. The most widely used indirect model updating method is the ‘Inverse Eigen Sensitivity Method’ [5-6]. These methods make use of the sensitivities associated with the modal test results with respect to selected updating parameters. After the determination of sensitivity matrices, the updating parameters are iteratively tweaked until the mathematical model results converge to their test counterparts. In recent years researchers started using neural networks as an alternative tool in model updating. An early work published by Levin et al. [7] showed that neural networks are working quite well with simple dynamic models, even in the presence of noise. Chang et al. [8] proposed an adaptive neural networks approach for model updating, and they developed a training sample selection methodology [9]. Yong et al. [10] proposed a two-level neural network approach which updates structural parameters in the first level and damping ratios in the second. More recently, Zapico et al. [11] reported their work on experimental validation and updating of a steel frame using neural networks. It should be noted that in the above mentioned model updating works, neural networks are used as the stand alone tool, whereas in this paper, it is used in conjunction with optimization methods to broaden the capability. Recent studies by the authors, Yumer [12] and Yumer et al. [13], where the method is applied to mistuning identification successfully, show that there is a promising capability increase in using neural networks and optimization tools together for identification purposes. The increased capability is demonstrated in this study by updating a sufficiently complex structure, namely a GARTEUR test bed. 2. UPDATING APPROACH In a modal updating problem, the known data is usually composed of modal parameters obtained through modal tests. The unknowns or the decision variables are the updating parameters which have correspondence in the mathematical model constructed. In this section, the proposed approach for model updating by using neural networks and optimization routines is presented. 2.1. The Neural Network Configuration The pattern of the neural network used in this study is a two layer feed-forward back-propagation configuration given in Figure 1. In this network a hidden layer of neurons, where a nonlinear function with weighting vectors and bias components are utilized, is receiving information from the input parameters. The second layer, which is named as output layer, receives information from the hidden layer, then via linear functions with weighting vectors and bias components calculate the outputs. In Figure 1, wi is the th vector of weighting factors and bi is the bias component of i neuron. With this type of neural networks, the number of neurons in the output layer is fixed by the number of outputs. However, the number of neurons employed in the hidden layer depends on the needs of the problem, since increasing the number of neurons used in the hidden layer enhances the capability of the network to some extend [14]. A neural network is simply a fitting function, where weighting factors and bias components need to be determined such that the network relates its inputs with the outputs coherently. The process of fitting a neural network to a problem is called ‘training’. In the training process, the network is fed by a data set of

287 inputs and corresponding outputs; and with back propagation, fitting of the network is accomplished. During the network fitting process, weights and bias components of each of the neurons are adjusted in order to fit the training data. In this study, Levenberg-Marquardt (L-M) algorithm is used for training purposes, whereas the error function utilized is the mean square error (MSE) as defined in Equation (1).

Figure 1. Neural Network Configuration

MSE is a way of quantifying the difference between estimation and the actual value . MSE is the second moment of error, which is defined as follows:

MSE=

1 n 2 ∑ ei n i =1

(1)

where

{e} = {y}A − {y}I

(2)

Here, {y}A is the vector of actual modal parameters and {y}I is the vector of modal parameters obtained from the mathematical model. Note that, since the square of each of the elements in the error vector is utilized in Equation (1), MSE accounts for both variance and bias of the error [15]. The ideal case is MSE being zero which means that the estimated values are perfectly matching the actual values.

288 2.2. Model Updating Strategy The flowchart of the model updating strategy developed in this study is given in Figure 2. First a neural network needs to be created and trained with a dataset obtained from the original mathematical model. A data set is a number of input-output pairs. For a model updating problem, inputs are natural frequencies and mode shapes, and the outputs are the updating parameters chosen from the system parameters used in constructing the mathematical model or values of the newly defined elements in the mathematical model. In the updating strategy used in this work, the input vector x and the output vector y are formed as follows:

{}

 M      ω i    ω i +1       M   {x} =    M    {u}i    {u}i +1   M    In (3),

ωi

and

{u}i

 M   p  {y} =  j   p j +1   M 

{}

(3)

are the ith natural frequency and corresponding modal vector of the structure,

respectively, and p j are the parameters updated. Note that, there is no restriction on the number of natural frequencies, the number of modes, and the number of degrees of freedom to be used in each of these modal vectors. This fact introduces an evident advantage, since it enables tuning of the input vector according to the modal tests conducted. After the neural network is created and trained with the initial data set, actual modal test results are fed into the neural network to obtain network identified updating parameters, which in turn defines the ‘Network Identified Mathematical Model’. Network identified mathematical model may or may not be sufficiently accurate; that is it may or may not yield sufficiently accurate modal data compared with experimental values. However, this model may be taken as a reasonably accurate initial estimate to be used in optimization.

Figure 2. Model Updating Flowchart

289 Then an optimization problem consistent with the selected optimization routine is set up. The goal of the optimization problem is to minimize the total error between the test results and the modal parameters calculated from the mathematical model at each step, which is defined through MSE given by Equation (1). Note that the network identified updating parameters are used as an initial estimate, and the initial estimate used is very important for convergence of an optimization problem where iterative procedures are employed. In this study, a genetic algorithm with the parameters given in Table 1 is used via MATLAB’s optimization toolbox. Neither the terminology of genetic algorithms nor the selection of these parameters is included here, since it is out of the scope of this work. However, interested readers are referred to the Optimization Toolbox User’s Guide of MATLAB [16] for further information about optimization by genetic algorithms, related terminology and parameter selection. Table 1. Genetic Optimization Parameters Population Size

480

Crossover Function

0.80

Mutation Function

Adaptive feasible

Crossover Function

Random

Migration Direction

Forward

Migration Fraction

0.15

Migration Interval

15

3. GARTEUR CASE STUDY In this section real test data taken from a benchmark structure designed to simulate the dynamics of an aircraft structure, namely GARTEUR SM-AG 19, is used. For the test bed employed in this study (Figure 3); the wing-fuselage, fuselage-vertical stabilizer and vertical stabilizer-horizontal stabilizer connections are joined by welding, instead of bolts which are used in the original GARTEUR model [17-18]. The test data is taken from Kozak et al. [6]. The modal tests were conducted by using accelerometers and a modal hammer. A total of 12 accelerometers, 36 impact points and 66 degrees of freedom were used throughout the tests. A Finite Element (FE) model, which is shown in Figure 4, is constructed by using 6-DOF beam, 2-DOF spring and rigid elements. To overcome the discontinuities in the mating junctions of the model, which is caused by the differences in the positions of the neutral axes of the beam elements, rigid multi point constraints are used. The first 10 natural frequencies of this initial FE model and the experimental results are given in Table 2. The theoretical and experimental mode shapes are also compared via Modal Assurance Criteria (MAC). Defining {U} A and {U }C as the actual (experimental) and calculated (theoretical) mode shape vectors, respectively, for a particular mode, MAC value is calculated as follows:

MAC=

{U }TA {U }C

2

({U } {U } )({U } {U } ) T A

A

T C

C

(4)

290 MAC matrices between the modes of the initial FEM model and test results are given in Figure 5.

Figure 3. GARTEUR Test Bed

Figure 4. GARTEUR Beam Model

Figure 5. MAC Matrix between the Experimental Modes and the Modes of the Initial FEM

291 Table 2. Natural Frequencies (NF) from Test, Initial FE Model (IFEM), and Final FE Model (FFEM) Test Mode No

Test NF (Hz)

IFEM Mode No

IFEM NF (Hz)

Error in IFEM NF (%)

FFEM Mode No

FFEM NF (Hz)

Error in FFEM NF (%)

1

5.65

1

5.66

0.2

1

5.66

0.2

2

15.73

2

16.52

5.0

2

16.86

7.2

3

36.79

5

37.02

0.6

3

36.43

-0.9

4

37.51

3

30.82

-17.8

4

37.57

0.2

5

37.65

4

30.91

-17.9

5

37.65

0

6

43.73

6

43.21

-1.2

6

43.22

-1.2

7

50.32

7

50.25

-0.1

7

49.81

-1.0

8

55.00

8

54.68

-0.6

8

54.71

-0.5

9

60.66

10

72.41

19.4

9

60.96

0.5

10

68.23

9

63.97

-6.2

10

68.30

0.1

th

th

th

th

As it is clear from Table 2, 4 , 5 , 9 , and 10 modes are the most poorly represented ones by the initial th th FE model. An investigation of the test mode shapes reveal that 4 and 5 modes are dominated by the th th torsional motions of the wings, whereas 9 and 10 are controlled by the torsional and bending motions of the vertical stabilizer. As a result, to achieve a minimum parameter updating model, 4 parameters are selected. These parameters are the thicknesses of the two elements which model the vertical stabilizer, two torsional stiffness values between the fuselage-wing joint and a wing end and another torsional stiffness placed between the fuselage-vertical stabilizer joint and vertical stabilizer-horizontal stabilizer joint. A data set of 20000 samples is generated where 4 updating parameters are modeled with random numbers taken from 4 different uniform random distributions. Generation of 20000 samples involves solution of that many eigenvalue problems. However, since this process is fairly automated and unsupervised, creation of these samples are straight forward. Parallel computing is also possible for larger problems where time is a primary concern, since these eigenvalue problems do not depend on each other. First two distributions, used for the thickness parameters, have minimum and maximum bounds as 5 mm and 20 mm. Bounds of the third distribution corresponding to the vertical stabilizer are 50 Nm/rad and 4000 Nm/rad. The last distribution which is used for the torsional stiffness of wings has bounds of 50 Nm/rad and 2000 Nm/rad. The mathematical model using the generated data set is solved for modal values, and the resulting mode shape vectors (corresponding to test DOFs) and the corresponding natural frequencies are used as the input vector set for the network, where the updating parameter vectors are used as the output vector set as defined in Equation (3). A network is created as shown in Figure 1, with 4 neurons (corresponding to the updating parameters) in the output layer and 76 neurons in the hidden layer. Afterwards, it is trained with the data set created as explained above, but the input vectors are polluted by adding random noise picked from a pool of uniform random distribution with ±5% variation of the mean value in order to include the effect of noise in measurements. The data set is divided between training, validation and test sets as 75%, 15%, 10% respectively.

292 The resulting linear regression between actual updating parameters and network outputs are 0.931 for training set, 0.932 for validation set, and 0.928 for the test set; which show very high correlation on a 0 to 1 scale. The network is then used to update the initial FE model with real test data. A second updating scheme, starting with the network updated FE model, is performed with genetic optimization where the parameters are set according to Table 1. The boundaries are not limited in the optimization problem. After 1652 iterations the relative error tolerance is reached and the optimization is stopped. The natural frequencies obtained from the resulting updated model are given in Table 2. Mode shapes from final FEM are also compared with those obtained from tests via MAC. MAC matrix between modes obtained from the final FEM model and the test results are given in Figure 6. The corresponding mode shapes are given in Figure 7. Note that the order of the modes is corrected and the errors associated are reduced remarkably compared to those of the initial FE model. Moreover the mode shapes are consistent with the test results.

Figure 6. MAC Matrix between the Experimental Modes and the Modes of the Final FEM 4. CONCLUSION This paper presents a new approach for model updating in structural dynamics by using neural networks and genetic optimization algorithms. The approach suggested is different from other model updating approaches using neural networks. In the method proposed, the model updating capability of neural networks is improved by a genetic optimization algorithm by guiding the optimization problem with neural network identification. The most important advantage of the proposed approach is the possibility of using different number of measuring points in each mode. That is, the proposed method can be used with incomplete test data. The application of the method is demonstrated with actual test data taken from a GARTEUR model. It is shown that the method is able to update the model so that the natural frequencies of erroneous modes are corrected, and more importantly, the correct order of the modes and the mode shapes in the dense mode region (3rd, 4th, and 5th modes in the case study) are captured.

293

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

Mode 7

Mode 8

Mode 9

Figure 7. Test (

Mode 10

), and Final Updated FE Model (

) Mode Shapes

294 REFERENCES [1]

Baruch, M., Itzhack, Y. B., “Optimal Weighted Orthogonalization of Measured Modes”, AIAA Journal, vol. 16, no. 4, pp. 346-351, 1978.

[2]

Berman, A., Nagy, E. J., “Improvement of a Large Analytical Model Using Test Data”, AIAA Journal, vol. 21, no. 8, pp. 1168-1173, 1983.

[3]

Sidhu, J., Ewins, D. J., “Correlation of Finite Element and Modal Test Studies of a Practical Structure”, Proceedings of the 2nd IMAC, pp. 394-401, Los Angeles, CA, 1986.

[4]

Carvalho, J., Datta, B. N., Gupta, A., Lagadapati, M., “A Direct Method for Model Updating with Incomplete Measured Data and without Spurious Modes”, Mechanical Systems and Signal Processing, vol. 21, pp. 2715-2731, 2007.

[5]

Lin, R. M., Ewins, D. J., “Model Updating Using FRF Data”, Proceedings of the 15th International Seminar on Modal Analysis, pp. 141-162, Leuven, Belgium, 1990.

[6]

Kozak, M. T., Öztürk, M., Özgüven, H. N., “A Method in Model Updating Using Miscorrelation Index Sensitivity”, Mechanical Systems and Signal Processing, vol. 23, no. 6, pp. 1747-1758, 2008.

[7]

Levin, R.I., Lieven, N. A. J., “Dynamic Finite Element Model Updating Using Neural Networks”, Journal of Sound and Vibration, vol. 210, no. 5, pp. 593-607, 1998.

[8]

Chang, C. C., Chang, T. Y. P., Xu, Y. G., “Adaptive neural networks for model Updating of Structures”, Smart Materials and Structures, vol. 9, pp. 59-68, 2000.

[9]

Chang, C. C., Chang, T. Y. P., Xu, Y. G., “Selection of Training Samples for Model Updating Using Neural Networks”, Journal of Sound and Vibration, vol. 249, pp. 867-883, 2002.

[10]

Lu, Y., Tu, Z., “A two-level neural network approach for dynamic FE model updating including damping”, Journal of Sound and Vibration, vol. 275, pp. 931-952, 2004.

[11]

Zapico, J. L., Gonzalez-Buelga, A., Gonzalez, M. P., Alonso, R., “Finite Element Model Updating of a Small Steel Frame Using Neural Networks”, Smart Materials and Structures, vol. 17, pp. 045016, 2008.

[12]

Yumer, M. E., “On the Non-linear Vibration and Mistuning Identification of Bladed Disks”, MSc Thesis, Middle East Technical University, 2010.

[13]

Yumer, M. E., Cigeroglu, E., Özgüven, H. N., “Mistuning Identification of Bladed Disks Utilizing Neural Networks”, Proceedings of ASME Turbo Expo 2010, Glasgow, United Kingdom, 2010.

[14]

Bishop, C. M., “Neural Networks for Pattern Recognition”, Oxford university Press, Oxford, UK, 1998.

[15]

Casella, G., Lehmann, E. L., “Theory of Point Estimation”, Springer, New York, USA, 1999.

[16]

The Mathworks, Inc., “Optimization Toolbox User’s Guide, Version 4”, Natick, MA, 2009.

[17]

Link, M., Friswell, M., “Working Group 1: Generation of Validated Structural Dynamic Models – Results of a Benchmark Study Utilizing the GARTEUR SM-AG19 Test-bed”, Mechanical Systems and Signal Processing, vol. 17, no. 1, pp. 9-20, 2003.

[18]

Goge, D., Link, M., “Results Obtained by Minimizing Natural Frequency and Mode Shape Errors of a Beam Model”, Mechanical Systems and Signal Processing, vol. 17, no. 1, pp. 21-27, 2003.

A Piezoelectric Actuated Stabilization Mount for Payloads Onboard Small UAS Katie J. Stuckel and William H. Semke Unmanned Aircraft Systems Engineering Laboratory School of Engineering and Mines University of North Dakota Grand Forks, ND 58202 ABSTRACT A piezoelectric actuated stabilization mount for payloads onboard small Unmanned Aircraft Systems (UAS) is designed, analyzed, and experimentally tested. The custom three degree of freedom system is capable of producing attitude corrections in pitch and roll as well as axial stabilization. The mount assists in many applications including Intelligence Surveillance and Reconnaissance (ISR) missions, airborne target identification and tracking, and narrow beam directional communications. The system consists of a mounting plate, piezoelectric actuators, precision position sensors, and a digital controller in a tailor-made compact lightweight package. Full analytical and finite element models are created to characterize the dynamic behavior of the system. Both the modal and transient analyses are in good agreement with experimental testing results. A closed-loop Proportional, Integral, Derivative (PID) controller is implemented and tuned to actively reduce base excitation to provide a much-improved stabilized platform. The stabilization mount is subjected to single disturbances as well as random excitations over large frequency ranges for assessment of performance. Results presented include analytical, numerical, and experimental test data as well as a tuned controller for an effective piezoelectric actuated stabilization mount for small UAS.

Nomenclature α β δ f F Fact I k keq L L m Qk T x y z Xa Ya Za ωn V V Y(s) zg

= = = = = = = = = = = = = = = = = = = = = = = = =

rotation about x-axis of stabilization system rotation about y-axis of stabilization system displacement due to expansion/contraction of actuator in meters (m) or micrometers (µm) frequency in Hertz (Hz) Force in Newtons (N) Force in actuator (N) 4 Moment of inertia (µm ) stiffness of spring in N/m equivalent k component for natural frequency calculation length in meters (m) Lagrangian mass in kilograms (kg) generalized forces in Lagrange’s method kinetic energy in Lagrange’s method 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 2 2 natural frequency in radians/second (rad/sec ) voltage (V) potential energy in Lagrange’s method output response in regards to transfer functions vertical displacement of the ring mount, located at the center of inertia (µm)

T. Proulx (ed.), Advanced Aerospace Applications, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 4, DOI 10.1007/978-1-4419-9302-1_25, © The Society for Experimental Mechanics, Inc. 2011

295

296 I. Introduction A piezoelectric actuated stabilization mount used to enhance imaging and communication systems during flight onboard small Unmanned Aircraft Systems (UAS) is presented. The active vibration stabilization utilizes piezoelectric stack actuators and a circular camera mount to steady the imaging system onboard the aircraft. A paper presented at the IMAC 2010 Conference titled “A High Frequency Stabilization System for UAS Imaging Payloads” encompasses the initial design, basic analytical modeling and initial static testing for the stabilization mount prototype [1]. The goal of the vibration compensator is to steady any targeting system or directional communication device used during flight. The Unmanned Systems Aircraft Engineering (UASE) team at the University of North Dakota (UND) uses a Bruce Thorpe Engineering (BTE) aircraft, known as the Super Hauler as the “small” unmanned aerial vehicle (UAV) [2,3]. This is shown in Figure 1. The platform is used to flight test multiple payloads for Intelligence, Surveillance, and Reconnaissance (ISR) missions, many of which would benefit greatly from an active stabilization mount.

Figure 1. BTE Super hauler Unmanned Aerial Vehicle Operated by the UASE Laboratory The piezoelectric driven stabilization mount with its coordinate system, angles and orientations is shown in Figure 2. 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 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.

Figure 2. Stabilization Mount Coordinate System The stabilization mount is made from aluminum, a piezoelectric stack actuator and spring combination along with a Proportional-Integral-Derivate (PID) controller. This combination allows for a lightweight, precise system that accounts for error with corrections. As the aircraft vibrates from engine rotations and high frequency turbulence, the precision sensors onboard the aircraft detect the change in position, sending a voltage through the controller system to the actuators, allowing for them to correct the change. During static testing, it was proven that both the hardware and software performed successfully, providing positive results for dynamic applications. All physical

297 components were able to communicate successfully with each other, as well as provide small error results. More details about the stabilization mount, controller process and static testing can be found in the thesis entitled “A High Frequency Stabilization System for UAS Imaging Payloads” by Katie J. Stuckel [4]. After sufficient laboratory testing and calculations, including static deflection, mathematical calculations, control parameter identification, and finite element analysis; dynamic testing of the stabilization mount is performed by placing the stabilization mount on a mechanical shaker. A steady sinusoidal wave was fed into the shaker, causing the ring base to move up and down. The results were recorded with the laser measuring device, analyzed, and compared to computer simulations. This testing identified the baseline response of the system under a single degree of freedom excitation. In depth computer simulations of the tilt motion simulates the capabilities of the high frequency stabilization mount. In these scenarios, each actuator will be allowed to act individually, using separate sensors, amplifier and controllers. It is important to note, however, that the deflections given to each actuator are related through the geometry of the system and can be given through two rotations about the x- and y-axes, as shown in Figure 2. Using Lagrange’s method and MATLAB for computer simulations, the effectiveness of the system will be demonstrated and analyzed. II. Equations of Motion The equations of motion for the vertical displacement scenario can be found using Newton’s method. In this situation, the actuators are all acting in parallel, receiving the same voltage at the same time. This produces a simple vertical displacement of the ring, with no rotation. By summing the forces in the vertical direction, the forces in the actuators and the forces in the springs are shown to counteract each other, settling at equilibrium. Figure 3 shows the schematic and free body diagram for vertical displacement.

Figure 3. Schematic and Free Body Diagram for a simple vertical deflection Assuming the input forces are those of the actuators and using the spring/deflection relationship, the final equation of motion can be found in Equation 1. 𝑚𝑧̈𝑔 + (𝑘1 + 𝑘2 + 𝑘3 )𝑧𝑔 = 𝐹𝑎𝑐𝑡1 + 𝐹𝑎𝑐𝑡2 + 𝐹𝑎𝑐𝑡3

(1)

𝑚𝑧𝑔̈ + 3𝑘𝑧𝑔 = 𝐹𝑎𝑐𝑡1 + 𝐹𝑎𝑐𝑡2 + 𝐹𝑎𝑐𝑡3

(2)

Since k1=k2=k3=k, this can be simplified into Equation 2. This also gives 𝑚𝑒𝑞 = 𝑚 and 𝑘𝑒𝑞 = 3𝑘 for transfer function development for use in the PID controller.

The equations of motion of the system during tilt compensation are developed using a Lagrangian approach. It is important to note that the deflections of each actuator are related through the geometry of the rigid body system and can be given through two rotations, about the x-axis, alpha (α), and y-axis, beta (β), and a translation in the zaxis. Figure 4 shows these rotations, and also demonstrates the third degree of freedom, zg.

298

ksp3

ksp1 ksp2

zg Fact3

Fsp3

y-axis

β

α x-axis

Fact1

Fsp1 zg

Fact3

Fact2

α x-axis

Fsp2

y-axis

β

Fact1

Fact2

Figure 4. Schematic and Free Body Diagram demonstrating degrees of freedom for tilt compensation By using the two rotations and the vertical displacement, Lagrange’s method for dynamic systems is used to find the equations of motion (EOM). These equations of motion are used to develop the transfer functions of the system. This allows modeling the response of the stabilization mount in a MATLAB computer simulation. Assumptions that can be made in advances are small angle approximations and a rigid body analysis, with no deflections in the ring itself. Equation 3 is the initial starting equation for Lagrange’s method, where L is the potential energy (V) of the system subtracted from the kinetic energy (T) of the system, as shown in Equation 4. The variable Qk represents the generalized forces in terms of the coordinate system, while qk designates the chosen degree of freedom. 𝑑

�

𝛿𝐿

𝑑𝑡 𝛿𝑞̇ 𝑘

�−

𝛿𝐿

𝛿𝑞𝑘

= 𝑄𝑘

(3)

Using energy methods, the free body diagram in Figure 4, and assuming k1=k2=k3=k, Equation 5 is designated as the kinetic energy of the system, while Equation 6 is the potential energy of the system. 𝐿 = 𝑇−𝑉 1 1 1 𝑇 = 𝑚𝑧̇𝑔2 + 𝐼𝛼̇ 2 + 𝐼𝛽̇2 2 1

1

2

1

2

𝑉 = 𝑘𝑧12 + 𝑘𝑧22 + 𝑘𝑧32 2

2

2

(4) (5) (6)

After taking the necessary derivates as designated in Equation 3, using Equations 4 through 6, the following EOMs are developed (Equations 7 -9). 𝐼𝛼̈ + 0.008164𝑘𝛼 − 0.04183𝑘𝑧𝑔 = 0.04949(𝐹1 + 𝐹3 ) − 0.05715𝐹2 𝐼𝛽̈ + 0.001633𝑘𝛽 = 0.028575𝐹1 − 0.028575𝐹3 𝑚𝑧̈𝑔 + 3𝑘𝑧𝑔 − 0.04183𝑘𝛼 = −𝐹1 − 𝐹2 − 𝐹3

(7) (8) (9)

As seen in Equations 7 and 9, two of the three equations of motions for the full system transfer function are coupled, where the rotation about the x-axis, α, and the vertical deflection, zg, depend on each other. Because of this, it is necessary to use state space analysis for computer simulation. The three degrees of freedom used to model the system are alpha, α, the rotation about the x-axis, beta, β, the rotation about the y-axis and zg, the vertical deflection of the ring, located at the center of gravity. The three input forces are located at the actuators, defined as Fact1, Fact2 and Fact3 (Figure 4). Putting the equations of motion into state space, the three degrees of freedom are assigned as x1, x2 and x3 and the system can be represented by the notation in Equations 10 and 11. 𝑥̇ = 𝐴𝑥 + 𝐵𝑢 𝑦 = 𝐶𝑥 + 𝐷𝑢

(10) (11)

By utilizing MATLAB code to develop transfer functions for the state space model, there are nine transfer functions develop in which each relate one input to one output.

299 III. PID Controller To begin the development of a PID controller it is important to distinguish the type of control system being used. Open-loop control for a system is where the input for the system is the actual signal. Closed-loop control can be best explained where the actual input is the difference between the input and the output through the feedback loop [5]. Feedback in the control loop allows for the system to monitor the output and compare the result to the input [6]. The control system then compensates for this error and continuously adjusts the input to minimize the error. Closed loop control is necessary for effective active vibration isolation and a block diagram of the control methodology is shown in Figure 5. G(s) R(s) +

Controller

Plant

Y(s)

Feedback H(s) Closed Loop System

Figure 5. Closed-loop control diagram illustrating the controller, plant, and feedback loop The PID controller will be used in the closed loop system to aid in stabilizing the system [7]. The PID transfer function with variables Kc, τi, and τd gives a final transfer function in Equation 12. 𝐺𝑃𝐼𝐷 (𝑠) =

𝐾𝑐 𝑠 2 +𝜏𝑖 𝑠+𝜏𝑑 𝑠

(12)

To find the necessary G(s) transfer function in Equation 19 for a closed loop system, the PID controller transfer function and the plant transfer function need to be combined. Since the controller and plant are in series with each other, simply multiplying the two transfer functions together will give the complete G(s) transfer function, as shown in Equation 13. 𝐺(𝑠) =

𝐾𝑐 𝑠 2 +𝜏𝑖 𝑠+𝜏𝑑

𝑠(𝑚𝑒𝑞 𝑠 2 +𝑏𝑠+𝑘𝑒𝑞 )

(13)

The next step in finding the complete transfer function, which includes both the PID controller and the feedback loop is combing all transfer functions as in Equation 13. With the feedback loop, H(s) equivalent to 1, the final transfer function for the system is shown in Equation 14. 𝑌(𝑠)

𝑅(𝑠)

=

𝐾𝑐 𝑠 2 +𝜏𝑖 𝑠+𝜏𝑑 3 𝑚𝑒𝑞𝑠 +(𝑏+𝑘𝑐 )𝑠 2 +�𝑘𝑒𝑞 +𝜏𝑖 �𝑠+𝜏𝑑

(14)

IV. Dynamic Testing and Computer Simulation During dynamic testing, all three actuators were hooked up in parallel so they all received the same voltage throughout the testing. This gave the ring on the stabilization mount a vertical movement only. The shaker was given a desired amplitude with an offset enough to avoid sending negative voltages to the actuator system. The initial frequency of 0.1 Hz was given to the shaker so the operators were able to observe the results on the laser controller. A closed loop, PID controlled system is used to provide stabilization. Computer simulation is performed with each step of the design process to help provide reliability. Vertical displacement computer simulation was performed in MATLAB, using the combinations of transfer functions based on the physical attributes of the system. By combining the transfer function of the PID controller with that of the physical system, and then including the feedback loop, the final transfer function is created and shown in Equation 15. 𝑌(𝑠) 𝐹𝑎𝑐𝑡

=

1000𝑠 2 +1𝑒 8 𝑠+2000

0.153𝑠 3 +1003𝑠 2 +1.002𝑒 8 𝑠+2000

(15)

300 By applying a step function to the vertical displacement transfer function, Figure 6 is produced. The important items to notice in the graph are the settling time and the percent overshoot. A settling time of 0.001 seconds shows that the system has a quick response, allowing the system to settle at a very fast rate. The percent overshoot shows a value of 0.67 micrometers, which is a slightly high value, but determined acceptable as the system does not overshoot by more than 1.0 micrometer.

Figure 6. Simulation step response in MATLAB with vertical displacement Physical dynamic testing of the high frequency stabilization mount was performed in the laboratory on a Vibrations Testing System (VTS) Incorporated mechanical shaker. The vibration stabilization mount was fixed to the top of the shaker with a custom made mount, while the laser measuring device was positioned above the machine. This set up is shown in Figure 7.

Laser Head Stabilization Mount

Shaker

Figure 7. Dynamic Experimental Set-Up Dynamic laboratory testing includes a custom designed LabVIEW PID Controller program and a National Instruments USB-6251 Data Acquisition (DAQ) Board for communication with the actuators. As the Keyence Laser Displacement Sensor 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. This is all shown in Figure 8. The actuators then expand or contract to the necessary height, seemingly so the stabilization mount never moved. This all needs to be completed nearly instantaneously to keep the ring mount stable and level.

301

Figure 8. Dynamic testing setup: laptop computer, DAQ board, laser output, and amplifier on left, dynamic shaker, frequency generator, and mounted laser on right To provide feedback in the system, and the use of a PID controller, a closed loop system was used. This closed loop system was tested and proven to work during the static testing, and provided useful feedback for the dynamic case. As the location of the ring came in as data points into the system through the DAQ assistant .vi, they were passed through the PID controller block and manipulated as necessary. The PID controller outputs the scaled voltages that are applied to the actuators. Again using the DAQ assistant .vi built into LabVIEW, the output was fed back through the DAQ board, into the amplifiers and ultimately to the actuators. This was done at the rate of 1000 times per second. During the first dynamic test, the shaker was set to have an amplitude of 7.5 micrometers (0.0075 millimeters on the graph in Figure 9) with an offset of 19 micrometers (0.019 millimeters). This offset prevented the controller from requesting tensile loads in the piezoelectric actuators and allowed for steady oscillations between 0.012 volts and 0.027 volts (equivalent to 0.012 micrometers and 0.027 micrometers, respectively). On the graph in Figure 9, the green path demonstrates the movement of the shaker. The blue path shows the response of the actuators by plotting the incoming data points, measuring the location of the ring directly above the actuator. The solid red line was added following the experiment to represent the setpoint of the system and to demonstrate where the blue line, which is the actual response of the system, should follow. The shaker was operating in a steady state condition at a constant frequency and the controller was activated at time t=0.

Figure 9. First sample displacement plot for dynamic testing of stabilization mount. The noise in the system without PID control is demonstrated in the width of the green path. These measurements are taken through the laser measuring device while the shaker is on, without the actuators activated, and is

302 especially noticeable when the shaker changes directions and the scatter of the points widen. The response of the ring, shown in the blue line, provides proof that the displacement is minimized as the plotted points remain close together. The average deviation from the setpoint is 0.158 micrometers and the total amplitude of the shaker is 15 micrometers. The maximum deviation from the setpoint is 0.866 micrometers. By taking the average deviation of the experiment and dividing it by the total amplitude of the shaker, the error for this experiment is found to be 1.05%. The low error is determined a very acceptable result. The second sample dynamic test was performed similar to the first test, with an amplitude of 6.5 micrometers and an offset of 17 micrometers (0.0065 millimeters and 0.017 millimeters, respectively). The graph in Figure 10 shows the results of this test. The maximum deviation for this second sample was 1.0 micrometers with the average deviation of 0.154 micrometers. Comparing the average deviation to the total travel of base gives a low error of 1.11% and because of this low error, a second successful trial. Comparing these experimental tests to the computer simulations shows the reliability of the system. The overshoot of the dynamic experiments are 0.8 and 0.7 micrometers, respectively, and when compared to the computer simulation for this scenario of 0.67 micrometers, it is proven that the experimental set up performed well within the predicted limits. The settling time of the system is significantly slower than the predicted value of 0.001 seconds, at approximately 0.6 and 0.5 seconds, respectively. This can be explained by delays in the system from the time that the voltage is calculated in LabVIEW program until it actually reached the actuators and they respond accordingly. This important result from the vertical displacement dynamic testing provides further proof of successful operation of the vibration stabilization mount.

Figure 10. Second sample displacement plot for dynamic testing of stabilization mount V. Tilt Compensation Simulation The next step in the development of the stabilization mount is to allow each actuator to act separately. Using the state space equations, the three degrees of freedom are assigned as x1, x2 and x3 and the system can be represented by the notation in Equations 9 and 10. By utilizing MATLAB code to develop transfer functions for the state space model, there are nine transfer functions produced. The nine combinations each relate one input to one output. The x matrix in the state space model is a column matrix listing the degrees of freedom used in the analysis: alpha (α), beta (β) and the vertical deflection (zg). The u matrix is also a column matrix used to list the input forces; Fact1, Fact2 and Fact3. These two matrices define the parameters used in the stabilization mount tilt compensation analysis, as shown in Equation 16. 𝛼 𝛽 𝐹𝑎𝑐𝑡1 ⎛𝑧 ⎞ 𝑔 𝒙 = ⎜ 𝛼̇ ⎟ and 𝒖 = �𝐹𝑎𝑐𝑡2 � (16) ⎜ ⎟ 𝐹 𝑎𝑐𝑡3 ̇ 𝛽 ⎝𝑧𝑔̇ ⎠

303 The matrix A is a square matrix which represents the mathematical model of the physical system. The values for the matrix can be found directly from the equations of motion. The matrix B is a set of values describing the ratios of the outputs to each input. Matrices C and D demonstrate the desired output signals expressed by the user. Matrices A and B shown in Equation 17, rewritten with the parameters from the high frequency stabilization mount. The u matrix is the external actuator forces acting on the system [8,9]. 0 ⎡ 0 ⎢ 0 ⎢ 0.08164𝑘 − ⎢ 𝑨= 𝐼 ⎢ 0 ⎢ ⎢ 0.04183𝑘 ⎣ 𝑚

−

0 0 0 0

0.001633𝑘

0

𝐼

0 0 0

0.04183𝑘 𝐼

0

−

1 0 0 𝑏 − 𝛼 0

3𝑘

0 1 0 0

𝐼

−

0

𝑚

0 0 1 0

0 0 0 ⎤ ⎡ 0 ⎤ 0 0 ⎥ ⎢ ⎥ 0 0 0 ⎥ ⎢ 0.04949 ⎥ 0.5715 0.04949 − ⎥, and 𝑩 = ⎢ 𝐼 ⎥ 𝐼 𝐼 ⎢0.028575 ⎥ 0.028575⎥ 0 ⎥ 0 − ⎢ 𝐼 ⎥ 𝐼 ⎢ ⎥ 𝑏𝑧𝑔 ⎥ 1 1 1 − − − ⎦ ⎣ −𝑚 ⎦ 𝑚 𝑚 𝐼

𝑏𝛽

0

𝐼

(17)

Choosing all degrees of freedom as desired outputs, matrix C and D are shown in Equation 18. Matrix D is a 3x3 of zeros due to the fact that none of the inputs led directly into the output [33]. 1 𝑪 = �0 0

0 1 0

0 0 1

0 0 0

0 0 0

0 0 0� and 𝑫 = �0 0 0

0 0 0

0 0� 0

(18)

A MATLAB code was used to develop transfer functions for the state space model resulting in nine transfer functions. Each combination is relating one of the inputs to one of the degree of freedom outputs. The final nine transfer functions for computer tilt simulation are listed in Equations 19 through 27: Transfer function for output of alpha, α, with the input force at actuator 1 location: 𝑌𝛼 (𝑠)

𝐹𝑎𝑐𝑡1

=

3172𝑠 2 +7.083𝑒 4 𝑠+2.839𝑒 9

𝑠 4 +137.7𝑠 3 +3.451𝑒 7 𝑠 2 +8.862𝑒 8 𝑠+3.848𝑒 13

(19)

Transfer function for output of beta, β, with the input force at actuator 1 location: 𝑌𝛽 (𝑠) 𝐹𝑎𝑐𝑡1

=

1832

𝑠 2 +51.58𝑠+6.652𝑒 6

(20)

Transfer function for output of vertical deflection, zg, with the input at actuator 1 location: 𝑌𝑧𝑔 (𝑠) 𝐹𝑎𝑐𝑡1

=

−6.536𝑠 2 −753.9𝑠−1.623𝑒 8

𝑠 4 +137.7𝑠 3 +3.451𝑒 7 𝑠 2 +8.862𝑒 8 𝑠+3.848𝑒 13

(21)

Transfer function for output of alpha, α, with the input force at actuator 2 location: 𝑌𝛼 (𝑠)

𝐹𝑎𝑐𝑡2

=

−3663𝑠 2 −8.179𝑒 4 𝑒 4 𝑠−5.679𝑒 9

𝑠 4 +137.7𝑠 3 +3.451𝑒 7 𝑠 2 +8.862𝑒 8 𝑠+3.848𝑒 13

(22)

Transfer function for output of beta, β, with the input force at actuator 2 location: 𝑌𝛽 (𝑠) 𝐹𝑎𝑐𝑡2

=0

(23)

Transfer function for output of vertical deflection, zg, with the input at actuator 2 location: 𝑌𝑧𝑔 (𝑠) 𝐹𝑎𝑐𝑡2

−6.536𝑠 2 −753.9𝑠−2.81𝑒 8

=

𝑠 4 +137.7𝑠 3 +3.451𝑒 7 𝑠 2 +8.862𝑒 8 𝑠+3.848𝑒 13

=

𝑠 4 +137.7𝑠 3 +3.451𝑒 7 𝑠 2 +8.862𝑒 8 𝑠+3.848𝑒 13

(24)

Transfer function for output of alpha with the input force at actuator 3 location: 𝑌𝛼 (𝑠)

𝐹𝑎𝑐𝑡3

3172𝑠 2 +7.083𝑒 4 𝑠+2.839𝑒 9

Transfer function for output of beta with the input force at actuator 3 location:

(25)

304 𝑌𝛽 (𝑠) 𝐹𝑎𝑐𝑡3

=

−1832

(26)

𝑠 2 +51.58𝑠+6.652𝑒 6

Transfer function for output of vertical deflection, zg, with the input at actuator 3 location: 𝑌𝑧𝑔 (𝑠) 𝐹𝑎𝑐𝑡3

=

−6.536𝑠 2 −753.9𝑠−1.623𝑒 8

𝑠 4 +137.7𝑠 3 +3.451𝑒 7 𝑠 2 +8.862𝑒 8 𝑠+3.848𝑒 13

(27)

By using the transfer functions in Equations 19 - 27, MATLAB is used to simulate the response of the system. These transfer functions are then used in combination with the tuned PID controller and a feedback loop in MATLAB to study the final results. It is important to note that the PID values have not been optimized, but rather tuned for an acceptable response level. The step responses are done individually with a single step input on a case by case basis for the nine scenarios. The controller was tuned by using the approximate values from the previous experiments and modifying them to produce a reasonable response. The simulations are the final transfer function with the plant transfer function combined with the PID transfer function and a feedback loop at a value of 1. Figure 11 illustrates the step response graphs with the step response given to alpha, α, beta, β, and the vertical displacement, zg, located at each actuator.

Alpha response at actuator 1

Beta response at actuator 1

Vertical displacement at actuator 1

Alpha response at actuator 2

Beta response at actuator 2

Vertical displacement at actuator 2

Alpha response at actuator 3

Beta response at actuator 3

Vertical displacement at actuator 3

Figure 11. Step response graphs for each combination of inputs and outputs in tilt compensation

305 Some important items to notice in the response graphs are the percent overshoot and settling time in each scenario. All tilt compensation situations have settling times that are similar to the settling time of the vertical displacement scenario. These results demonstrate that further testing of the vibration stabilization mount would result in successful implementation. Also, noticeable is the percent overshoot for each situation. All nine scenarios have an overshoot of less than one micrometer, which again provides confidence that the stabilization mount will react in a predictable manner. One scenario unique to the tilt compensation mount that further proves the predictability of the situation is the response of the beta rotation about the y-axis of the system with a force applied at the second actuator location. This situation is unique because the actuator is located on the specific axis. Because of this, there will be no rotation about the y-axis. Anything multiplied by this transfer function will still results in the original transfer function or zero, creating no response in the system. VI. Conclusion The computer simulation for tilt compensation demonstrates a reliable system that will respond accordingly to reduce rotational and translational vibration. From all the findings, it is shown that the stabilization mount will react as necessary in each situation, providing confidence that this vibration compensator will work in the field. The reliability of the computer simulation compared to actual testing procedures demonstrates the accurateness of the simulations. By showing that the tilt compensation will work according to simulation, therefore strengthening the confidence in the ability of the actual system to work when put through physical testing. Accordingly, the next goal for the vibration stabilization mount should include tilt compensation testing, where each actuator is allowed to move separately from the other actuators. VII. 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. References [1] Stuckel, K., Semke W., Baer, N., Schultz, R., “A High Frequency Stabilization System for UAS Imaging Payloads,” IMAC SEM Conference, February 2010. [2] 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,” Proceedings of 2007 ASME International Mechanical Engineering Congress and Exposition, IMECE2007-43620, November 2007. [3] Lendway, M., Berseth, B., Tandem, S., Schultz, R., and Semke, W., “Integration and Flight of a UniversityDesigned UAV Payload in an Industry-Designed Airframe,” Proceedings of the AUVSI Conference, 2007. [4] Stuckel, K., M.S. Mechanical Engineering, University of North Dakota, “A High Frequency Stabilization System for UAS Imaging Payloads,” May 2010. [5] Emanuel., P., Leff, E., Introduction to Feedback Control Systems [6] Wescott, T., “PID with a PhD”, Embedded Systems Programming, http://www.embedded.com/2000/0010/0010feat3.htm [7] Guda, V., M.S. Mechanical Engineering, University of North Dakota, “A Precision Positioning Actuator System Using a PID Controller,” 2007 th

[8] Dorf, R., Bishop., “Modern Control Systems,” Prentice Hall, 11 edition, 2008. [9] Ogata, K. “MATLAB for Control Engineers,” Prentice Hall, 2008.

Extraction of Modal Parameters From Spacecraft Flight Data George H. James, Timothy T. Cao, Vincent A. Fogt, Robert L. Wilson Loads and Structural Dynamics Branch NASA Johnson Space Center Houston, Texas 77058 Theodore J. Bartkowicz The Boeing Company Space Exploration Division Houston, Texas 77059

ABSTRACT The modeled response of spacecraft systems must be validated using flight data as ground tests cannot adequately represent the flight. Tools from the field of operational modal analysis would typically be brought to bear on such structures. However, spacecraft systems have several complicated issues: 1. 2. 3. 4. 5.

High amplitudes of loads; Compressive loads on the vehicle in flight; Lack of generous time-synchronized flight data; Changing properties during the flight; and Major vehicle changes due to staging.

A particularly vexing parameter to extract is modal damping. Damping estimation has become a more critical issue as new mass-driven vehicle designs seek to use the highest damping value possible. The paper will focus on recent efforts to utilize spacecraft flight data to extract system parameters, with a special interest on modal damping. This work utilizes the analysis of correlation functions derived from a sliding window technique applied to the time record. Four different case studies are reported in the sequence that drove the authors’ understanding. The insights derived from these four exercises are preliminary conclusions for the general state-of-the-art, but may be of specific utility to similar problems approached with similar tools. INTRODUCTION The spacecraft launch environment is a highly complex event that is characterized by high amplitude input forces, highly variable loads, a wide spectrum of responses, constantly changing vehicle mass, active control interactions, and staging. At the same time, structural response analyses and loads estimations must be performed with models that are only partially validated using ground test data. In fact in modern design cycles, the access to diagnostic and environmental ground tests is also limited. To compound matters, project managers tend to reduce uncertainty factors designed to protect for loads increases and model unknowns. As a result, the designs progress rapidly before loads and structural problems are uncovered. This means that there are very few tools available to recover from structural dynamics issues in such a highly dynamic environment without costly redesigns late in the design cycle or in early operations. One tool is the further reduction of any uncertainty factors that cover model unknowns, which are driven strongly by the need to know the modal frequencies. Since the ability to obtain good modal frequency information is limited by ground test availability, the flight data from early test flights and early operational flights must be used to estimate frequency values. Another of the tools available is to adjust the assumed damping values for the flight environment. Most structural dynamics specialists realize that damping is not a parameter that can be determined with any certainty (even in high controlled laboratory situations). However, vigorous discussions about damping in the flight environment are becoming a more common occurrence. The industry as a whole is very limited on tools and techniques to estimate damping from flight data. This paper contains information about early attempts to understand and develop frequency and damping estimation in the flight environment. The obvious starting point for estimating modal parameters from a structure in the field is the current technology of Operational Modal Analysis (OMA) [1]. This rapidly advancing field of study is developing and applying techniques to es-

T. Proulx (ed.), Advanced Aerospace Applications, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 4, DOI 10.1007/978-1-4419-9302-1_26, © The Society for Experimental Mechanics, Inc. 2011

307

308 timate modal parameters for operational structures with response measurements but without a measure of the input forces. These techniques are focused on ground-based structures that are stable and long measurement times are generally available. There were attempts at operational modal analyses as early as the 1960’s. However, it was in the 1990’s before the early generalized techniques were available [2, 3]. There are two general classes of algorithms for performing OMA: (1) time history-based techniques that are generally related to Stochastic Subspace Identification (SSI) [1-6] and frequency domain-based techniques that are related to Frequency Domain Decomposition (FDD) [1, 4, 5, 6]. The technical basis for the early time domain approaches involved converting measured responses into auto and crosscorrelation functions and processing with standard time domain modal analysis routines [3]. However, the more general SSI techniques directly integrate the correlation calculations and modal processing algorithms into a single step rooted in discrete time system identification theory [4, 6]. The earliest manifestations of FDD were peak picking and half-power bandwidth estimation schemes operating on the Power Spectral Density (PSD) functions [5]. However the advanced FDD algorithms refine the modal parameter estimates using powerful tools like the Singular Value Decomposition (SVD) [1, 5]. An interesting direction for frequency domain approaches involves the use of Hilbert transforms applied to PSD’s to obtain biased Frequency Response Function (FRF) estimates [1, 7]. There is another direction in operational testing that is offering some hope for more fully understanding system dynamics is to estimate the forces acting on the system. This would allow more traditional FRF-based approaches to be used for system identification. These forces can be estimated via known mass changes to the system [8] or via hybrid analytical/experimental data [9]. There have been a limited number of reported attempts to analyze flight data to extract modal parameter information, although there are certainly many other unreported attempts. The time domain approaches based on correlation and SSI are generally used for flight data analyses as the rapidly changing vehicle properties do not allow the full advantages of the frequency-domain approaches to be realized. The responses are generally broken into a series of sliding windows, each of short duration, which are processed individually. If the loading and system characteristics are fairly constant over each window, then estimates of the changing parameters can be obtained as a function of flight time [10-17]. This paper will include four “case studies” with different levels of complexity to exercise these techniques. A lander case study is provided first, which provides some early insight with less complexity to cover the issues. The second case study is a pointer to a launch system analysis that has shown some of the same unusual features that the first case study contained but contains all of the complexities expected in a launch system. The third case study is a series of very simple analytical studies using single degree-of-freedom systems to gain insight into the most basic aspects of operational flight data analysis seen in the earlier exercises. The final case study is the analysis of a very rapidly changing system during an abort test, which is the most difficult launch system that one can envision. The insights gained from the observations of the earlier case studies has driven the approach to this very difficult data analysis. The end result of this work are preliminary conclusions for further advancement of the field of operational analysis of rapidly changing systems and specific application recommendations for similar problems. TECHNICAL BACKGROUND The procedure used in this work will be to calculate auto and cross-correlation functions from response data obtained from the vehicle under launch conditions. It has been shown that these correlation functions are composed of decaying sinusoids with the appropriate modal frequency and modal damping. These modal parameters are then extracted from the correlation functions using time domain extraction algorithms [3]. Since the vehicle and the launch environment are rapidly changing systems, a sliding window approach is used to repetitively analyze short segments of data and collect the information to track changing frequency and damping [10-13]. This utilizes automated environment for the time domain modal parameter extraction using a technique (called AUTO-ID) [18, 19]. The addition of this technique eases the computational burden of extracting parameters in a consistent manner from the multitude of correlation functions calculated from the sliding window segments of the random time histories to allow a rapid assessment of the data. It is expected that in-depth data analysis will follow using a combination of digital filtering [10, 11], time-zooming [17], or advanced SSI techniques [15].

309 CASE STUDY 1 – LANDER SYSTEM FLIGHT The RR1 vehicle is a lander test craft built by Armadillo Aerospace. It was flown in a tethered hover flight for over 30 seconds to produce the data set used herein (see Figure 1). The RR1 flight data was unique in the respect that it was real flight data with high amplitude excitation, an active control system, and mass changes due to fuel use. However, this data set did not contain significant changes in natural environments or forward velocity effects as it was a hover test. The RR1 data set consisted of 31,000 samples of three low frequency accelerometers that were sampled at 1,000 samples per second and useful for 1.4 to 350 Hz (see Figure 2). The channels were ranged for +/- 16g with a .007g resolution.

Figure 1. RR1 Vehicle in Static and Tethered Flight Configurations

Figure 2. RR1 Vehicle Accelerometers and Coordinate System The first use of the data was to assess the stability of the modal properties using a sliding window analysis with 2,000 point (2 sec.) windows. A 50% overlap or one thousand points (1 sec.) were skipped for each sliding window move. This allowed 30 different windows to be generated out of the full 31,000 sample time records. Nine cross and autocorrelation functions were calculated from the three sensors. AUTO-ID with the Eigensystem Realization Algorithm (ERA) was used as the modal engine [19]. The Consistent Mode Indicator (CMI) [20] was used with a 50% cut-off. Frequencies with damping values outside of 0% to 20% were excluded. Figure 3 shows the original data from the three sensors and the frequencies determined for each of the 30 different sliding windows. The data plots in Figure 3 show a strong Z direction mode near 30 Hz. Additional dynamic content is seen between 125 and 225 Hz with some content above 225 Hz (Y direction). The sliding window analysis shown in Figure 3 mirrors these observations. An important point is that neither the sensor magnitudes nor the extracted frequencies appear to change over the times selected.

310

Figure 3. Three Axis RR1 Data and Sliding Window Analysis Results Using 30 Two Second Windows Figure 4 shows the variations in frequency and damping for modes near 30, 130, 150, and 200 Hz. In the Figure 4 data near 30 Hz, reasonable stability in the frequencies and typical variations in damping is observed. The Figure 4 mode near 130 Hz plot, also shows reasonable stability in damping and frequency. In fact the frequency does show a minor increasing trend and the damping is more stable than the mode near 30 Hz. The modes near 150 and 200 Hz show the unusual characteristics of having stable damping but more variability in frequency. The important information here is that (in spite of the 130 Hz frequency) there is little apparent change in modal parameters over flight time. The next logical step in the analysis of this data is to filter in around a frequency and use a limited dataset to more completely define specific frequency and damping. For this exercise, the Y direction sensor was used to attempt to refine the assessment of the mode near 30 Hz. A 4th order Butterworth forward/reverse 10-45 Hz bandpass filter was applied to the data. Figure 5 provides the resulting autocorrelation function (and autospectrum) as well as the frequency/damping results of processing each 1,000 lag window of the correlation function. It is important to note that the correlation function contains an unexpected “beating” after an initial decay, which manifests itself as instability in the frequency domain. The correlation function in Figure 5 shows that the first part (approximately 2 seconds) of the trace looks like the expected decaying sinusoid, although the rest is dominated by the “beating”. Table 1 shows the results of a more detailed assessment of the first 1,000 lags (one second). Analyses of the first, second, third, and fourth 250 lags are provided as well as an analysis of the first 1,000 lags (which envelopes all of the four 250 lag sets). It can be seen that a fairly consistent 28.2 to 28.3 Hz mode is seen. The short segment analyses have damping values that vary between 0.4% and 1.1%

311 with the full 1,000 lag analysis suggesting a damping value of 0.8%. It should be noted that the 1,000 lag analysis is the first data point on sliding window 1,000 lag analysis plots of Figure 5. Hence Figure 5 can be used to understand how the analysis results vary for subsequent 1,000 lag analyses. Table 1 also shows that there are additional higher damped loads in the early half second of the correlation functions. Therefore this data suggests that the first part of the correlation functions decay as expected and contain higher damped modes – but this is not conclusive. Additional insights will be gained using the numerical studies that will be discussed in an upcoming section.

Figure 4. RR1 Frequency and Damping Sliding Window Variations for Modes near 30, 130, 150, and 200 Hz CASE STUDY 2 – LAUNCH SYSTEM FLIGHT The Ares1-X vehicle was a full-up launch system that flew first stage flight including motor ignition, liftoff acoustics, aerodynamic flight including maximum dynamic pressure, and motor burn-out. This flight data has a full suite of the characteristics seen in flight data and thus presents a more difficult analysis than the lander data in case study 1 above. The fifth author has performed a detailed independent analysis of this launch data and the reader is referred to reference [17] for details. As detailed in [17], the analyst identified the same problems with respect to unexpected “beating” at the higher time lags in the correlation functions. CASE STUDY 3 – SIMPLIFIED NUMERICAL STUDY The recent experience (discussed in previous sections) with the analysis of flight data has prompted the need to answer some basic questions concerning the data content and processing requirements. A simplified numerical study was performed to address the two most pressing issues seen in the case studies mentioned above: how does amplitude dependent

312 damping manifest itself in the analysis, and what is the source/mitigation of the beating phenomena. A single degree-offreedom system was developed with a known frequency (10 Hz) and known damping values. Constant damping (2%) and amplitude-dependent damping were both modeled. Both impulse response and random excitations were applied to the system. Additive random noise of 0%, 5%, and 10% of maximum input force was added to the constant damping cases as a side study. The variable damping schedule for the impulse excitations was 5% for amplitudes above fifty percent of maximum amplitude, 2% for amplitudes between twenty five and fifty percent of maximum amplitude, 1% for amplitudes between ten and twenty five percent, and 0.5% below ten percent. The variable damping schedule for the random excitations was 5% for displacement amplitudes above .002 inches and 0.5% below .002 inches. A NewmarkBeta scheme was used to integrate the system (with an effective sample frequency of 500 samples per second) and develop traces that were five seconds long for impulse loading and twenty seconds long for random loading. For the random excitation cases, the damping for each step was set based on the displacement magnitude of the intermediate steps.

Figure 5. 30 Hz Variations for Sliding 1,000 Lag Window Analysis of Filtered Full-Length RR1 Autocorrelation Table 1. Detailed Frequency and Damping for Early Lags of 30Hz Detailed Analysis of RR1 Data Lags Used 1st 250 2nd 250 3rd 250 4th 250 1st 1000

Frequency #1 (Hz) 26.9 27.4 -

Damping #1 (%) 3.5 5.1 -

Frequency #2 (Hz) 28.3 28.2 28.3 28.3 28.2

Damping #2 (%) 0.4 1.0 0.6 1.1 0.8

The same analysis process was employed on this data as was used on the previous case studies (develop correlation functions and process with a time domain parameter estimation scheme). Figure 6 shows the impulse response (free decay) for the system with the variable damping (regions of the different damping levels are noted) as well as the associated correlation function. Table 2 shows the results of these impulse response data sets. It can be seen that the constant case with 0% added noise recovers the correct damping and frequency. Noise has a noticeable but not significant effect on the extracted values for this simple study. The variable damping data produced estimated modes at 10.1 Hz (6.2% damping) and 10.0 Hz (0.8% damping) when the correlation function was processed the same as the constant 0% noise case. The split modes with different damping values is expected but the over-compensation of the high damping levels is unexpected. Additionally, the variable damping correlation function was broken into 24 .2 second (100 sample) segments. The extracted damping for this part of the study was dependent on the segment analyzed (see Table 2). The early windows are strong in the higher damping and the later windows reproduce the lowest damping.

313 Figure 7 provides two functions. The left plot is the autocorrelation of a single time record under random loading with a constant damping. Notice the significant “beating” phenomena that is seen in the longer lags. This is a reproduction of the same effect seen in the first two case studies mentioned above. The right hand plot of Figure 7 shows an average of 20 of these correlation functions, each with a different random input. Notice that the “beating” phenomena is significantly reduced after averaging. In this study, averaging provided the most dramatic reduction in the “beating”. Table 2 clearly shows the adverse effects of this beating on the damping estimates (1.4 to 3.0 for each of the 20 autocorrelation functions processed individually). This is not an unexpected finding but is critically important to understanding and potentially mitigating some difficulties in estimating damping from flight data. Most of the low frequency bands in flight data are not amenable to temporal averaging to reduce this inherent noise as the system is changing too rapidly. The alternatives are to use only the early part of the correlation functions, perform spatial averaging of several sensors, or utilizing force estimation procedures to attempt to reduce the random component of the responses.

Figure 6. Impulse Response and Autocorrelation Functions for 10 Hz Case with 5.0% to .5% Variable Damping Figure 8 shows the displacement response of the 10 Hz system with variable damping and a random input. Three different segments of the time history are denoted on the plot. The first exercise with the random data was to process these segments separately to understand how the damping might manifest itself. Auto-correlation functions associated with each segment are provided in the other plots of Figure 8. Table 2 shows the results of processing the correlation functions associated with these three segments. It can be seen that the modal frequency is extracted reasonably well but the damping values are different than expected (neither 5 Hz nor .5 Hz are found). This is an indication that the simulation of such a system is non-trivial and the approach used herein may be too simplistic. The hypothesis going into this study was that the early lags of the correlation function would be enhanced in the highly damped responses and the later lags would be enhanced in the lightly damped responses (as was seen with the impulse response loading). However, if this effect does exist it may not be manifested in this simplified approach. The next numerical study performed was to assess the results of processing the complete autocorrelation function for the random loading/variable damping case. Table 2 shows the results from processing a correlation function and the results using an average of 20 correlation functions each with a different random input. The comparison seen in Figure 7 shows the effects of this averaging on the correlation functions. The comparative assessments are performed using a sequential series of .2 second windows taken from the single or average correlation functions. The results from the first three windows are very similar between the single and the average functions. Also, neither plot shows the expected 5% damping or the alternative 0.5% damping of the intended system. The resulting 1.3% is some combination of the two and is similar to the earlier results seen in processing different segments of the time record. The later windows, which utilize longer time lags, show much variation in damping as the numerical “beating” effect dominates the damping estimation. It is not

314 clear from the data provided in Table 2, but the average correlation function does produce consistent damping values for more windows than the single correlation function. However, once the windows of the average case enter the “beating” region, the damping values vary as much as those extracted from the single correlation function. The major conclusions are that it is advantageous to average correlation functions and focus on the early lags of the system. There are indications from the impulse data that the correlation function can maintain information on variable damping. However, this work has not yet shown that correlation functions of random excitation cases can retain that same information. Table 2. Results of Numerical Study Excitation Type Impulse Impulse Impulse Impulse

True Frequency (Hz) 10.0 10.0 10.0 10.0

True Damping (%) 2.0 2.0 2.0 5.0 to 0.5

Impulse Impulse Impulse Impulse

10.0 10.0 10.0 10.0

5.0 to 0.5 5.0 to 0.5 5.0 to 0.5 5.0 to 0.5

Random Random Random Random

10.0 10.0 10.0 10.0

2.0 2.0 5.0 and 0.5 5.0 and 0.5

Random

10.0

5.0 and 0.5

Random

10.0

5.0 and 0.5

Random

10.0

5.0 and 0.5

Study Parameters 0% noise 5% noise 10% noise same parameters as constant case windows 1,2,3 windows 4,5,6 windows 7,8,9 segments 10,11 to 24 20 single traces 20 trace average segment 1, 2, 3 windows 1,2,3 (single trace) windows 4-23 (single trace) windows 1,2,3 (average trace) windows 4-23 (average trace)

Estimated Frequency (Hz) 10.0 10.0 10.0 10.1 and 10.0

Estimated Damping (%) 2.0 2.1 2.3 6.2 and 0.8

10.0, 10.0, 10.0 10.0, 10.0, 10.0 10.0, 10.0, 10.0 10.0, 10.0, …, 10.0 9.9 to 10.1 10.0 10.0, 10.0, 9.9 10.0, 10.0, 10.0

4.7, 4.5, 1.75 0.9, 1.0, 1.0 1.0, 1.0, .9 0.6, 0.5, …, 0.5

9.8 – 10.0

0.3 – 2.1

10.0, 10.0, 10.0

1.3, 1.3, 1.3

10.0 – 10.0

-0.3 – 1.9

1.4 to 3.0 2.0 6.0, 1.0, 1.7 1.3, 1.2, 1.2

Figure 7. Autocorrelation Function of a Single Random Response and 20 Case Average (Constant Damping)

315

Figure 8. Displacement History and Acceleration Autocorrelation Function with Variable Damping CASE STUDY 4 – ESCAPE SYSTEM FLIGHT DATA The fourth case study is one of the most difficult applications of operation analysis that is available and the analysis efforts attempted to date have used insights gained from the case studies mentioned previously. The Pad Abort 1 test flight was a test flight of an escape system for a manned vehicle, which occurred in May 2010 at the U.S. Army’s White Sands Missile Test Range near Las Cruces, NM. The test flight is characterized by very short and energetic excitations, rapidly depleting fuel, a wide range of excited frequencies, and high assumed flight damping estimates. Figure 9 shows the PA-1 vehicle in static and flight. The vehicle includes a Command Module (CM) test article with active flight control and data acquisition systems. The Launch Abort System (LAS) containing the motors for abort flight, steering control, and jettison is also an active part of the system. The focus of the initial studies of this flight data have been on the very challenging early burn phase of flight, which is the first few seconds of flight with the highest excitation levels and the most rapidly changing system. A subset of 19 accelerometers covering both the CM and the LAS were used for these studies. Figure 10 provides example data from two of these sensors. These two sensors have been used as the “reference channel” in all the early burn analyses to date. Note that the time axis is given in terms of % of early burn and the frequency axis is provided in terms of % of sampling frequency to avoid any issues with current government-imposed data restrictions. This approach is intended to protect the sanctity of the data yet providing information that experts in the field can use to assess the results of this work. Figure 11 provides the cross-correlation function between the two sensors shown in Figure 10 in the left hand plot. Also included in the plot is the sum of this cross-correlation function with of the other 18 correlation functions in the right hand plot (the sensor trace in the right hand plot in Figure 10 was used as the reference channel for all 19 correlations). This average of the 19 correlation functions is intended to implement the lessons learned in analytical studies shown previously. There are not enough time records (and the system is changing to fast) to average the data temporally, so this spatial averaging of data from a distributed set of sensors is used. Although neither of the functions shown in Figure 11

316 looks like the damped sinusoidal response expected, the sum of all correlation functions does show more distinct dynamic phenomena at the higher frequencies.

Figure 9. PA-1 Vehicle and Test Flight.

Figure 10. Two PA-1 Accelerometer Traces with Time and Frequency Domain Representations

317

Figure 11. Cross-Correlation Function between Two PA-1 Responses with the Sum of 19 Correlations Functions The most consistent analysis of the PA-1 data in this study has been focused on first three primary bending modes of the structure using 4th order Butterworth filters (used in a forward/reverse mode) around each assumed mode. The bandwidth of the filters is 1.2 % of the sampling frequency wide. A sliding window analysis is then used to assess the variability of the frequency and damping in each filtered dataset. Each sliding window is 21% of the total early burn time and is incremented by 1.3% of the early burn time to allow 60 separate analyses over the early burn flight time. Figure 12 shows an example of the results for the analysis around the first bending. The results for the sliding window analysis are provided in the left hand plot as well as the extracted results for a potential mode for the latter part of the burn (the right hand plot). In the sliding window analysis there are a significant number of noise modes, but more significantly are the consistent trends of the potential bending modes. Obtaining such consistent trends has been a primary focus of this flight data analyses to-date. The modal frequency has a reasonable and consistent trend for similar analyses. The damping has a trend but is not consistent (although similar to past flight damping estimates [10, 11]). The damping could be affected by the natural environment changes, control system forcing functions, or inconsistencies in the local fit. The next phase of this analysis would entail going to these specific windows between 63 and 71 and performing more detailed analyses. Regardless of additional work to understand this unusual trend, the results provided would suggest that typical values of 1% to 2% should be maintained for the damping in this mode for future flight analyses. Another finding is that the dynamic properties are changing too rapidly for the entire early burn to be used as a monolithic analysis. Figure 13 contains the sliding window analysis results for a narrow band of frequencies around the second bending mode of the system in the left hand plot. There are several dynamic phenomena that show distinct trends. The mode that is likely one of the second bending modes is shown in the right hand plot. The frequency is showing a fairly consistent trend of increasing frequency as the fuel in the abort motor is quickly burned in this flight phase. The damping is showing quite a bit of variation throughout the flight (from 0.5% to 4%). Although this is similar to trends seen in previous flight data analyses [10, 11], the numerical study provided earlier suggests that the noise in the data could be significantly affecting damping. However, the assessment of the data if taken at face value is that typical damping values of 1% to 2% should be maintained for this frequency range until additional analyses can be performed. Figure 14 provides the sliding window analyses for the region around the third bending mode. As expected, there are an increasing number of dynamic phenomena as the frequency band increases. There are also more consistent trends and fewer noise modes in this analysis than at the lower modes. This is potentially due to the fact that there is more apparent excitation in the frequency bands (see Figures 10 and 11) and there are more cycles of the damped sinusoids in each window of correlation data analyzed. However the frequency and damping trends are similar to what was seen in the earlier frequency bands. The increasing modal density suggests that it would be useful to perform shape fitting to the correlation functions of the individual sensors to extract information help in determining the true nature of the dynamic phenomena captured in these results.

318

Figure 12. PA-1 Early Burn Results for Sliding Window Analysis around the First Bending Mode

Figure 13. PA-1 Early Burn Results for Sliding Window Analysis around the Second Bending Mode

Figure 14. PA-1 Early Burn Results for Sliding Window Analysis around the Third Bending Mode

319 CONCLUSIONS Although the information reported herein is a work in progress, there are several preliminary conclusions that could be drawn from the four case studies mentioned herein. First the most prevalent feature of the analyses performed herein is the “beating” phenomena seen in the random excitation data that is typically reduced by averaging in non-flight data sets. There were three tools used to help clear up the analyses issues reported in this paper (including the “beating” effect): spatially averaging several sensors correlation functions, using the earliest lags in the correlation functions, and applying fairly tight bandpass filters to the data. Also, additional analysis of local time segments is needed to find the true cause for the variations in damping seen in the extracted modes. There are several additional suggestions for forward work that have been suggested: continue to understand the complexities of simulating random excitation/variable damping systems; implement, apply, and assess more advanced operational analysis tools; perform shape estimations as the next step in such analyses; and apply force reconstruction techniques to reduce some of the detrimental noise/forcing function effects. In addition, a widely distributed set of time-synchronized flight sensors should be developed, flown, and used for analyses to expand the state-of-the-art. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

Brinker R and Kirkegaard P (2010) Special issue on operational modal analysis. Editorial in Mechanical Systems and Signal Processing, Vol. 24, No. 5, pp. 1209-1212. Carne T and James G (2010) The inception of OMA in the development of modal testing technology for wind turbines. Mechanical Systems and Signal Processing, Vol. 24, No. 5, pp. 1213-1226. James G, Carne T, and Lauffer J (1995) The Natural Excitation Technique (NExT) for modal parameter extraction from operating structures. SEM International Journal of Analytical and Experimental Modal Analysis, Vol. 10, No. 4. Brownjohn J and Carden P (2007) Reliability of frequency and damping estimates from free vibration response. Proceedings of the 2nd International Operational Modal Analysis Conference, pp 23-30. Zhang L, Wang T, and Tamura Y (2010) A Frequency-Spatial Domain Decomposition (FSDD) method for operational modal analysis. Mechanical Systems and Signal Processing, Vol. 24, No. 5, pp. 1227-1239. Magalhăes F, Cunha A, Caetano E, and Brinker R (2010) Damping estimation using free decays and ambient vibration tests. Mechanical Systems and Signal Processing, Vol. 24, No. 5, pp. 1274-1290. Agneni A, Crema L B, and Coppotelli G (2010) Output-only analysis of structures with closely spaced poles. Mechanical Systems and Signal Processing, Vol. 24, No. 5, pp. 1241-1249. Bjerg I, Hansen S, Brinker R, and Aenlle M L (2007) Load estimation by frequency domain decomposition. Proceedings of the 2nd International Operational Modal Analysis Conference, pp 669-676. James G, Carne T, and Wilson B (2007) Reconstruction of the Space Shuttle roll-out forcing function. Proceedings of the 25th International Modal Analysis Conference. James G, Carne T, and Edmunds R (1994) STARS missile – modal analysis of first flight data using the Natural Excitation Technique (NExT). Proceedings of the 12th International Modal Analysis Conference. James G, Carne T, and Marek E (1994) In-situ modal analysis of STARS missile flight data and comparison to pre-flight predictions from test-reconciled models. Proceedings of the 15th Institute of Environmental Sciences Aerospace Testing Seminar. Kim H, VanHorn D, and Doiron H (1994) Free-decay time-domain modal identification for large space structures. Journal of Guidance, Control, and Dynamics, Vol. 17, No. 3, pp. 513-519. James G (2003) Modal parameter estimation from Space Shuttle flight data. Proceedings of the 21st International Modal Analysis Conference. Le Gallo V, Goursat M, and Gonidou L (2007) Damping characterization and flight identification. Proceedings of the 25th International Modal Analysis Conference. Goursat M, Döhler M, Mevel L, and Andersen P (2010) Crystal Clear SSI for Operational Modal Analysis of Aerospace Vehicles. Proceedings of the 28th International Modal Analysis Conference. Fransen S, Rixen D, Henricksen T, and Bonnet M (2010) On the operational modal analysis of solid rocket motors. Proceedings of the 28th International Modal Analysis Conference Bartkowicz T and James G (2011) Ares 1-X in-flight modal identification. Submitted to the American Institute of Aeronautics and Astronautics Structures, Structural Dynamics, and Materials Conference. Pappa R, James G, and Zimmerman D (1999) Application of autonomous modal identification of the Space Shuttle tail rudder. AIAA Journal of Spacecraft and Rockets, Vol. 35, No. 2, pp. 163-169. James G, Chhipwadia K, and Zimmerman, D (1999) Application of autonomous modal identification to traditional and ambient data sets. Proceedings of the 17th International Modal Analysis Conference. Pappa R, Elliott K, and Schenk A (1993) Consistent mode indicator for the eigensystem realization algorithm. Journal of Guidance, Dynamics, and Control, vol. 16, no. 5, pp. 852-858.

Dynamic Characterization of Satellite Components through Non-Invasive Methods David Macknelly1, Josh Mullins2, Heather Wiest3, David Mascarenas4, Gyuhae Park4 1 Dept. of Mechanical Eng., Imperial College London, UK 2 Dept. of Civil and Environmental Eng., Vanderbilt University, Nashville, TN 37240 3 Dept. of Mechanical Eng., Rose-Hulman Inst. of Tech., Terre Haute, IN 47803 4 The Engineering Institute, Los Alamos National Laboratory, Los Alamos, NM 87545

Abstract The rapid deployment of satellites is hindered by the need to flight-qualify their components and the resulting mechanical assembly. Conventional methods for qualification testing of satellite components are costly and time consuming. Furthermore, full-scale vehicles must be subjected to launch loads during testing. This harsh testing environment increases the risk of component damage during qualification. The focus of this research effort was to assess the performance of Structural Health Monitoring (SHM) techniques as a replacement for traditional vibration testing. SHM techniques were applied on a small-scale structure representative of a responsive satellite. The test structure consisted of an extruded aluminum space-frame covered with aluminum shear plates, which was assembled using bolted joints. Multiple piezoelectric patches were bonded to the test structure and acted as combined actuators and sensors. Various methods of SHM were explored including impedance-based health monitoring, wave propagation, and conventional frequency response functions. Using these methods in conjunction with finite element modelling, the dynamic properties of the test structure were established and areas of potential damage were identified and localized. The adequacy of the results from each SHM method was validated by comparison to results from conventional vibration testing. LA-UR 10-07115.

1

Introduction

The current worldwide security climate requires surveillance capabilities at short notice, which in turn demands more responsive satellite launches. Current satellite missions can take years between inception and launch (Yachbes, Roopnarine, Sadick, Arrit, & Gardenier, 2008), by which time, the situation which demanded the satellite may have evolved. It is the goal of the Air Force Research Laboratory (AFRL) to be able to reduce this lead time to six days. In order to accomplish this, a new type of satellite construction and testing methodology is required. A new, modular, plug-and-play (PnP) construction architecture has been pioneered with small research satellites called Cubesats (Voss, Coombs, Fritz, & Dailey, 2009). These satellites are physically small, with the 1U standard measuring 100 x 100 x 100 mm. The Cubesats can be ‘piggy-backed’ onto traditional satellite launches, dramatically reducing the costs associated with placing the hardware into orbit. Traditional satellites, however, require a strict regime of vibration testing in order to qualify them for launch. The satellite is first exposed to a sine-sweep to establish its vibration T. Proulx (ed.), Advanced Aerospace Applications, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 4, DOI 10.1007/978-1-4419-9302-1_27, © The Society for Experimental Mechanics, Inc. 2011

321

322

characteristics and then to a random spectrum replicating launch loads. Finally the sine-sweep is re-applied to verify that the random spectrum did not significantly alter the dynamic response of the satellite, which would indicate damage had occurred. This testing regime is not only potentially damaging to the satellite but also increases the lead-time between order and launch. With this in mind, AFRL intends to use SHM techniques to qualify satellites for launch. The monitoring would start during construction, establishing a ‘baseline’ measurement which can be used for comparison during the satellites operational lifetime. The preferred method of SHM is piezoelectric (PZT) patches which act as combined actuators and response sensors (PiezoSystems, 2008). The patches are cheap, can be easily attached to the structure, and are lightweight to minimize mass loading. The PZT patches actuate structures with lower amplitude vibrations compared to modal hammers or electrodynamic shakers, greatly reducing the risk of damage to the structure under test. Whilst this technique reduces the risk of damage due to its low power, it also increases the difficulty in actuating the structure with enough power to enable global dynamic characterisation. The aim of this project is to use PZT patches to establish the dynamic characteristics of a representative structure. The representative structure consists of an extruded aluminium box frame 300 x 300 x 300 mm, with 3 mm thick aluminium shear plates bolted to the frame, as illustrated in Figure 1.

Figure 1: Illustration of construction of cube structures

The primary aim of the project is to excite and measure the first three modes of the structure using the PZT patches. The project will proceed in phases. The first phase will be to characterise the structure using traditional excitation and sensor methods. The next phase will be to correlate the results from the traditional method with a Finite Element Model (FEM) and use the results to determine the optimal position for PZT patch placement. Different PZT excitation methods will be evaluated in order to find the optimal method of exciting and

323

measuring the first three global modes. Bolt loosening detection will also be explored using the same PZT excitation method, but at a higher frequency.

2

Preliminary Testing

Two modal hammer tests were performed on the aforementioned representative test structure. The first test was performed on the entire cube structure while the second test was performed on a single panel section of the cube. 2.1 Set-up and Procedure for First Modal Test Due to the symmetry of the test structure, only three adjacent faces of the cube were used in the first modal test. These faces were the top and two sides of the test structure. A diagram of the test cube showing the panel locations is shown in Figure 2 for reference in the rest of this paper.

Figure 2: Panel reference diagram for experiments and FE models

Figure 3: Set-up of first modal hammer test on entire test structure

324

The set-up used in the first modal test is shown above in Figure 3. A grid consisting of 16 points was drawn on each of the three sides for a total of 48 measurement points. The dimensions of the grid spacing used in this test are shown below in Figure 4.

Figure 4: Dimensions of grid spacing on one face of the cube used in first modal test

Three accelerometers were used to record the measurements from the first modal test. One accelerometer was placed on each of the three sides tested. The accelerometers were placed at pointson plates 2, 4, and 6. The cube was placed on top of two foam blocks as well as two rolled pieces of bubble-wrap to create a free-free boundary condition. During data acquisition, the Nyquist frequency was set to 2500 Hz, 4096 samples were taken, and 3 impacts were averaged at each point. A National Instruments PXI data acquisition system was used to collect the data. The plastic tip of the modal hammer was covered with a rubber jacket to ensure that higher modes were excited while minimizing the chance of double impacts. 2.1.1 Set-up and Procedure for Second Modal Test For the second modal test, the cube test structure was disassembled and only one section was used to take measurements. The section consisted of one of the aluminum shear plate and the part of the extruded aluminum frame that was bolted flat against the plate. The complete set-up used in the second modal test is shown below in Figure 5.

325

Figure 5: Set-up of second modal hammer test on a section of the test structure

A grid consisting of 49 points was drawn on this section of the cube, and the dimensions of the grid spacing used in this test are shown below in Figure . Three accelerometers were used to record the measurements for the second modal test. The accelerometers were placed in the configuration shown in Figure 6. In order to create a free-free boundary condition, two bungee cords were used to hang the section. The settings on the National Instruments PXI data acquisition system were as follows: the Nyquist frequency was set to 1500 Hz, 2048 samples were taken, and 4 hits were averaged at each point. Once again, a rubber covering was used over the plastic tip of the modal hammer.

Figure 6: Grid spacing on the section of cube used in second modal test

2.2 Testing Using Piezoelectric Sensors PZT patches were applied to the same panel section of the whole cube to explore the viability of using PZT actuation to excite the predetermined modes of the structural component. PZT

326

patches were also used to determine the feasibility of exciting all modes on the full cube structure. 2.2.1 Set-Up and Procedure for PZT Excitation of Single Plate One piezoceramic patch was placed on the single panel test structure. It was fully bonded to the plate along its 2.85 inch length and 1.5 inch width. A second PZT patch made from a macro fiber composite (MFC) material was also placed on the plate and was similarly bonded to the surface. An accelerometer was also placed on the plate adjacent to the MFC for comparison. This configuration is shown in Figure 7.

Figure 7: PZT configuration on single panel section of the test structure

For the purposes of this test, the piezoceramic patch served as the actuator, and the MFC served as the sensor. A chirp signal with amplitude of 5.4 V was applied over a frequency range of 10 to 2000 Hz to the PZT actuator. A total of 4096 samples were taken at a sampling frequency of 4000 Hz with the chirp actuation applied for 80 percent of the sampling period. Data was acquired via RTPro software using a Dactron system. 2.2.2 Set-up and Procedure for PZT Excitation of Full Cube This initial test of the full cube with PZT sensors was performed with two piezoceramic patches on adjacent faces. The actuator patch was located on plate 6 while the sensing patch was located on plate 3. A sine chirp ranging from 10 to 1000 Hz with 1V amplitude and 70 percent period was used as the input. Data was acquired via RTPro software using a Dactron system. In order to create a free-free boundary condition for this testing, each cube was hung using rubber tubing. The tubing was hung over a piece of an 80-20 bar which rested on two identical stools. The set up used is shown below in Figure 8.

327

Figure 8: Set-up of full cube test with PZT excitation

2.2.3 Set-up and Procedure for PZT Excitation of Single Plate with 4 Small Sensors The first experiment with small circular PZT patches was performed on a single plate of the cube, as shown in Figure 9. Four piezoceramic patches, each 1 cm in diameter, were used and were placed on the plate in the configuration shown in Figure 9. The plate was hung in the same configuration as described in section 2.1.2. Data was acquired via RTPro software using a Dactron system. The excitation for this single plate test was a sine chirp with 5V amplitude with a 90 percent period over a frequency range of 100 Hz to 20 kHz.

Figure 9: PZT sensor configuration for single plate test

2.3

Results of Proof-of-Concept Testing From these preliminary tests, a basis for further testing and FE modeling was established. As a result of these tests, the rubber tubing set-up as described in section 2.2.2 was determined to be the best configuration for testing the full cube. PZT sensor locations used in these tests proved that modes shapes and frequencies for the entire cube and single plate could be excited and were comparable to those measured during the modal tests.

328

3

Finite Element Modelling

To help characterize the cube structure, a number of Finite Element (FE) models were constructed using the commercial software package Abaqus. These models were used to predict the dynamic response of the structure and to find the optimal PZT actuator/sensor location. 3.1 Beam and Shell (Full Cube) The first model that was constructed used beam elements to model the 80/20 extruded aluminum sections and shell elements for the shear plates. The bolted interfaces were represented using tied contacts between nodes on the beam elements to nodes on the shell plate. The aluminum angle brackets were assumed to not significantly contribute to the dynamic response of the structure and so were ignored. The results of this model showed multiple close modes in three separate frequency ‘clusters’. With the limited modal hammer results from the test on three sides of one cube structure it was difficult to validate the FE model to confirm accurate representation of the structures. 3.2 Beam and Shell (Single Frame) To better understand the results from the FE model compared to experimental data, it was decided to test and model a single side of the cube (as discussed in section 2.2). This model also used beam elements to represent the 80/20 extruded sections. To create the beam sections it was necessary to input certain parameters into the Abaqus program to define the geometry of the beam, such as cross-sectional area and area moments of inertia. The warping constant and sectorial moment were also required to calculate the response of the beam to torsion. These constants were not available from the manufacturer so were initially assumed to be insignificant. The results from the model did not correlate to the experimental results well enough to give confidence in the predictive ability of the model. The results indicated that failing to include warping constants in the beam geometry definition resulted in a model that could not accurately predict the dynamic response of the structure. 3.3 Full Shell Model (Single Frame) In order to fully capture the torsional properties of the 80/20 beam sections, it was decided to model them using thick shell elements. This is shown (green) in Figure 10 compared to the exact cross-section (grey).

329

Figure 10: Shell element approximation of 80/20 aluminum (left) compared to exact cross section (right)

Whilst the cross-section shows the shell element model to be a very rough approximation of the geometry of the 80/20, it gives a very good estimation of the natural frequencies of the 80/20 beam in isolation (described in section 3.3.1) and when integrated into a full model correctly describes the torsional response of the beam. In addition to modelling the 80/20 extruded sections using shell elements, the brackets that connect the beam sections together were also modelled using shell elements. This allowed for a more realistic bolting constraint between extruded beam sections, again enabling better dynamic response prediction. Figure 21 shows the shell elements used to create the single frame, to which a shear plate (also modelled using shell elements) was attached using small regions of tied nodes to represent bolts.

Figure 21: Shell element model of 80/20 single panel frame structure

330

This model correlated well with the experimental data, with natural frequency estimations within 3% of experimental results (shown in Table 1) and modes shapes that were visually identical. At the time it was not possible to quantify the quality of the modes shapes using the Modal Assurance Criterion (MAC). Table 1: Comparison of natural frequencies between FE model and Experimental results for single panel

Mode # Experiment 1 (Hz)

Experiment 2 (Hz)

Experimental Difference %

FE Results (Hz) FE to Experiment 1 Difference (%)

FE to Experiment 2 Difference (%)

1

121

118

2.54

119

1.68

0.8

2

325

322

0.93

331

1.81

2.7

3

615

603

1.99

622

1.13

3.1

4

618

610

1.31

626

1.28

2.6

5

974

968

0.62

996

2.21

2.8

6

1050

1040

0.96

1019

3.04

2.1

3.3.1.

Shell Element Model Verification

To ensure that the shell element models were accurate physical representations, a number of verification studies were performed. First the model of the 80/20 was modified to include cantilever beam boundary conditions (zero rotations and displacements at nodes on one end). The model was then run through a natural frequency analysis and compared to theoretical results (Avitabile, 2006).

Figure 32: First three mode shapes of cantilever beam 80/20 section

Figure 32 shows the cantilever beam in its initial state and then the first three mode shapes. Table 2 details the natural frequencies of the FE model compared to theoretical values from simple beam theory. The first two natural frequencies were estimated to within a few percent of the theoretical value which was sufficiently accurate for the model. The third natural frequency shows a large error compared to the theoretical result, indicating that the shell model requires adjustment. The large error occurs at a frequency out of the range of interest and so was assumed not to significantly affect the quality of the model.

331 Table 2: Cantilever beam verification of shell element representation

Mode #

Theoretical Value (Hz)

Finite Element Result (Hz)

Error (%)

1

195.64

206.45

5.23

2

1226.09

1193

-2.77

3

3433.12

2983

-15.09

A similar natural frequency study was performed on the shear plate and found to match theoretical predictions to within 1%. 3.4 Full Shell Model (Full Cube) After verifying and validating the FE model of the single plate attached to 80/20 frame, a model of the full cube was developed in Abaqus. The model used multiple instances of the single frame parts to create the full cube and included the same boundary and contact conditions. The 80/20 frame model for the full cube is shown in Figure 13. Section 4.3 details the comparison between the results from this model and experimental data.

Figure 43: Abaqus model of 80/20 extruded Aluminum frame modelled using shell elements

3.5 Piezoelectric Patch Placement Calculation With a full FE model of the cube structure, it was possible to perform calculations to determine preferred locations for the PZT patch actuators/sensors. As the PZT patches work using the principal of straining the surface they are bonded to, it was decided to examine the strain mode shapes of the structure. A crude estimation of an optimal actuator/sensor location is where there is a maximum amount of strain energy over the modes of interest. To accomplish this, the strain mode shapes were computed in Abaqus using the solution from the mass-normalised displacement mode shapes.

332

These in-plane strain mode shapes were then linearly averaged over the modes of interest to give an indication of strain distribution over multiple modes, shown in Figure 54.

Figure 54: Strain mode shapes averaged to indicate optimal piezoelectric sensor/actuator placement (red indicates areas of maximum strain)

Using this information, combined with engineering intuition, preferred actuator/sensor locations were determined and later used for PZT patch testing.

4

Unit Variability

4.1 Full Cubes Three nominally identical cubes were used as test structures for this research. Each cube was characterized using a modal hammer test, and the first six modes were evaluated. These results were then compared with a cross MAC using LMS Polymax software. Unit-to-unit variability testing was also performed on a single plate from one of the cubes. 4.1.1.

Set-up and Procedure for Full Cube Modal Test

The complete set-up used in each full cube test of the unit-to-unit variability testing is shown in Figure 65 below.

333

Figure 65: Set-up of full cube test for unit-to-unit variability

The same grid as described in section 2.1.1 was drawn on all six sides of each of the three cubes for a total of 96 measurement points per cube. Three accelerometers were used to record the measurements from these three tests. One accelerometer was placed on each of the three sides tested. The accelerometers were placed on plates 2, 4, and 6. The free-free boundary condition for this testing was created by hanging each cube using rubber tubing as described in section 2.2.2. During data acquisition, the Nyquist frequency was set to 3500 Hz, 4096 samples were taken, and 5 impacts were averaged at each point. A National Instruments PXI data acquisition system was used to collect the data. The plastic tip of the modal hammer was not covered to ensure that all modes of the cube could be excited. 4.1.2.

Unit-to-Unit Comparison for Full Cube

The cross MAC’s for the six mode shapes show low variation from cube to cube. Modes 3, 4, and 5 were harder to differentiate because of their close proximity in frequency. 6 show the cross MAC’s between each of the cubes. Table 3 shows the natural frequency figures and percentage errors from the variability tests. Table 3: Unit variability data for full cubes Mode #

Cube 1 (Hz)

Cube 2 (Hz)

Cube 3 (Hz)

Mean

Max % Error

1

269

267

268

268

0.37

2

275

278

276

276

0.60

3

304

305

306

305

0.33

4

314

311

311

312

0.64

5

338

340

337

338

0.49

6

551

558

555

555

0.66

334

Figure 76: Cross-MAC plots between different cube structures

4.2 Single Panel A single panel of one cube was also used in unit-to-unit variability testing. In both tests, three accelerometers were used to measure the data; however, the configuration is different between the two tests. The set-up for these tests was as described in section 2.1.2. Accelerometers were placed two different, evenly spaced, configurations. Little variation between frequencies and mode shapes measured during the two tests was observed. 4.3 Full Cube FE Validation The results from the unit variability data were compared to the mode shapes generated by the FE model, shown using MAC plots in Figure 87. The quality of the MAC plots is generally good, with 50% correlation between the 4th and 5th modes. This is due to the 4th and 5th modes being double modes (at the same frequency, with a phase shift) which have almost identical mode shapes.

Figure 87: MAC plots for FE to Cube 1, 2 & 3 respectively

The good match between the FE data and the experimental data further confirms that the FE model can accurately describe the dynamic response of the structure. The model can be utilised to extrapolate full mode shapes from a limited test data set and to also validate a subset of experimental data.

335

5

Full Cube PZT Testing

5.1 Low Frequency Baseline In order to determine the mode shapes of the cube test structure, three 1 cm diameter circular piezoceramic patches were placed on each face. The location of the patches was determined using information from the strain mode shapes as discussed in section 3.5 The cube was placed in the same hanging configuration as described in section 2.2.2. During this low frequency testing, the cube was considered to be healthy. A healthy cube is defined as all bolts each face of the cube tightened to a torque of 40 in-lbs. The excitation was a sine chirp input with amplitude of 5V and was applied for 90 percent of the sampling period over a frequency range of 100 to 1200 Hz. The excitation was applied using one of the PZT patches on plate 6, and the response was measure using all three PZT patches on each face. The configuration of the three patches on each face of the cube is shown below in Figure 9.

Figure 9: PZT sensor locations on three sides of full cube

5.2 High Frequency Baseline for SHM The high frequency baseline testing was performed on a healthy cube with the same hanging configuration as used in the low frequency baseline testing. The excitation was applied over a frequency range of 100 Hz to 20 kHz while all other excitation characteristics remanded the same as the previous baseline test. The excitation was applied using the PZT patch located at point 26 on plate 4. Measurements were taken at points on plates 3, 5, and 6. 5.3 Loose Bolt Detection Testing All of the test settings and configuration described in the high frequency baseline testing were used in the loose bolt detection testing. A loose bolt for this testing was defined as loose as possible without falling out. Each loose bolt was able to rattle around without coming out of the nut. Each run of the loose bolt detection testing consisted of only one loose bolt on the cube. The loose bolts were located next to the following locations: interface between plate 4 and 6 on plate 6, interface between 4 and 6 on plate 4, interface between 4 and 5 on plate 4, interface between 3 and 5 on plate 5, interface between 3 and 4 on plate 3.

336

5.4 Correlation Analysis After high frequency testing data was collected, SHM was performed via high frequency correlation in MATLAB. The imaginary portion of the FRF data over a band from approximately 10-20 kHz was selected because it has been shown that high frequency modes are more sensitive to local structural changes (Park, Rutherford, Wait, & Nadler, 2005). At this high frequency range, a correlation coefficient between the current state and the baseline state was calculated for each of three transfer functions for each damage condition. A damage index equal to 1 minus the correlation coefficient was then used to compare the changes in the transfer functions; the relative magnitudes of the damage index were then used for damage localization. The results of testing for eight damaged states and two baseline states are shown in the Table 4. Table 4: Results of bolt loosening detection tests and correlation analysis

Damage Index of Transfer Functions vs. Baseline State (Test 1) Plate 4 to Plate Plate 4 to Plate Plate 4 to Plate Test State Damaged Plate 6 3 5 1 Healthy None 0.000 0.000 0.000 2 Healthy None 0.001 0.001 0.001 3 L40 6 0.472 0.174 0.114 4 L24 4 0.507 0.296 0.195 5 L25 4 0.503 0.405 0.700 6 L17* 4 0.428 0.320 0.366 7 L76 5 0.485 0.327 0.463 8 L73 5 0.312 0.353 0.343 9 L64* 3 0.298 0.291 0.166 10 L57 3 0.276 0.455 0.197 *Denotes the bolt is in a corner of the plate. Repeated baseline tests show very little change in the damage index, so any bolt damage can be readily shown to impact the damage index well outside the realm of natural variation. In general, the transfer functions which cross the interface along the damaged bolt line show the largest magnitude of damage. For example, for Test 3 with the loose bolt on plate 6 along the interface between plates 4 and 6, the first transfer function crosses that interface and shows the largest damage index, shown in Figure 10.

337

Figure 10: Diagram showing loose bolt in L40 location

The second transfer function shows the next largest index since at least one path of the vibration crosses the damaged interface; meanwhile, the third transfer function shows the smallest indication of damage since its energy path is largely unchanged. In addition, the largest magnitudes of the damage index are typically observed when the damage is near the point where excitation is applied. Thus, when comparing two bolts on the same interface (ie: Test 3 and Test 4), the larger indications of damage occur when the damaged bolt is on the same face as the excitation (as in Test 4). The corner bolts are a bit more difficult to distinguish since they are along two interfaces, and the stiffness of the frame reduces the impact of the bolt stiffness. Thus, tests in multiple directions would be required to localize this damage. However, by taking advantage of these observations and the symmetry of the cube, the damage could be fully isolated to one interface line on a particular plate. To do so, at least one PZT on each plate is needed so that transfer functions can be calculated across each interface of the cube.

6

Conclusions

SHM techniques are being explored as an alternative to traditional vibration testing for the qualification of satellite components. The evidence provided in the paper shows that PZT devices are a versatile tool that can be used to accomplish this goal. A robust sensing system used to achieve multiple objectives within the lifecycle of the satellite has been developed as a result of this research. While each element of the methodology used in this paper is unremarkable in isolation, together they serve to greatly reduce the time period needed to qualify satellites. The results of this paper indicate that a small about of input energy is sufficient to excite low frequency global modes of the system which previously required traditional modal hammer testing. In order to effectively identify the interface along which a bolt is loose or other areas of potential damage, sensors will need to placed on each face of the satellite to ensure that a transfer function across the area in question can be created and

338

compared with the healthy baseline transfer function. If this system is further developed, not only the interface but also the exact location of a loose bolt or area of damage could be detected.

7

Acknowledgements

Funding for this research was provided by the Department of Energy through the Los Alamos National Laboratory. The following companies generously provided various software packages to aid in modelling and data analysis: Vibrant Technologies, SIMULIA, and The Mathworks. The authors would also like to give special thanks to Pete Avitable for his guidance in modal analysis and data acquisition, D.J. Luscher for his advice on finite element modelling, Nick Lieven for his Matlab hints, and Ramón Alejandro Silva for his help with piezoelectric sensors, and Derek Doyle at the Air Force Research Lab for his insights regarding the challenges facing responsive space. Finally, the authors would like to thank Dr. Charles Farrar and the Los Alamos Dynamics Summer School for the opportunity to conduct this research.

8

References

Avitabile, P. (2006). Cantilever Beam Experiment. Retrieved 2010 йил 8-August from University of Massachusetts Lowell: http://faculty.uml.edu/pavitabile/22.403/web_downloads/Final_Project_Cantilever_101806.pdf LANL. (2008 йил November). Cube satellite being prepared for launch. Retrieved 2010 йил 8August from Currents - Employee Monthly Magazine: http://www.lanl.gov/news/currents/2008/nov/cube_sats.shtml Park, G., Rutherford, A., Wait, J., & Nadler, B. (2005). High-Frequency Response Functions for Composite Plate Monitoring with Ultrasonic Validation. AIAA Journal , 43 (11), 2431-2437. PiezoSystems. (2008). Introduction to Piezoelectric Transducers. Retrieved 2010 йил 8August from Piezo Systems: http://www.piezo.com/tech2intropiezotrans.html Voss, D., Coombs, J., Fritz, T., & Dailey, J. (2009). A novel spacecraft standard for a modular nanosatellite bus in an operationally responsive space environment. 7th Responsive Space Conference (pp. 1-11). Los Angeles: AIAA. Yachbes, I., Roopnarine, Sadick, S., Arrit, B., & Gardenier, H. (2008). Rapid Assembly of Spacecraft Structures for Operationally Responsive Space. 6th Responsive Space Conference (pp. 1-10). Los Angeles: AIAA.

An Inertially Referenced Non-contact Sensor for Ground Vibration Tests Authors: B Allen, Moog CSA Engineering, C Harris and D Lange, Edwards Flight Test Center

  This document has been approved for public release by the USAF with unlimited distribution,  AFFTC‐PA‐10543 

Abstract: A sensor is introduced that enables new methods for modal testing. Noncontact inertiareferenced velocity (NIRV) sensors may be mounted to a flexible stand without concern for measurement contamination due to motion of the stand. Non-contact sensors that are stand mounted may be setup without the test article, saving large amounts of the test article’s time, particularly for high-value articles and repetitive sequences on similar layouts. In cases where Computer Aided Design models of the test article are available, dimensions of the stand and measurement degrees-of-freedom may be identified without taking measurements directly from the test article. Specifications of the sensor are presented along with results comparing its output to that of traditional accelerometer data and examining differences in reduction methods. Data is presented that demonstrates insensitivity to motion at the NIRV sensor mount. Ground Vibration Test (GVT) results are presented from an aircraft with both NIRV and accelerometer data to demonstrate the interchangeability.

T. Proulx (ed.), Advanced Aerospace Applications, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 4, DOI 10.1007/978-1-4419-9302-1_28, © The Society for Experimental Mechanics, Inc. 2011

339

340

Figure 1 – A GVT on an aircraft with NIRV sensors for response DOFs

341

Introduction Accelerometers are a common response sensor for modal analysis. While their inertiareferenced response is practical for purposes of data reduction and interpretation, their installation can be cumbersome and can require large down times for test articles. This paper introduces a new non-contact sensor that can reduce setup times, minimizing down times for the test article and generating large savings for articles with limited availability. Moog CSA Engineering developed the Non-contact Inertially Referenced Velocity (NIRV) sensor as a part of a research contract with the United States Air Force (USAF.) Efforts were funded out of the Flight Test Center at Edwards Air Force Base. The primary objective was to develop a non-contact sensor that could reduce down times to aircraft during Ground Vibration Tests (GVTs). Moog CSA has a patent pending on the NIRV sensor technology. A NIRV sensor measures inertially referenced velocity and has been designed for modal analysis and vibration measurement. A system of 80 NIRV sensors has been delivered to the USAF to acquire response degrees-of-freedom (DOFs) during GVTs on aircraft at Edwards Air Force Base. Two full-scale ranges of sensors were delivered. NIRV 157 sensors have a stand-off distance of 5.9 inches and a peak-to-peak stroke range of 3.2 inches, and NIRV 87 have a stand off of 3.2 inches and operate across a range of 1.2 inches. This paper demonstrates their performance with a test results on the sensor component as well as with a GVT that was performed at Edwards Air Force Flight Test Center. NIRV sensors are compared to data from PCB type 333B32 accelerometers. Performance demonstrated in this paper refers to sensors generated for the USAF and may or may not indicate a limitation of the technology. Efforts are continuing to develop sensors that are fully integrated by a single manufacturer with minimum packaging size and cost.

The Sensor The NIRV sensor is a non-contact velocity sensor that has been designed specifically for applications in modal analysis. Figure 1 is a photograph of a GVT using NIRV sensors, and Figure 2 shows a NIRV sensor and its stand from close up. The sensor indicates motion using a class II laser, and is only sensitive in the direction of the emitted beam. Motion of the stand to which it mounts is rejected, avoiding risks that readings will be biased by stand vibration. Sensor ranges for the USAF system are between 1.2 and 3.2 inches maximum stroke, and they output analog signals which are between -10 and 10 volts and are proportional to velocity. Common modal analysis software can acquire and reduce data from NIRV sensors with the same processes that are typical to accelerometers. Because there is no appreciable phase lag or delay in the NIRV sensor output, they may be intermixed with other analog sensors. LMS taught a GVT master class at Edwards Air Force Base using NIRV sensors. The class demonstrated random, swept sine, stepped sine, and sine dwell excitation each using feedback from the NIRV sensor where required.

342

Figure 2 - A NIRV sensor acquiring data on an aircraft The NIRV sensor produces an inertially referenced velocity signal using a combination of signals from two conventional sensors and a special combining circuit. An accelerometer is integrated with a non-contact laser displacement sensor and is used to correct for motion of the laser sensor. The laser sensor used in the USAF system is a Keyence LK-G displacement sensor. It measures displacement using a laser and triangulation principles. It has excellent dynamic range, a relatively long stand-off distance, superior calibration stability with time, and is insensitive to changes in color and texture of targets because of an adaptive circuit that increases the laser intensity on surfaces yielding lower light return. Its performance is unaffected by errors in angle of incidence to greater than 20 degrees, and by surface roughness and reflectivity changes that occur on the aircraft and most other structures. NIRV sensors output velocity so modal software will store Frequency Response Functions (FRFs) in the form of mobility. Mobility data generated by the NIRV sensor reflects identical structural dynamic behavior as that of an accelerometer, and is merely rotated 90 degrees in the complex plane relative to the more common Inertance function. To demonstrate this point, several locations on the left wing were instrumented with both accelerometers and lasers. Figure 3 shows an overlay of the FRFs from each, where the mobility has been multiplied by (jω). Results are identical with the exception of low frequencies, where the noise floor of the NIRV sensor is lower than the accelerometer.

343 FRF Magnitude from Left Wing Tip in Vertical Direction, LWG:6F

-3

10

-4

10

-5

G's/lbf

10

-6

10

-7

10

FRF from accel FRF from NIRV sensor -8

10

0

2

4

6

8 10 12 Frequency (Hz)

14

16

18

20

Figure 3 – Overlay of Mobility with Collocated Accelerometer Data

Noise properties of NIRV sensors are defined dominantly by the non-contact laser sensor output. Because the sensor for this application is a displacement sensor, its noise properties outperform accelerometers at low frequency, and fall behind accelerometers at high frequencies. The cross occurs at different frequencies depending on the NIRV and accelerometer. Figure 4 is a overlay of noise floors between an LKG-32 sensor and PCB 333B32 accelerometer, compared in the form of acceleration.

344

-6

Noise overlay between PCB333B32 accel and LKG32 displacement sensor

10

-8

PSD of Acceleration (G2/Hz)

10

-10

10

-12

10

-14

10

PCB333B32 LKG32

-16

10

0

10

20

30

40 50 60 Frequency (Hz)

70

80

90

100

Figure 4 – Overlay of Noise Floors from a LKG-32 and an accelerometer

Calibration stability of the NIRV sensor is excellent due to its construction. It is inherently a digital device that refers to dimensions of a CCD matrix for its calibration. In contrast, piezo-electric accelerometers rely on the charge of a piezo crystal, a quantity that decays as a function of time. NIRV sensors are insensitive to motion of their stand. Moog/CSA documents their capability to reject motion with a quantity called rejection ratio. It examines sensor output when vibrating and sighting on a stationary target. Each sensor manufactured by Moog/CSA is characterized for its rejection ratio, and a plot of typical performance for the sensors delivered to the USAF is shown in Figure 5. These units targeted high rejection between 1 and 100 Hz. A ratio of 40 dB indicates 100 to 1 attenuation of sensor motion.

345

Figure 5 - Rejection ratio of a NIRV sensor

Testing with NIRV Sensors Insensitivity of the NIRV sensor simplifies requirements for a stand. Requirements are only to locate the NIRV within the operational range of the displacement sensor. Moog/CSA has a standard set of stand framing and joints that are appropriate for test article sizes between 5 and 100+ feet. Two-inch diameter aluminum tubing is the primary structure. Sensors are mounted off of the primary assembly with a 1-inch tubing and a standardized articulating assembly. Figure 6 shows a frame generated for an aircraft ground vibration test (GVT). Larger frame segments roll on casters until final positioning. Casters are replaced with fixed feet in a single operation that requires no modifications or additional parts. Articulating mounts for sensors can accommodate any orientation of tubing and function adequately with distances between 0.5 and 3 feet between the stand and test article. Swaying motions of the stand are rejected by NIRV sensors.

346

Figure 6 - Frame supporting an aircraft GVT

Sensors were typically mounted off of the stand, yet remote locations were sometimes mounted with a single tripod. Biaxial and triaxial measurements are common in modal analysis and in some cases surfaces of the target structure did not support each required direction. An example of a similar biaxial measurement is shown in Figure 7. Angle sections of aluminum were attached to the test article with hot glue in order to generate a suitable laser target.

347

Figure 7 – Biaxial measurement using an angle aluminum target

Results from NIRV Sensors Several tests have demonstrated performance of the NIRV sensor, yet the largest test was performed at Edwards Air Force Base, and results of this test are documented within this paper. This test was also the subject of a GVT Master Class taught by LMS on site. As is common with GVTs for flutter studies, FRFs are reprocessed into modal parameters which are then compared to analytical predictions. Table 1 presents modal parameters that were reduced from measurements using random excitation that was applied near the wing tips at a zero to peak levels of around 20 lbs. Rigid body modes are not shown. Table 1 – Parameters from lowest 6 modes with strain energy in aircraft body

Aircraft GVT Mode Table Mode #

Frequency (Hz)

Damping % Critical

Mode Description

1

7.886

0.4

Wing Bending, symmetric

2

10.65

0.7

Wing Bending, anti-symmetric

3

11.709

1.3

Fuselage Bending

4

12.716

0.5

Empennage Mode

5

13.467

0.4

Wing Torsion, symmetric

6

13.885

0.5

Wing Torsion, anti-symmetric

348

Mode shapes were also reduced from the same test sequence. They are overlaid with an undeformed wireframe in Figure 8. Steps for data reduction were unchanged from that with accelerometers. Today’s multiple reference curve fitters require no changes in operations between mobility and inertance frequency response functions.

Figure 8 – Deformation Shapes

349

Conclusions This paper presents a practical method to reduce down times for test articles using a noncontact laser sensor. The sensor and stand construction techniques make non-contact modal analysis both practical and accurate. Applications on structures of similar shape will save even larger amounts of setup, since no tear down is required between tests with non-contact sensors.

Reliability of Experimental Modal Data Determined on Large Spaceflight Structures Anton Grillenbeck, Stephan Dillinger IABG mbH, Einsteinstrasse 20, 85521 Ottobrunn, Germany

[email protected], [email protected] Abstract Launcher components, large payloads – and particularly in the past – also all major Space Shuttle payloads have required modal survey tests to determine the modal parameters with high accuracy and reliability to validate the respective dynamic structural models for coupled loads analysis and flight piloting, but also for system level structural qualification. The modal survey test methods applied usually consist of the classical ground vibration test using tuned sine excitation and the modal analysis of measured frequency response functions. While the tuned sine excitation directly results in well understood and established modal parameters – typically of the fundamental or primary modes, the modal analysis is used to complement the modal model with respect to the secondary modes. This two-folded approach serves both for high accuracy of the modal model considering the fundamental modes as well as to minimize test effort to complement the modal model in the higher frequency bands. Almost all available modal analysis tools – besides of very specific ones – which are available today are based on linear models and on a more or less statistical approach to estimate the modal parameters. With respect to real measured transfer functions, in particular when modal density increases and some non-linearity is present, which is typical of large space structures, the modal analysis results inherently suffer from a considerable amount of scatter or uncertainty. In this paper, application cases are studied to provide insight in the actual challenges to extract reliable modal parameters from large spaceflight components. Based on this, a consolidated strategy is discussed to extract the best match between experimental data and experimental modal model. In addition, the remaining uncertainty may be quantified as well.

Introduction For a period longer than two decades, in Europe, modal testing has been performed regularly on ARIANE main launcher components, involving even the almost completed launcher on its launch pad, as well as on all European payloads for the Space Shuttle and on heavy payloads of ARIANE like the Automated Transfer Vehicle (ATV). Besides of the almost unchanged ground vibration technology during these years, the modal analysis of frequency response data has seen considerable change mainly driven by the increasing computational power, the refined modal analysis theories and concepts, as well as the more and more sophisticated software tools. This evolution went hand in hand with the higher complexity of the structures to be modally tested and analyzed. At a first glance, today’s most advanced modal analysis tools seem to offer a simple platform to easily extract modal parameters in even complex situations which are characterized by high modal density and some light effects of non-linearity. Indeed, the effectiveness of these tools for extracting the modal parameters of clear and well excited modes, also considering the benefit of multi-point excitation techniques, is convincing and without doubt. In reality, when dealing with complex structures like a launcher upper stage in the mid frequency range, let’s say between 50 and 150 Hz, the situation still is not as simple. There are several reasons for this complexity, in particular on upper stages and heavy payloads. The most important reasons originating from the structure point of view are – but not exhaustively:  These structures are very complex: there are many large and small functional parts connected to each other, huge feed lines and smaller functional lines, a number of more or less identical parts of similar attachment stiffness and mass, actuators and joints, specific dampers and many other dynamically relevant features …  The modal density may become very high: up to 200 predicted modes in the frequency range of interest is considered a typical number.

T. Proulx (ed.), Advanced Aerospace Applications, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 4, DOI 10.1007/978-1-4419-9302-1_29, © The Society for Experimental Mechanics, Inc. 2011

351

352 

The modal mass and stiffness of the individual modes varies considerably such that global and local modes are mixed up in a random sequence and the damping of these individual modes may vary considerably.  Similar or repeated structures result in bundles of similar modes in narrow frequency bands.  Modes with higher response magnitude tend to obscure modes of less response magnitude making their identification quite difficult.  Specific damping mechanisms, flexible joints, bellows and long lines, etc. introduce some level of non-linearity.  Filled or empty tanks and their pressurization add further complexity to this picture due to their mass singularity and the production of numerous tank shell modes. The latter point has one important implication on the accurate experimental modal analysis of upper stages since the decreasing fuel mass or varying pressurization during flight is significantly impacting the modal parameters. In consequence, the tracking of individual modes or even bundles of similar modes during a modal test on a stage with several tank fill levels and pressurization conditions adds an additional challenge to the precision of the experimental modal analysis. Besides of this, some reasons for the modal analysis complexity originating from the test set-up may add:  A limited number of sensors, respectively an insufficiently derived measurement plan are inhibiting modes from direct observation, an effect which is often experienced in conjunction with local modes.  The limited access for a sufficient excitation: At high modal density, various excitation points for a multi-point excitation may be desirable, however in reality, the accessible points with adequate stiffness and suitable dynamic loading path are rare. Often only one exciter is used because it best represents the excitation of the stage engine and heavy masses inside of the structure cannot be excited directly. It has to be pointed out that this complexity is less relevant, if negligible at all, when considering the main modes of such structures which are relevant for flight piloting and coupled loads analysis. To this respect both the phase resonance and the phase separation test approach result in well understood modes as it will be shown later. However, all secondary modes which in particular are relevant for the functioning and structural integrity of the structure under test are more or less affected. Accordingly, the modal analysis on these secondary modes will be specifically addressed in this paper focusing on the question how to extract the optimal and best substantiated modal parameters which best represent the measured frequency response data. In literature, the modal analysis process itself is well reflected considering the use of the various extraction methods in praxis, e.g. [1], and the general understanding of the modal analysis tools and their relation to each other has been clarified in the unified matrix polynomial approach (UMPA) [2]. In parallel, the use of stabilization and of consistency diagrams as a tool to easily identify the physical roots and to estimate the uncertainty of the extracted roots attained considerable maturity and consent, e.g. [3], [4]. In [5] and [6] use is made of additional information to extract besides of the roots also the best and most consistent estimate of the residues, modal participation factors and finally the mode shapes. These new perspectives and concepts reflect very well the needs and the trends for the extraction of modal parameters in the complex scenarios described above, and are considered essential for the consistent modal analysis in such cases. The authors were confronted with the complex modal analysis scenario long before these concepts were available, not to mention that any commercial code was available to support the practical modal analysis work to this respect. Therefore, in a retrospective, the concepts used to cope with the complex modal analysis scenario in the practical application are revisited and discussed promoting the development of useful automatic modal analysis code. In this paper, the typical accuracy requirements to experimental modal parameters are summarized in section 1, and in section 2, typical conditions of complex modal analysis scenarios are described. In section 3 the application of state-of-the-art modal analysis tools is reviewed. In section 4 modal quality indicators for optimizing the selection of the most suitable residues are recalled such that in section 5, an alternative, but universal concept of identifying the optimal modal parameters can be presented.

1. Normative Accuracy Requirements to Experimentally Determined Modal Parameters With scope on modal characterization of space vehicles, two documents, namely the MIL-HDBK 340A [7] and the ECSSE-ST-32-11C [8] provide some details on the requirements to the accuracy of experimentally determined modal parameters. Without limiting the definition of specifically agreed requirements for a modal survey test, the general requirements of the ECSS to experimentally determine modal parameters may be summarized as follows:  Frequency accuracy  0.5 %

353  Mode shape  5 %  Mode indicator value for global modes > 0.9 (using Breitbach’s definition of the mode indicator value)  Auto-orthogonality < 0.1 in the off-diagonal elements In contrast to this, the MIL-HDBK does not provide requirements to the experimentally determined modal parameters, but requirements are given with respect to the correlation between the experimental data and the modal model to be validated and updated. These requirements are almost in line with those given in the ECSS:  Frequency correlation of primary modes (more than 10% effective mass) < 3%  Cross-MAC of primary modes > 0.9  Cross-orthogonality of primary modes > 0.95  Off-diagonal elements of the cross-MAC and cross-orthogonality < 0.1 For secondary modes (less than 10% effective mass), the ECSS relaxes the frequency correlation to < 5% and the autoMAC to > 0.85, while the MIL-HDBK in general allows exceptions when physical reasons can be stated. All these definitions in general assume a linear, time invariant structure under test, viscous and almost proportionally damped eigenmodes and the absence of gyroscopic effects. Interestingly, both standards do not provide accuracy requirements to modal damping nor on modal and effective mass. But obviously, in both standards, the orthogonality of extracted mode shapes as well as their modal purity and a good normal mode approximation is considered an important ingredient in assessing the experimental modal results. Besides of this, a de facto standard has emerged in the practical execution of modal survey tests as they have been performed in the past by the authors. Based on customer requirements and on specific analysis of the error budget of a modal survey test for the phase resonance method and the phase separation method, the following accuracies are targeted: Tab. 1 Targeted modal parameter accuracy during a modal survey test

Uncertainty of … or Quantity Eigenfrequency Modal (Viscous) Damping Max. Mode Shape Element Modal Mass Auto-MAC off-diagonal Auto-Orthogonality off-diagonal

Phase Resonance (primary target modes) < 0.1 % < 10 % <3% < 15 % < 0.1 < 0.1

Phase Separation (secondary target modes) < 0.2 % < 15 % <5% < 25 % < 0.1 < 0.1

The primary target modes considered in Tab. 1 are the modes with more than 10% effective mass and selected secondary modes of particular interest. In fact, the uncertainty on the modal mass is quite high due to the error propagation and the way the modal mass is determined, i.e. usually by applying the energy input method. Accordingly, the uncertainty of the modal damping and the uncertainty of the driving point response and of the maximum response point response contribute significantly to this uncertainty in modal mass. The consequence for the performance of a modal test, in particular also when performing a modal analysis, is that besides of using the stability or consistency diagrams, the extracted mode shapes have to be assessed in addition with respect to modal purity, modal complexity, residues and modal damping. Based on such an assessment, in many cases the actual uncertainty may be rated to be less than stated in Tab. 1. As a proof of evidence, the fit quality of narrow-band sweeps around tuned modes or of well isolated modes in a modal analysis can be used to attribute an accuracy class to each individual mode. From our practical experience, the following accuracy classes as

354 given in Tab. 2 considering the damping and the generalized mass have proven to be a useful classification. The application of this table will be discussed in the next section. Tab. 2 Accuracy rating of extracted modes

Accuracy Class

Uncertainty of the Uncertainty of the Viscous Damping Generalized Mass <2% < 10 % <5% < 12 % < 10 % < 15 % Less accurate than target given in Tab. 1

I II III IV

2. Typical Application Scenarios The test data of two modal test programs on large structural models are introduced which reflect a relatively straightforward and a more complicated modal test scenario.  STM A: 8 metric ton model of a Space Shuttle payload, frequency range of interest 10 – 60 Hz, about 40 predicted modes, structure test model being specifically configured for a modal test, i.e. linearity and separated modes are enforced in the STM design, up to 14 exciters and 620 vibration response measurements used  STM B: heavy payload structure model, frequency range of interest 10 – 100 Hz, more than 100 predicted modes, many similar components in a flight typical configuration, 4 exciters and up to 520 response measurements The modal behavior of both structural models is presented in Fig. 1 in terms of frequency response functions given without frequency scale, but well representing the situation for the modal extraction task. Frequency response function

Frequency response function

1.5

2.7E-03

1.0

2.0E-03

0.0

0.0E+00

-1.0

-2.0E-03 -2.7E-03 2.6E-03

-1.6 1.7

2.0E-03

1.0

1.0E-03 1.2E-10

0.0 -1.0E-03 -2.2E-03

-1.0 Frequency (Hz)

Frequency Hz

Fig. 1 (a) Frequency response functions of STM A (left) and (b) STM B (right) each based on four excitation points

STM A was specifically designed for the modal updating task. In order to ease this task, the structural model was designed and optimized to provide almost only well separated modes and it did not incorporate functional elements such as lines, cables or other equipment in order to reduce the amount of secondary modes. The achievement of this objective is well documented in its frequency response functions presented in Fig. 1(a), clearly showing separated individual modes in the overlay plot of all relevant frequency response functions. Also, it can be seen that the appearance of almost all modes is linear. In addition, the STM A enabled access for a considerable number of exciters to guarantee the excitation of any target mode of interest. Accordingly, the objective of the modal test was the identification of the complete modal model in the frequency range of interest and the phase resonance method was used extensively. In contrast to this, STM B was designed both for a modal characterization and a vibration qualification test campaign. In consequence, the structural model was designed to well represent the flight configuration comprising lines and cables, various storage and payload installations as well as the propulsion section. In consequence, the STM B well reflects almost all opera-

355 tional modes including the primary modes as well as many overlaid secondary modes. In consequence, the coupling of modes, some non-linear behavior and the considerable high modal density can be observed in the overlay plot of the frequency response functions, see Fig. 1(b). A particularly complicating issue for the modal test was the limited access to major structural parts of the STM B for direct excitation. Nevertheless, such a scenario is typical for large structure parts, and therefore can be considered as a representative example. Accordingly, the focus of the modal characterization is limited to the extraction of primary modes and of significant secondary modes, and the modal analysis based on phase separation methods plays a more important role.. But the modal parameter extraction task becomes more challenging, since a mixture of well and not well excited modes is encountered. In consequence much effort has to be put on assessing the quality of the modal results obtained. In either test case, the basic assumption for the determination of the modal parameters is that the structures are linear and time invariant. As the modal test has to provide experimental data for a correlation with the respective analytical dynamic model, it is the primary objective of a modal test to provide the best possible approximation of undamped modal frequencies and of normal modes.

3. Extracted Modal Parameter of Test Case STM A Since the baseline test approach for STM A was a classical ground vibration test, complete sets of both tuned mode measurements and frequency response data of the measurements for preliminary mode identification are available. Subjecting the frequency response data to a modal analysis, both data sets can be used for an assessment and comparison with respect to modal parameter results accuracy based on phase resonance and phase separation methods. After the identification of target modes and an estimation of the initial force pattern, the phase resonance method basically consists of the mode tuning and the performance of narrow-band sweeps around the isolated resonance. The mode tuning is balancing the damping forces of the structure and the measured frequency and the modal responses directly represent a very good approximation of the undamped modal frequency and of the normal mode. This information easily can be correlated with the respective analytical modal model. In consequence, a mode indicator value based on the phase relationship between excitation force (real) and the resonance responses (almost imaginary) is the driver for assessing the quality of the tuned modal frequency and measured modal vector. Since the mode indicator value is optimized during the mode tuning process, the measured modal parameters can be considered as the optimum experimental solution with respect to the available or chosen exciter configuration. In Fig. 2(a) the blue bars indicate the mode indicator values obtained by applying the phase resonance method on STM A. The mode indicator exceeds 0.95 for all primary modes, and for all other modes, the mode indicator is better than 0.9. Also, the cross-orthogonality check given in Fig. 2(b) already indicates a very good correlation with the (not yet updated) analytical model and almost fulfills the normative requirements stated in section 1. Besides of this, also the modal damping and the generalized mass, to be calculated from the injected energy compared to the damped modal response, need to be assessed with respect to accuracy. This easily can be done using the narrow-band sweep around the tuned modal resonances. Fig. 3 shows two examples of a narrow-band fit which yield a different grade of approximation by the linear model of a single DOF oscillator to the real structural behavior. The deviation from the ideal linear oscillator response can be interpreted as the uncertainty of the extracted modal damping and of the measured modal response. Using such reference fit results based on simulation, it is possible to attribute to each modal result an accuracy class as indicated in Tab. 2. Thus, a very conclusive picture on the accuracy and reliability of tuned modes is achieved and this provides a scale bar for assessing the results of a modal analysis on the same structure, but using the frequency response functions. For almost each modal extraction method used in practice today, the stabilization or consistency diagrams as depicted in Fig. 4(a) and 4(b) are used to find stable physical roots or poles. For each model order the correlation matrix is solved, the complex eigenvalue and shape vector variation from the one to the next model order is graphically displayed and repeated stable poles indicate the existence of a resonance frequency of the system investigated. Not well established stability lines

356 may indicate not well excited or non-linear physical roots as well, or simply may be computational roots without physical meaning, but accounting for noise or methodological shortcomings. However, the stability diagram is just one but incomplete tool for the selection of the optimum modal parameters for the description of the measured data. Cross-Orthogonality Check

Indicator Values Tuned Measurements vs. Modal Analysis 1.00 0.98

1,00 0,90

In d ic a t o r V a lu e

0.96

0,80 0,70

0.94

MIV-Tuned MIV-Analysis

0.92

0,60 0,50 0,40

0.90

0,30 0,20

0.88

0,10 0,00

0.86 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 Mode No.

1

Test

6

11

16

21

6

11

16

21

Analysis

1

Fig. 2 (a) Mode indicator value of STM A modes obtained from phase resonance (blue) and phase separation (red); (b) cross-orthogonality check of the phase resonance modes with the not yet updated analytical model (correlation better than 0.9 is indicated with red bars)

Fig. 3 Narrow-band fit simulation model to assess the uncertainty of a linear single degree-of freedom estimator (dashed line) approximating the real test data (solid line). (a) on the left hand side, the uncertainty in damping is less than 2 %, (b) on the right hand side it is less than 10%

The stabilization diagrams indicate that the estimated poles vary between user definable intervals, but they are too abstract for practical use or interpretation and in particular exact reasons why a certain mode is chosen amongst others cannot be answered satisfactorily. Consistency diagrams may provide better insight, for instance by graphing the modal damping as a function of the modal frequency, see Fig. 4(b). Here, areas of cumulating roots provide both information on the existence of physical roots, but also directly an indication of the actual variance or uncertainty of the taken choice. This approach also is very practical, since the median or centroid of such an accumulation of roots can be taken for extracting the experimental modal parameters and their uncertainty, respectively variance, out of the diagram. But the question remains which choice of roots shall be taken for the determination of the median or centroid and how this selection could be documented. These methods work pretty well in the case of STM A (see Fig. 4a and b). Nevertheless, when comparing the results of the modal analysis to the results of the modes tuning, see Fig. 2(a) and Fig. 5, some interesting consequences become evident. In terms of modal phase purity, i.e. the indicator value, both approaches yield almost the same result. Difference may be attributed to excitation levels (higher excitation level results in lower MIF), or not well excited or observable modes in the case

357 of performing frequency response measurements based on a subset of exciters. This is resulting in the fact, that modes 6, 7, 20 and 21 could not be extracted from the frequency response data with sufficient phase purity. Pole Surface Density

1.2

1

Zeta (%)

0.8

0.6

0.4

0.2

28

Density

Model Iteration

Consistency Diagram 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 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 15

20

25

30

35 Frequency (Hz)

40

45

50

20 15 10

0

5 1

55

15

20

25

30

35 Imaginary (Hz)

40

45

50

55

Fig. 4 (a) Stabilization diagram of the RFP-Z algorithm (left) and (b) consistency diagram of the RFP-Z algorithm (right) applied to STM A data set showing damping variation versus variation of the eigenfrequency in a consistency and density diagram Complex Mode Indicator Function

-6

Pole Surface Density

10

1.4

1.2

1 -7

10

Zeta (% )

Magnitude

0.8

0.6

-8

10

0.4

Density

47

-9

10

40

45

50

55

60

65 Frequency (Hz)

70

75

80

85

20 10

90

0.2

30

0

40

1

42

44

46 Imaginary (Hz)

48

50

52

54

Fig. 4 (c) CMIF diagram for the STM-B (left) and (d) consistency and density diagram of the PTD-Z algorithm (right) applied to STM B data set for the first third of the frequency range showing high modal density but still considerable mode stabilization Generalised Mass, Tuned Measurements vs. Modal Analysis

Viscous Damping Tuned Measurements vs. Modal Analysis 1.40

1.8 1.6

1.20

1.2 0.80

DAMP.-Tuned DAMP.-Analysis

0.60

G en . M ass

Visco us Dam pin g (% )

1.4 1.00

1 0.8 0.6

0.40

0.4 0.20

0.2 0

0.00 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

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

Mode No.

Mode No.

Fig. 5 (a) Modal damping of STM A modes obtained from phase resonance (blue) and phase separation (red), the first mode suffered from gapping which was blocked for mode tuning; (b) modal mass obtained from phase resonance (blue) and phase separation (red) compared to the predicted normal modal mass = 1

358 It may be noted, that the consistency diagram given in Fig. 4(b) shows some significant scatter in the damping of some modes. However, by carefully selecting the roots in the curve fitting process, the uncertainty in damping can be settled to some extent and modal damping values well comparable to the tuned results can be achieved. However, for some modes the difference in the damping estimate compared to the tuned modal result is considerable and exceeds 10 %. Regarding the modal mass, the differences between the tuned results and the modal analysis results vary dramatically. The most relevant reason for this is that both the driving point residue and the selected modal response for mode shape vector scaling have to be assessed in the curve fitting process. Often the modal response at the driving point is comparatively small such that the modal response at the driving point cannot be assessed accurately. This is due to the fact that the residue may appear as the difference of two numbers of similar magnitudes. Accordingly, good modal mass estimates only can be expected when a well-established or well isolated modal response at the driving point exists. Although the STM A example seems to have clear and well separated resonance frequencies, the phase separation methods, compared to mode tuning, obviously do not result in the same accurate extraction of the modal parameters. We may expect that in more complicated test cases like STM B, the situation may become even worse. Fig. 4(c) presents the modal density of STM-B, while Fig. 4(d) shows the density diagram of just the first third of the frequency band investigated. The high modal density resulting in significant modal response overlapping becomes evident. Although the consistency diagram here also helps selecting in the average good estimates for the poles, this is not so clear with respect to the mode shapes. Moreover, in order to perform a transparent and well substantiated extraction of modes by using phase separation methods, obviously a much more detailed selection process is required to ensure that the optimum choice of modal parameters and mode shapes is found which best represent the measured data. In the next sections, based on mode shape quality indicators, a structured modal parameter selection process is proposed.

4. Mode Quality Indicators In literature, a number of mode quality indicators – besides of mode indicator value and the orthogonality checks – have been proposed to better support the selection of modal parameters during the modal analysis process. The quality indicators to be named here are modal phase collinearity, mean phase deviation and mean phase, as well as the deviation in generalized mass estimate. Modal Phase Collinearity (MPC, [0,1]): This indicator checks the complexity of a mode by measuring the strength of the linear functional relationship between the real and imaginary parts of the mode shape. MPC values between 0.95 to 1.0 must be expected for global modes and lightly damped modes, otherwise, such modes should not be considered for real mode approximation and modal updating. For local modes - which usually show only for a part of the structure reasonable responses – MPC values down to 0.8 to 0.85 may be acceptable, but these modes always should be used with care. When modes are to be selected in the modal analysis process, the chosen mode shall have a maximum, ideally, however a value of 1. Mean Phase Deviation (MPD, [degree]): Based on the real and imaginary part of each mode shape element, the phase with respect to the excitation force may be calculated. Taking the average (or weighed average) over all response points, the mean phase of a mode is calculated. Based on this mean value, the scatter or deviation of all responses may be derived with respect to this mean phase. The average value of this scatter, the MPD, is an indicator for the degree an extracted complex mode shape is affected by non-phase contributions (like noise, unresolved local modes or poor excitation levels in some part of the structure). During the modal analysis process, the MPD is used to select the mode with the least phase scatter, and for the further comparison with analytical modes, this value also may be used to decide whether the mode shape may be useful for comparison or not. Global modes are expected to have a MPD below 30° (best below 15°), local modes will show up with a MPD between 20° and 40°. Mode shapes with higher MPD are in fact existing modes, however, their excitation and consequently their extraction quality is poor and they should be handled with care. Often, the mean phase deviation is displayed together with a mode phase diagram by displaying the real and imaginary part of measured responses in a polar diagram. Normal Mode Indicator Value (MIV, [0,1]): This indicator has the same meaning as the normal mode indicator function applied to frequency response functions, but applied here to a single frequency point. Hence it has the same meaning like the mode indicator value used in the mode tuning process: In the modal analysis, modes with a maximum mode indicator value shall be selected. It should be noted that for complex modes, the MIV depends on the modal phase at the reference point which may be well off 90° for local modes. To remedy this situation, it is a good practice to perform a complex scaling of the mode such that the mean phase of the mode will be close to 90°. Deviation of the generalized mass estimate (DGM, [%]): Using the complex residues, instead of the generalized mass, modal A is the theoretically correct property. However, since we are interested in normal mode approximation, also the gen-

359 eralized mass has to be approximated. Using the energy input method, the generalized mass for a mode can be calculated by using either the imaginary parts only or the magnitude of the respective modal responses. If both results are in a good agreement, it can be concluded that the general mass estimate also is well represented by the approximated normal mode. The use of these quality indicators during the modal analysis eases the selection of roots in the modal analysis process and it provides an additional means to justify the choice. This idea is incorporated in the proposed structured approach.

5. Structured Approach Over many years, the following structured approach has been applied in simple as well as in many complex modal analysis scenarios. In simple cases, this approach has helped to identify almost the same quality of modal results as phase resonance techniques would have provided. In more complex scenarios, an optimization of the accuracy of the modal results was achievable, in any case at least a reasonable assessment on the quality of the extracted modes. The idea for such a structured approach was driven by:  Extracting the optimum parameter estimate from a data base with respect to curve fit, mode quality indicators and mode linear independence or orthogonality  Expanding in a reasonable manner modal analysis to complex cases  Applying the most suitable modal analysis method  Ensuring almost the same quality level of a modal analysis, independent from the analyst  Standardizing and documenting the modal analysis process such that the result can be traced and reconstructed  Being less dependent from modal analysis methods or software  Performing benchmarks The structured approach then is implemented as follows:  All relevant frequency response functions are inspected for resonance frequencies and absence of defects that may need specific treatment (e.g. non-linearities) or hamper the analysis (e.g defective sensors); global assesment functions like MIF, CMIF, etc. are used to perform a first identification of frequencies.  Characterization of these modes either being of global or of local nature and determination of reasonably excited modes as target modes for the subsequent modal analysis; non-target modes are noted since they may become relevant when curve fit results on target modes have to be assessed.  The modal analysis is performed by applying a suitable analysis method and by compilation of the correlation matrix.  By inspecting the stability and error diagrams, the analysis parameters like frequency range, correlation matrix size and selection of responses to be included in the analysis diagrams is adapted such that optimum condition of stability is achieved for each of the identified target mode. It is advised –depending on the extraction method used – that the number of physical roots does not exceed 10 to 15 per analysis set.  Based on the stabilization diagram like in Fig. 4(a), all or a subset of root solutions are calculated and all mode shapes exceeding a certain threshold, e.g. the MCF, are calculated and stored together with the parameter tables. In this manner, a large table of parameters and an array of mode shapes is created which includes all relevant solutions of the correlation matrix.  Using all these parameters and mode shapes, the mode quality indicators are calculated and arranged in an assessment table as provided in Tab. 3. This table includes root number, modal frequency and damping, the mode indicator value (MIV), modal phase collinearity (MPC), mean phase (MPH) and means phase deviation (MPD), the mode indicator value after complex normalization by the mean phase (MIP), the location and magnitude of the maximum modal response (MAX), the generalized mass (GMass) as well as the deviation of the generalized mass estimate (DGM).

360 







Using this table, the mode parameter set with the optimum but preliminary parameters can be selected by iterating between MPC, MPD, MIP and DGM. Usually and fortunately, the optimal parameter set found for a resonance frequency yields certain stability in eigenfrequency and modal damping, as well as a minimum of the phase deviation (MPD) and a maximum of the MIP. Then this identification is checked against the measured frequency response functions and for linear independence of the eigenmodes. This work is relatively easy since all modal parameters and eigenmodes are available for these cross-checks and the preliminary parameters can be reconsidered to improve the curve fit and the linear independence or orthogonality. Also, suitable roots of non-target modes can be included to verify the fit quality of roots chosen to meet the previously identified target modes. Finally, this cross-checks end with a refined choice of modal parameters. The curve fit in particular is checked against the measured frequency response functions for the excitation point and the points of maximum modal response. In this manner the validity of the residues is proven and assures the accurate estimation of the generalized mass. The modal analysis process then is concluded by assembling the modal model after this process.

Tab. 3 Example of an assessment table on the stability of two modes Root

Mode

Freq [Hz]

DMP [%]

MIF [-]

MPC [-]

MPH [deg]

MPD [deg]

MIP [-]

DGMas Node Dir [%]

21 23 25 27 29 31 33 33 35 35

7 32 57 84 114 146 180 181 215 216

115,49 115,42 115,41 115,40 115,40 115,40 115,39 116,76 115,38 116,44

0,57 0,57 0,55 0,54 0,52 0,52 0,52 2,21 0,50 1,84

0,81 0,70 0,68 0,65 0,65 0,66 0,64 0,13 0,63 0,22

1,00 1,00 1,00 1,00 1,00 1,00 1,00 0,16 1,00 0,89

-10,8 -16,9 -18,3 -19,9 -20,3 -19,9 -20,8 19,8 -22,0 35,6

6,0 8,0 7,8 6,4 4,7 3,5 2,7 64,3 2,2 46,8

0,97 0,98 0,97 0,97 0,98 0,99 0,99 0,27 0,99 0,57

-1,8 -7,5 -6,0 -5,6 -7,7 -8,0 -8,4 34,8 -9,2 57,5

36 35 35 35 35 35 35 36 35 36

1 1 1 1 1 1 1 1 1 1

21 23 25 27 29 31 33 35

10 33 58 85 116 148 182 218

120,72 120,68 120,68 120,66 120,66 120,64 120,61 120,61

0,64 0,61 0,59 0,58 0,59 0,60 0,60 0,58

0,92 0,92 0,94 0,94 0,95 0,91 0,87 0,88

0,99 0,99 1,00 1,00 1,00 1,00 1,00 1,00

2,3 -0,5 0,5 -1,3 -1,7 -4,3 -6,7 -5,8

8,5 9,7 7,4 6,7 5,5 5,0 6,0 6,2

0,92 0,92 0,94 0,95 0,95 0,94 0,93 0,94

39,2 41,2 11,2 0,7 1,1 0,2 -2,9 -2,4

27 27 27 27 27 27 27 27

2 2 2 2 2 2 2 2

With this process in mind, also the remaining uncertainty of the modal analysis process can be described clearly. Eigenfrequency: Typically, the uncertainty is 0.2 % of value. The modal analysis tools in general provide a quite stable estimate of the eigenfrequency in the case of well excited modes and moderate modal density. This in particular is well justified because each of the extracted modes is used to generate synthetic frequency response functions which visually, mode by mode, have been compared with the originally measured data (curve fitting validation). One reserve remains for modes at high modal density when individual modes cannot clearly be distinguished any more, and the curve fit validation only is based on the general shape of the frequency response functions – instead of individual and isolated resonances. Consequently the accuracy on the eigenfrequency typically has to be relaxed. Modal Damping: The uncertainty typically is 15% of the damping value. Since the modal extraction process delivers the complex eigenvalue, i.e. the eigenfrequency composed with the damping at a time, in principle the same assessment holds as stated above for the eigenfrequency. The higher relative uncertainty for the damping of 15% versus the 0.2 % for the frequency is justified by the much higher scatter in the stabilization diagrams. However, the stabilization diagrams together with the curve fitting process can be used to justify a reduced uncertainty of the damping value. Mode Shape: The typical uncertainty of the maximum shape element is 5% of value. This accuracy in fact depends on two major factors: first on the calibration of the measurement chain and second on the separation of modes by the analysis process. Practically, this means, the better a mode has been excited and the better a mode may be distinguished from any other mode, the more accurate a mode shape can be extracted. Here, the mode shape quality indicators are useful to characterize the shape quality. As long as MPD and MPC values as indicated in section 4 are achieved – for global and local modes – the

361 claimed standard uncertainty will hold, otherwise higher uncertainty has to be considered, in particular when the complexity of the mode is high. Generalised Mass Values: The typical uncertainty of the generalized mass is 25% of value. This uncertainty is relatively high, because it depends on all the uncertainties of the modal parameters stated above, however, and in particular on the accuracy on the mode shape elements used for the calculation of the generalized mass. To this respect, the driving point shape element is the most critical, because it often is quite small when compared with some dominating modal responses within the driving point frequency response function (e.g. in such a case, the generalized mass with respect to the driving point is very high). Consequently, in the case of good MPD and MPC values as stated above for a good mode shape extraction, and relatively low driving point generalized mass, the uncertainty on the generalized mass will be less than 25 % and a classification analogue to Tab. 2 could be used.

6. Conclusion Based on practical experience of many modal analysis applications, a structured procedure has been developed and used since many years. This resulted in well traceable and transparent way to extract modal parameters and mode shapes and was in particular independent from commercial modal analysis code, but the authors are delighted that these ideas are emerging more and more in today’s commercial codes. But nevertheless, the workload for the analyst still is quite high and there is still some benefit if such a process could be further automated.

References [1] N.M.M. Maia, J.M.M. Silva (Eds): Theoretical and Experimental Modal Analysis; Research Studies Press Ltd., Baldock, 1997 [2] R.J. Allemang, A.W. Philips: The Unified Matrix Polynomial Approach to Understanding Modal Parameter Estimation: An Update; Proc. Int. Conf. on Noise and Vibration Engineering, Katholieke Universiteit Leuven, Belgium, 2004 [3] P. Verboven et al.: Stabilization Charts and Uncertainty Bounds for Frequency Domain Linear Least Squares Estimators; Proc. of the IMACXXI, Kissimmee, FL, 2003 [4] A.W. Philips, R.J. Allemang: Data Presentation Schemes for Selection and Identification of Modal Parameters; Proc. of the IMAC-XXIII, Orlando, FL, 2005 [5] T. Simmermacher, R. Mayes: Estimating the Uncertainty in Modal Parameters Using SMAC; Proc. Of the IMAC-XXVII, Orlando, FL, 2009 [6] A.W. Philips, R.J. Allemang: Application of Modal Scaling to the Pole Selection Phase of Parameter Estimation; Proc. Of the IMAC-XXVIII, Jacksonville, FL, 2010 [7] MIL-HDBK 340A: Test Requirements for Launch, Upper Stage, and Space Vehicles; 1999 [8] ESA-ECSS-E-ST-32-11C: Space Engineering - Modal Survey Assessment; 2008

Abbreviations CMIF DOF MAC MCF MIF, MIV, MIP MPD, MPH MPC UMPA PTD RFP-Z

Complex mode indicator function Degree of freedom Modal assurance criterion Modal confidence factor Mode indicator function or value Mean phase deviation, mean phase Modal phase collinearity Unified matrix polynomial approach Polyreference time domain modal analysis method Rational fraction polynomial in the z-domain modal analysis method

Operational Modal Analysis of a Spacecraft Vibration Test Mark O’Grady and Raj Singhal, Canadian Space Agency, David Florida Laboratory, Ottawa, Ontario

Abstract CASSIOPE is a small Canadian spacecraft carrying two payloads: the e-POP payload consisting of eight scientific instruments and the CASCADE payload - a high speed Ka-Band communication technology demonstrator. As part of the Cassiope environmental test campaign, the spacecraft was subjected to sine vibration testing for structural qualification, and to random vibration testing for workmanship verification. The purpose of the spacecraft sine vibration test was twofold: subject the spacecraft to protoflight launch environment and to provide a test verified structural model for final coupled loads analysis with the launch vehicle. In this paper, we employ an operational modal analysis technique to the ground vibration sine qualification test of the Cassiope spacecraft. The purpose of this exercise was to determine the applicability of operational modal processing to vibration qualification tests in cases where interface forces between the vibration shaker and the test article are not able to be determined in the course of the test and traditional modal techniques are not applicable. This paper will examine the ability of operational modal techniques in separating complex vibration dynamics and modal interactions present in sine vibration qualification tests and in aiding the determination of required test notching. Cassiope (CASCADE Smallsat and Ionospheric Polar Explorer) is a Canadian smallsat scheduled to be launched in 2011[1]. The three goals of the Casssiope mission are to demonstrate the high speed Ka-Band store and forward capability of the CASCADE CX payload (provided by MacDonald, Dettwiler and Associates Ltd.) [2], to investigate the atmospheric and plasma flows and related wave particle interaction and radio wave propagation in the topside ionosphere with a suite of eight e-POP (Enhanced Polar Outflow Probe) science instruments provided by the University of Calgary, and to develop a Canadian Smallsat Bus (provided by Bristol Aerospace Ltd). Development programs for space hardware generally envision two phases of testing. The first stage would be a modal survey test consisting of excitation by multiple small shakers and optimized response channels throughout the spacecraft. The purpose of this test is to allow updating of the finite element models to allow for better estimates of the flight loads through coupled loads analysis and also better predictions of the vibration test response levels and estimation of notching profiles. However, in many cases, due to budgetary or schedule constraints, the modal test is often omitted. The second stage of testing is qualification testing consisting of sine vibration testing, random vibration testing and acoustic testing. The sine vibration testing is closed loop controlled over a limited frequency band, with notching appropriate for the loads predicted during launch (these may be base shear or bending moment, c.g. acceleration, etc). The random vibration testing is often called for as a workmanship test or to capture low frequency content not easily excited by an acoustic test. This paper first looks at an overview of the Cassiope spacecraft. This is followed by an outline of the vibration test setup and sequence. The next section looks at the use of operational modal technique in processing the sine vibration test data. The modes in each axis are determined and their interactions are discussed. The applicability of the modal results to aid in the prediction of sine test notching is presented. Cassiope Spacecraft Overview The structure of the Cassiope Spacecraft is shown in Figure 1. The spacecraft consists of vertical aluminum honeycomb panels in a hexagonal configuration, tied together with machined corner posts. There are also three aluminum honeycomb sandwich panels; the lower being the e-POP payload panel, a middeck panel supporting momentum wheels and magnetorque rods, and an upper CX panel under which the CASCADE payload is mounted. The launch vehicle standoffs attach to the spacecraft hexagonal side panels directly through gussets that penetrate through the lower e-POP panel. The horizontal panels are then attached to the hexagonal structure through bolted plate support brackets. Five solar arrays, covering the full top panel, two full side panels and two half side panels are mounted to the spacecraft. The solar array panels are constructed from graphite composite facesheets and aluminum cores to better match the coefficient of thermal expansion (CTE) of the mounted solar cells. The solar panels are mounted offset from the spacecraft structure via titanium blade flextures in order to

T. Proulx (ed.), Advanced Aerospace Applications, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 4, DOI 10.1007/978-1-4419-9302-1_30, © The Society for Experimental Mechanics, Inc. 2011

363

364 thermally decouple the solar panels from the spacecraft bus structure, provide good fatigue properties, and to allow from the CTE mismatch between the graphite facesheets of the solar arrays and the aluminum facesheets of the bus. Initially, in the preliminary spacecraft design, flexture-type isolators were incorporated into each mounting foot in order to protect some sensitive instruments from mid-frequency launch loads. However, in the end a change in launch vehicle resulted in a reduced launch spectrum allowed their removal

Solar Arrays (5)

Ka-Band CX Antenna

S-Band Antennae (4)

GPS Antennae (4)

CX Panel

SEI Boom

CERTO Boom

RRI Instrument

IRM Boom ePOP Panel

+YMB

Sun Sensors (6)

+XMB

+ZMB

Launch Interfaces (6)

MGF Boom

+YMB

Star Trackers (2)

+XMB +ZMB

Fig. 1. CASSIOPE spacecraft in the vibration test configuration

Traditional Modal and Vibration tests There are some differences between a traditional modal test and a ground vibration test in terns of modal analysis. In the traditional modal test, a number of individual shakers may be optimally located in order to excite all of the required modes. In the ground vibration tests, excitation is confined to three orthogonal axes, one axis at a time, resulting in suboptimal excitation for some modes, particularly torsion modes. Another advantage of dedicated modal survey test is an optimized set of accelerometer locations may be chosen to resolve the modes in the frequency range of interest. In the vibration test, the choice of response locations is usually governed by the determination of responses at critical locations and the protection of critical instruments and components. The accelerometer location set for a vibration qualification test may not be ideally suited to resolve all of the modes for the purpose of finite element model correlation. Vibration Test Setup and Sequence The Cassiope spacecraft vibration testing was performed at the David Florida Laboratory of the Canadian Space Agency (CSA). The spacecraft was subjected to both sine and random vibration testing. The sine vibration tests were conducted to determine that the spacecraft was able to survive the protoflight level sine vibration environment of the Space-X Falcon-9 launch vehicle through meeting the interface shear forces and bending moments predicted by the preliminary coupled loads analysis of the spacecraft and launch vehicle. The protoflight sine tests were conducted from 5 to 80 Hz for the lateral axes (X and Y) but were extended from 5 to110 Hz in the vertical axis (Z) in order to capture the CX panel mode. The sweep rate employed was 4 octaves/min. The acceptance level random vibration testing was also performed on the satellite to verify workmanship in the construction. Prior to the initial sine vibration in each axis, a low level (0.1g) sine survey was performed from 5 - 500 Hz as a resonance search and to provide an initial prediction for any notching required for subsequent runs. Following the protoflight sine and acceptance random tests in each axis, the resonance search is repeated to confirm no permanent structural effects on the spacecraft resulting from the tests. The test setup on the shaker is given in Figure 2. for each of the test axes. For the lateral axes, the spacecraft was installed with an interface plate attached to the vibration slip table. For the vertical, the interface plate was mounted through a spacer to a vertical guidance fixture to provide adequate overturning moment capability. The spacecraft was instrumented with 150

365 accelerometers, of which 60 were employed in control and notching. An average control strategy was employed over 6 accelerometers located on the interface plate, next to each spacecraft standoff.

Fig. 2 The CASSIOPE spacecraft on the vibration shaker in X, Y And Z Axis Configurations (CCW from top left)

The primary limiting strategy for the lateral X and Y axes was to limit on the base shear force at a level to meet the coupled loads analysis with margin. The limiting strategy for the Z axis was to limit on the acceleration response of the horizontal panels (CX, mid-deck and e-Pop panels) at a level that covered their respective response predictions from the coupled loads analysis. Additional limits were placed on the C.G. accelerations in the X and Y (lateral) axes as well as on representative solar panels, and the CX and e-POP payload components. The pretest analysis of the finite element model of the Cassiope spacecraft indicated that the first bending modes in each of the X and Y axes would account for about 85% of the modal mass in their respective axes. As a result, it was predicted that the intermediate level sine runs would require significant notching and that the initial low level (0.1g) sine survey would be the only test level that would not be notched. It was this initial low level sine survey that was employed in this study of the operational modal analysis. Operation Preprocessing Time history data was recorded for all channels in the course of the sine vibration testing. Auto and Cross-Power spectra were computed from these time histories using 16384 time lags and a 90% exponential window resulting in a frequency resolution of 0.098 Hz. The control accelerometer C1 phase reference channel for the vibration test was chosen as the reference channel for the cross product calculations. The resulting spectra were computed over a frequency range of 0 to 800 Hz, while the low level sine test was conducted over 5 to 500 Hz. When the cross powers were generated, there appeared to be a rigid body mode around 5 Hz in each axis (as shown in Figure 3 and in the stabilization diagram for the X axis in Figure 4). These modes were found to be very useful in verifying the functionality and correct orientation of the accelerometers for modal processing. Operation Modal Analysis The cross power spectra obtained from the time histories of the low level sine runs were processed using an LMS Operational Polymax [3], a least square complex frequency domain algorithm. Each axis was processed separately, focusing over the

frequency ranges of the protoflight sine vibration tests; 5-80 Hz (X & Y axes) and 5 to 110 Hz (Z axis) over which the notching levels will be determined, employing this low level sine data.

Fig. 3 Rigid Body Modes in X,Y and Z axes

The stabilization diagram for the X axis processing is given in Figure 4. The first bending mode of the structure was determined to be 58.8 Hz. There was also a full size side solar panel mode at 98.8 Hz and an MGF boom mode on the e-POP panel at 112.4 Hz. In addition, a number of modes whose primary motion was in one of the other axes could be determined but the resultant mode shapes were poor due to the limited cross axis excitations. The first Y bending mode could also be determined from the X axis low level runs and was found to be 57.3 Hz. In addition, both of the primary bending modes appeared to be coupled with the first mid-plane mode. These complex dynamics proved to provide challenges in subsequent runs with a number of test aborts occurring in the X and Z axis runs in this frequency range.

367 which, while not detrimental to the wheel itself, did require a notch in order to limit the middeck responses to the coupled loads levels.

Fig. 5 Y axis primary bending mode at 58.8 Hz and middeck mode at 63.2 Hz

In the Z Axis, the three primary horizontal panel modes as well as the top solar panel mode were found. The lower e-POP panel's primary mode was at 89.8 Hz, the middeck panel mode was at 62.9 Hz and the upper CX panel mode was at 105.8 Hz. In addition to the horizontal panel modes, there was also a top solar panel mode at 74.8 Hz. As an illustration, the right portion of Figure 6 shows the CX panel mode at 105.8 Hz. Notches were required in the Z axis protoflight spectrum at each of these panel modes. These notches are clearly seen in the control spectrum of the full level Z axis protoflight sine test on the left of Figure 6. 2.00

1.00

CHU g Log

Amplitude

E-POP Mid-deck

CX

F F

Spectrum AvgCtrl Spectrum Reference

0.30

0.00 5.00

Hz

110.00

Fig. 6 Z axis Average Control for PFM Sine Vibration and corresponding CX Panel Mode

Table 1 is a summary of the modal processing of the data from each of the low level sine surveys in each axis and their associated frequencies. The same modes were found at different frequencies in different axis tests. These frequency shifts prevent the three axes from being processed together. This was not a sign of a changing structure as the pre and post test resonance surveys showed no frequency shifts or magnitude changes in any of the resonance peaks in the sine qualification frequency range.

368 Modes 'Rigid Body' Camera Head Unit (CHU) Bending Y Bending X Middeck1 Top Solar Array Middeck2 e-POP Panel Camera Head Unit2 Large Side Solar Panel CX Panel MGF Boom Top Solar & Gap A

X 4.9 57.3 58.8 73.5 79.0 90.4 98.8 105.0 112.4 120.9

Y 5.0 58.8 63.2 72.9 77.4 88.8 99.1 104.8 119.4

Z 5.0 53.0 62.9 74.8 80.4 89.7 105.8 -

Table 1 Modes and frequencies for each test axis

Summary Operational Modal Analysis was employed in the processing of low level sine vibration qualification test data for the Cassiope spacecraft. The purpose of this exercise was to determine the applicability of operational modal processing for vibration qualification tests where interface forces are not able to be determined in the course of the test. The operational modal results are useful in aiding in separating complex vibration dynamics and modal interactions present in sine vibration qualification tests. As a result, they can be employed as an aid in the determination of required notching levels. Future planned work includes the processing of the test data through traditional modal analysis methods, employing the interface forces as measured by force sensors as references, and comparison with the results determined through operational modal processing and finite element predictions. Acknowledgements The authors would like to thank Andrew Woronko, of MDA, Phillipe Tremblay of Maya HTT and Yvan Soucy of CSA for their lead of the Cassiope vibration test campaign. We acknowledge the excellent contributions and the dedication of the David Florida Laboratory Structural Qualification Facilities personnel. References 1.

Giffin, G.B., Ressl, V., Yau, A., King, P., “CASSIOPE: A Canadian SmallSAT-Based Space Science and Advanced Satcom Demonstration Mission”, Proceedings of the 18th AIAA/USU Conference on Small Satellite, Paper SSC04-VI-5, August 2004.

2.

Giffin, G.B., Magnussen, K., Wlodyka, M., Duffield, D., Poller, B., Bravman, J. “CASCADE: A Smallsat System Providing Global, High Quality Movement of Very Large Data Files”, Proceedings of the IAC 2004 Conference, Paper IAC-04-M.4.06, September 2004.

3.

Guillaum, P., Verboven, P., Vanlanduit, .S, Van Der Auweraer, H., and Peeters, B. A poly-reference implementation of the least-squares complex frequency-domain estimator. In Proceedings of IMAC 21, the International Modal Analysis Conference, Kissimmee (FL), USA, February 2003.Giffin, G.B.,